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3

Noncommutative localization in topology

Andrew Ranicki

Introduction

The topological applications of the Cohn noncommutative localization con-sidered in this paper deal with spaces (especially manifolds) with infinitefundamental group and involve localizations of infinite group rings andrelated triangular matrix rings Algebraists have usually considered non-commutative localization of rather better behaved rings so the topologicalapplications require new algebraic techniques

Part 1 is a brief survey of the applications of noncommutative localizationto topology finitely dominated spaces codimension 1 and 2 embeddings(knots and links) homology surgery theory open book decompositions andcircle-valued Morse theory These applications involve chain complexes andthe algebraic K- and L-theory of the noncommutative localization of grouprings

Part 2 is a report on work on chain complexes over generalized free prod-ucts and the related algebraic K- and L-theory from the point of view ofnoncommutative localization of triangular matrix rings Following Bergmanand Schofield a generalized free product of rings can be constructed as anoncommutative localization of a triangular matrix ring The novelty hereis the explicit connection to the algebraic topology of manifolds with a gen-eralized free product structure realized by a codimension 1 submanifoldleading to noncommutative localization proofs of the results of Waldhausenand Cappell on the algebraic K- and L-theory of generalized free prod-ucts In a sense this is more in the nature of an application of topologyto noncommutative localization But this algebra has in turn topologicalapplications since in dimensions gt 5 the surgery classification of manifoldswithin a homotopy type reduces to algebra

1

Part 1 A survey of applications

We start by recalling the universal noncommutative localization of PMCohn[5] Let A be a ring and let Σ = s P rarr Q be a set of morphismof fg projective A-modules A ring morphism A rarr R is Σ-invertingif for every s isin Σ the induced morphism of fg projective R-modules1otimes s RotimesAP rarr RotimesAQ is an isomorphism The noncommutative localiza-tion A rarr Σminus1A is Σ-inverting and has the universal property that any Σ-inverting ring morphism A rarr R has a unique factorization A rarr Σminus1A rarr RThe applications to topology involve homology with coefficients in a non-commutative localization Σminus1A

Homology with coefficients is defined as follows Let X be a connectedtopological space with universal cover X and let the fundamental groupπ1(X) act on the left of X so that the (singular) chain complex S(X) is a freeleft Z[π1(X)]-module complex Given a morphism of rings F Z[π1(X)] rarr Λdefine the Λ-coefficient homology of X to be

Hlowast(X Λ) = Hlowast(ΛotimesZ[π1(X)] S(X))

If X is a CW complex then S(X) is chain equivalent to the cellular freeZ[π1(X)]-module chain complex C(X) with one generator in degree r foreach r-cell of X and

Hlowast(X Λ) = Hlowast(ΛotimesZ[π1(X)] C(X))

11 Finite domination

A topological spaceX is finitely dominated if there exist a finite CW complexK maps f X rarr K g K rarr X and a homotopy gf ≃ 1 X rarr XThe finiteness obstruction of Wall [31] is a reduced projective class [X] isinK0(Z[π1(X)]) such that [X] = 0 if and only if X is homotopy equivalent toa finite CW complex

In the applications of the finiteness obstruction to manifold topologyX = M is an infinite cyclic cover of a compact manifold M ndash see Chapter 17of Hughes and Ranicki [13] for the geometric wrapping up procedure whichshows that in dimension gt 5 every tame manifold end has a neighbourhoodwhich is a finitely dominated infinite cyclic cover M of a compact manifoldM Let f M rarr S1 be a classifying map so that M = flowastR and letM

+= flowastR+ The finiteness obstruction [M

+] isin K0(Z[π1(M )]) is the end

2

obstruction of Siebenmann [27] such that [M+] = 0 if and only if the tame

end can be closed ie compactified by a manifold with boundary

Given a ring A let Ω be the set of square matrices ω isin Mr(A[z zminus1])

over the Laurent polynomial extension A[z zminus1] such that the A-module

P = coker(ω A[z zminus1]r rarr A[z zminus1]r)

is fg projective The noncommutative Fredholm localization Ωminus1A[z zminus1]has the universal property that a finite fg free A[z zminus1]-module chaincomplex C is A-module chain equivalent to a finite fg projective A-modulechain complex if and only if Hlowast(Ω

minus1C) = 0 (Ranicki [21 Proposition 139])with Ωminus1C = Ωminus1A[z zminus1]otimesA[zzminus1] C

Let M be a connected finite CW complex with a connected infinite cycliccover M The fundamental group π1(M) fits into an extension

1 rarr π1(M ) rarr π1(M) rarr Z rarr 1

and Z[π1(M)] is a twisted Laurent polynomial extension

Z[π1(M)] = Z[π1(M )]α[z zminus1]

withα π1(M ) rarr π1(M) g 7rarr zminus1gz

the monodromy automorphism For the sake of simplicity only the untwistedcase α = 1 will be considered here so that π1(M) = π1(M )timesZ The infinitecyclic cover M is finitely dominated if and only if Hlowast(M Ωminus1Z[π1(M)]) = 0with A = Z[π1(M)] and Z[π1(M)] = A[z zminus1] The Farrell-Siebenmann ob-struction Φ(M) isin Wh(π1(M)) of an n-dimensional manifold M with finitelydominated infinite cyclic cover M is such that Φ(M) = 0 if (and for n gt 6only if) M is a fibre bundle over S1 ndash see [21 Proposition 1516] for theexpression of Φ(M) in terms of the Ωminus1Z[π1(M)]-coefficient Reidemeister-Whitehead torsion

τ(M Ωminus1Z[π1(M)]) = τ(Ωminus1C(M)) isin K1(Ωminus1Z[π1(M)])

12 Codimension 1 splitting

Surgery theory asks whether a homotopy equivalence of manifolds is homo-topic (or h-cobordant) to a homeomorphism ndash in general the answer is noThere are obstructions in the topological K-theory of vector bundles in the

3

algebraic K-theory of modules and in the algebraic L-theory of quadraticforms The algebraic K-theory obstruction lives in the Whitehead groupWh(π) of the fundamental group π The L-theory obstruction lives in oneof the surgery groups Llowast(Z[π]) of Wall [32] and is defined when the topo-logical and algebraic K-theory obstructions vanish The groups Llowast(Λ) aredefined for any ring with involution Λ to be the generalized Witt groups ofnonsingular quadratic forms over Λ For manifolds of dimension gt 5 thevanishing of the algebraic obstructions is both a necessary and sufficientcondition for deforming a homotopy equivalence to a homeomorphism SeeRanicki [20] for the reduction of the Browder-Novikov-Sullivan-Wall surgerytheory to algebra

A homotopy equivalence of m-dimensional manifolds f M prime rarr M splits

along a submanifold Nn sub Mm if f is homotopic to a map (also denoted byf) such that N prime = fminus1(N) sub M prime is also a submanifold and the restrictionf | N prime rarr N is also a homotopy equivalence For codimension m minus n gt 3the splitting obstruction is just the ordinary surgery obstruction σlowast(f |) isinLm(Z[π1(N)]) For codimension m minus n = 1 2 the splitting obstructionsinvolve the interplay of the knotting properties of codimension (m minus n)submanifolds and Mayer-Vietoris-type decompositions of the algebraic K-and L-groups of Z[π1(M)] in terms of the groups of Z[π1(N)] Z[π1(MN)]

In the case m minus n = 1 π1(M) is a generalized free product ie eitheran amalgamated free product or an HNN extension by the Seifert-vanKampen theorem Codimension 1 splitting theorems and the algebraic K-and L-theory of generalized free products are a major ingredient of high-dimensional manifold topology featuring in the work of Stallings Brow-der Novikov Wall Siebenmann Farrell Hsiang Shaneson Casson Wald-hausen Cappell and the author Noncommutative localization providesa systematic development of this algebra using the intuition afforded by thetopological applications ndash see Part 2 below for a more detailed discussion

13 Homology surgery theory

For a morphism of rings with involution F Z[π] rarr Λ Cappell and Shaneson[3] considered the problem of whether a Λ-coefficient homology equivalenceof manifolds with fundamental group π is H-cobordant to a homeomor-phism Again the answer is no in general with obstructions in the topolog-ical K-theory of vector bundles and in the homology surgery groups Γlowast(F)which are generalized Witt groups of Λ-nonsingular quadratic forms over

4

Z[π] Vogel [28] [29] identified the Λ-coefficient homology surgery groupswith the ordinary L-groups of the localization Σminus1Z[π] of Z[π] inverting theset Σ of Λ-invertible square matrices over Z[π]

Γlowast(F) = Llowast(Σminus1Z[π])

and identified the relative L-groups Llowast(Z[π] rarr Σminus1Z[π]) in the localizationexact sequence

middot middot middot rarr Ln(Z[π]) rarr Ln(Σminus1Z[π]) rarr Ln(Z[π] rarr Σminus1Z[π]) rarr Lnminus1(Z[π]) rarr

with generalized Witt groups Llowast(Z[π]Σ) of nonsingular Σminus1Z[π]Z[π])-valued quadratic linking forms on Σ-torsion Z[π]-modules of homologicaldimension 1

14 Codimension 2 embeddings

Suppose given a codimension 2 embedding Nn sub Mn+2 such as a knot orlink Let Σminus1A be the localization of A = Z[π1(MN)] inverting the set Σof matrices over A which become invertible over Z[π1(M)] By Alexanderduality the Σminus1A-coefficient homology modules

Hlowast(MN Σminus1A) sim= Hn+2minuslowast(MN Σminus1A) (lowast 6= 0 n + 2)

are determined by the homotopy class of the inclusion N sub M The A-coefficient homology groups Hlowast(MN A) and their Poincare duality prop-erties reflect more subtle invariants of N sub M such as knotting See Ranicki[21] for a general account of high-dimensional codimension 2 embedding the-ory including some of the applications of noncommutative localization

15 Open books

An (n + 2)-dimensional manifold Mn+2 is an open book if there exists acodimension 2 submanifold Nn sub Mn+2 such that the complement MN isa fibre bundle over S1 Every odd-dimensional manifold is an open bookQuinn [17] showed that for k gt 2 a (2k + 2)-dimensional manifold M isan open book if and only if an asymmetric form over Z[π1(M)] associatedto M represents 0 in the Witt group This obstruction was identified inRanicki [21] with an element in the L-group L2k+2(Ω

minus1Z[π1(M)][z zminus1]) ofthe Fredholm localization of Z[π1(M)][z zminus1] (cf section 11 above)

5

16 Boundary link cobordism

An n-dimensional micro-component boundary link is a codimension 2 embedding

Nn =⋃

micro

Sn sub Mn+2 = Sn+2

with a micro-component Seifert surface in which case the fundamental groupof the complement X = MN has a compatible surjection π1(X) rarr Fmicro

onto the free group on micro generators Duval [8] used the work of Cap-pell and Shaneson [4] and Vogel [29] to identify the cobordism group ofn-dimensional micro-component boundary links for n gt 2 with the relative L-group Ln+3(Z[Fmicro]Σ) in the localization exact sequence

middot middot middot rarr Ln+3(Z[Fmicro]) rarr Ln+3(Σminus1Z[Fmicro]) rarr Ln+3(Z[Fmicro]Σ) rarr Ln+2(Z[Fmicro]) rarr

with Σ the set of Z-invertible square matrices over Z[Fmicro] The even-dimensionalboundary link cobordism groups are L2lowast+1(Z[Fmicro]Σ) = 0 The cobordismclass in L2k+2(Z[Fmicro]Σ) of a (2k minus 1)-dimensional micro-component boundarylink cupmicroS

2kminus1 sub S2k+1 was identified with the Witt class of a Σminus1Z[Fmicro]Z[Fmicro]-valued nonsingular (minus1)k+1-quadratic linking form on Hk(XZ[Fmicro]) gener-alizing the Blanchfield pairing on the homology of the infinite cyclic coverof a knot The localization Σminus1Z[Fmicro] was identified by Dicks and Sontag [7]and Farber and Vogel [11] with a ring of rational functions in micro noncommut-ing variables The high odd-dimensional boundary link cobordism groupsL2lowast+2(Z[Fmicro]Σ) have been computed by Sheiham [26]

17 Circle-valued Morse theory

Novikov [15] proposed the study of the critical points of Morse functionsf M rarr S1 on compact manifolds M The lsquoNovikov complexrsquo C(Mf)over Z((z)) = Z[[z]][zminus1] has one generator for each critical point of f andthe lsquoNovikov homologyrsquo

Hlowast(C(Mf)) = Hlowast(M Z((z)))

provides lower bounds on the number of critical points of Morse functions inthe homotopy class of f generalizing the inequalities of the classical Morsetheory of real-valued functions M rarr R Suppose given a Morse functionf M rarr S1 with M = flowastR such that π1(M) = π1(M ) times Z (for the sakeof simplicity) Let Σ be the set of square matrices over Z[π1(M )][z] which

6

become invertible over Z[π1(M)] under the augmentation z 7rarr 0 There is anatural morphism from the localization to the completion

Σminus1Z[π1(M)] rarr Z[π1(M)] = Z[π1(M)][[z]][zminus1]

which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also

features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]

18 3- and 4-dimensional manifolds

See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds

Part 2 The algebraic K- and L-theory of general-

ized free products via noncommutative localization

A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of

the Z[π1(M)]-module chain complex C(M) of the universal cover M is the

7

assembly of an A-module chain complex constructed from the chain com-

plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24

21 The algebraic K-theory of a noncommutative localiza-

tion

Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution

0 Ps

Q T 0

such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]

K1(A) rarr K1(Σminus1A) rarr K1(AΣ) rarr K0(A) rarr K0(Σ

minus1A)

was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo

TorAi (Σminus1AΣminus1A) = 0 (i gt 1)

then

(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B

(ii) the localization exact sequence extends to the higher K-groups

middot middot middot rarr Kn(A) rarr Kn(Σminus1A) rarr Kn(AΣ) rarr Knminus1(A) rarr middot middot middot rarr K0(Σ

minus1A)

with Kn(AΣ) = Knminus1(H(AΣ))

8

22 Matrix rings

The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings

Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring

A =

(A1 B0 A2

)

An A-module can be written as

M =

(M1

M2

)

with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection

A1 timesA2 rarr A (a1 a2) 7rarr

(a1 00 a2

)

induces isomorphisms of algebraic K-groups

Klowast(A1)oplusKlowast(A2) sim= Klowast(A)

The columns of A are fg projective A-modules

P1 =

(A1

0

) P2 =

(BA2

)

such that

P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)

HomA(P1 P2) = B HomA(P2 P1) = 0

The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring

Σminus1A = M2(C) =

(C CC C

)

with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1

sim= Σminus1P2 The Morita equivalence

Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L

9

induces isomorphisms in algebraic K-theory

Klowast(M2(C)) sim= Klowast(C)

The composite of the functor

A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M

and the Morita equivalence is the assembly functor

A-modules rarr C-modules

M =

(M1

M2

)7rarr (C C)otimesA M

= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)

inducing the morphisms

Klowast(A) = Klowast(A1)oplusKlowast(A2) rarr Klowast(Σminus1A) = Klowast(C)

in the algebraic K-theory localization exact sequence

There are evident generalizations to k times k matrix rings for any k gt 2

23 HNN extensions

The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with

α(s)z = zβ(s) isin R lowastαβ z (s isin S)

Define the triangular 2times 2 matrix ring

A =

(R Rα oplusRβ

0 S

)

with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with

σ1 =

((1 0)0

) σ2 =

((0 1)0

) P1 =

(R0

)rarr P2 =

(Rα oplusRβ

S

)

The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies

Σminus1A = M2(R lowastαβ z)

10

Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected

M = M1 cupNtimes01 N times [0 1]

N times [0 1]

M1

By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1

π1(M) = π1(M1) lowastαβ z

so thatZ[π1(M)] = Z[π1(M1)] lowastαβ z

As above define a triangular 2times 2 matrix ring

A =

(Z[π1(N)] Z[π1(M1)]α oplus Z[π1(M1)]β

0 Z[π1(M)]

)

with noncommutative localization

Σminus1A = M2(Z[π1(M1)] lowastαβ z) = M2(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are

injective and the universal cover M is a union

M =⋃

gisin[π1(M)π1(M1)]

gM1

11

of translates of the universal cover M1 of M1 and

g1M1 cap g2M1 =

hN if g1 cap g2z = h isin [π1(M) π1(N)]

g1M1 if g1 = g2

empty if g1 6= g2 and g1 cap g2z = empty

with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms

M zminus2M1 zminus1M1 M1 zM1 z2M1

zminus1N N zN z2N

The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps

iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)

iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)

defining a fg projective A-module chain complex

(C(M1)

C(N)

)with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)

)

= C(M)

by the Mayer-Vietoris theorem

Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms

R rarr R lowastαβ z S rarr R lowastαβ z

A =

(R Rα oplusRβ

0 S

)rarr Σminus1A = M2(R lowastαβ z)

12

are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C

0 R lowastαβ z otimesS Eiαminusziβ

R lowastαβ z otimesR D C 0

with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(RS α β)α β 01 1 0

Kn(A) = Kn(R)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R lowastαβ z) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(Rα β)(αminusβ)oplus0

Kn(R) Kn(R lowastαβ z)

In particular for α = β = 1 S = R rarr R the HNN extension is justthe Laurent polynomial extension

R lowastαβ z = R[z zminus1]

and the Mayer-Vietoris exact sequence splits to give the original splitting ofBass Heller and Swan [1]

K1(R[z zminus1]) = K1(R)oplusK0(R)oplus Nil0(R)oplus Nil0(R)

as well as its extension to the Quillen higher K-groups Klowast

24 Amalgamated free products

The amalgamated free product R1 lowastS R2 is defined for any ring morphismsi1 S rarr R1 i2 S rarr R2 with

r1i1(s) lowast r2 = r1 lowast i2(s)r2 isin R1 lowastS R2 (r1 isin R1 r2 isin R2 s isin S)

13

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

obstruction of Siebenmann [27] such that [M+] = 0 if and only if the tame

end can be closed ie compactified by a manifold with boundary

Given a ring A let Ω be the set of square matrices ω isin Mr(A[z zminus1])

over the Laurent polynomial extension A[z zminus1] such that the A-module

P = coker(ω A[z zminus1]r rarr A[z zminus1]r)

is fg projective The noncommutative Fredholm localization Ωminus1A[z zminus1]has the universal property that a finite fg free A[z zminus1]-module chaincomplex C is A-module chain equivalent to a finite fg projective A-modulechain complex if and only if Hlowast(Ω

minus1C) = 0 (Ranicki [21 Proposition 139])with Ωminus1C = Ωminus1A[z zminus1]otimesA[zzminus1] C

Let M be a connected finite CW complex with a connected infinite cycliccover M The fundamental group π1(M) fits into an extension

1 rarr π1(M ) rarr π1(M) rarr Z rarr 1

and Z[π1(M)] is a twisted Laurent polynomial extension

Z[π1(M)] = Z[π1(M )]α[z zminus1]

withα π1(M ) rarr π1(M) g 7rarr zminus1gz

the monodromy automorphism For the sake of simplicity only the untwistedcase α = 1 will be considered here so that π1(M) = π1(M )timesZ The infinitecyclic cover M is finitely dominated if and only if Hlowast(M Ωminus1Z[π1(M)]) = 0with A = Z[π1(M)] and Z[π1(M)] = A[z zminus1] The Farrell-Siebenmann ob-struction Φ(M) isin Wh(π1(M)) of an n-dimensional manifold M with finitelydominated infinite cyclic cover M is such that Φ(M) = 0 if (and for n gt 6only if) M is a fibre bundle over S1 ndash see [21 Proposition 1516] for theexpression of Φ(M) in terms of the Ωminus1Z[π1(M)]-coefficient Reidemeister-Whitehead torsion

τ(M Ωminus1Z[π1(M)]) = τ(Ωminus1C(M)) isin K1(Ωminus1Z[π1(M)])

12 Codimension 1 splitting

Surgery theory asks whether a homotopy equivalence of manifolds is homo-topic (or h-cobordant) to a homeomorphism ndash in general the answer is noThere are obstructions in the topological K-theory of vector bundles in the

3

algebraic K-theory of modules and in the algebraic L-theory of quadraticforms The algebraic K-theory obstruction lives in the Whitehead groupWh(π) of the fundamental group π The L-theory obstruction lives in oneof the surgery groups Llowast(Z[π]) of Wall [32] and is defined when the topo-logical and algebraic K-theory obstructions vanish The groups Llowast(Λ) aredefined for any ring with involution Λ to be the generalized Witt groups ofnonsingular quadratic forms over Λ For manifolds of dimension gt 5 thevanishing of the algebraic obstructions is both a necessary and sufficientcondition for deforming a homotopy equivalence to a homeomorphism SeeRanicki [20] for the reduction of the Browder-Novikov-Sullivan-Wall surgerytheory to algebra

A homotopy equivalence of m-dimensional manifolds f M prime rarr M splits

along a submanifold Nn sub Mm if f is homotopic to a map (also denoted byf) such that N prime = fminus1(N) sub M prime is also a submanifold and the restrictionf | N prime rarr N is also a homotopy equivalence For codimension m minus n gt 3the splitting obstruction is just the ordinary surgery obstruction σlowast(f |) isinLm(Z[π1(N)]) For codimension m minus n = 1 2 the splitting obstructionsinvolve the interplay of the knotting properties of codimension (m minus n)submanifolds and Mayer-Vietoris-type decompositions of the algebraic K-and L-groups of Z[π1(M)] in terms of the groups of Z[π1(N)] Z[π1(MN)]

In the case m minus n = 1 π1(M) is a generalized free product ie eitheran amalgamated free product or an HNN extension by the Seifert-vanKampen theorem Codimension 1 splitting theorems and the algebraic K-and L-theory of generalized free products are a major ingredient of high-dimensional manifold topology featuring in the work of Stallings Brow-der Novikov Wall Siebenmann Farrell Hsiang Shaneson Casson Wald-hausen Cappell and the author Noncommutative localization providesa systematic development of this algebra using the intuition afforded by thetopological applications ndash see Part 2 below for a more detailed discussion

13 Homology surgery theory

For a morphism of rings with involution F Z[π] rarr Λ Cappell and Shaneson[3] considered the problem of whether a Λ-coefficient homology equivalenceof manifolds with fundamental group π is H-cobordant to a homeomor-phism Again the answer is no in general with obstructions in the topolog-ical K-theory of vector bundles and in the homology surgery groups Γlowast(F)which are generalized Witt groups of Λ-nonsingular quadratic forms over

4

Z[π] Vogel [28] [29] identified the Λ-coefficient homology surgery groupswith the ordinary L-groups of the localization Σminus1Z[π] of Z[π] inverting theset Σ of Λ-invertible square matrices over Z[π]

Γlowast(F) = Llowast(Σminus1Z[π])

and identified the relative L-groups Llowast(Z[π] rarr Σminus1Z[π]) in the localizationexact sequence

middot middot middot rarr Ln(Z[π]) rarr Ln(Σminus1Z[π]) rarr Ln(Z[π] rarr Σminus1Z[π]) rarr Lnminus1(Z[π]) rarr

with generalized Witt groups Llowast(Z[π]Σ) of nonsingular Σminus1Z[π]Z[π])-valued quadratic linking forms on Σ-torsion Z[π]-modules of homologicaldimension 1

14 Codimension 2 embeddings

Suppose given a codimension 2 embedding Nn sub Mn+2 such as a knot orlink Let Σminus1A be the localization of A = Z[π1(MN)] inverting the set Σof matrices over A which become invertible over Z[π1(M)] By Alexanderduality the Σminus1A-coefficient homology modules

Hlowast(MN Σminus1A) sim= Hn+2minuslowast(MN Σminus1A) (lowast 6= 0 n + 2)

are determined by the homotopy class of the inclusion N sub M The A-coefficient homology groups Hlowast(MN A) and their Poincare duality prop-erties reflect more subtle invariants of N sub M such as knotting See Ranicki[21] for a general account of high-dimensional codimension 2 embedding the-ory including some of the applications of noncommutative localization

15 Open books

An (n + 2)-dimensional manifold Mn+2 is an open book if there exists acodimension 2 submanifold Nn sub Mn+2 such that the complement MN isa fibre bundle over S1 Every odd-dimensional manifold is an open bookQuinn [17] showed that for k gt 2 a (2k + 2)-dimensional manifold M isan open book if and only if an asymmetric form over Z[π1(M)] associatedto M represents 0 in the Witt group This obstruction was identified inRanicki [21] with an element in the L-group L2k+2(Ω

minus1Z[π1(M)][z zminus1]) ofthe Fredholm localization of Z[π1(M)][z zminus1] (cf section 11 above)

5

16 Boundary link cobordism

An n-dimensional micro-component boundary link is a codimension 2 embedding

Nn =⋃

micro

Sn sub Mn+2 = Sn+2

with a micro-component Seifert surface in which case the fundamental groupof the complement X = MN has a compatible surjection π1(X) rarr Fmicro

onto the free group on micro generators Duval [8] used the work of Cap-pell and Shaneson [4] and Vogel [29] to identify the cobordism group ofn-dimensional micro-component boundary links for n gt 2 with the relative L-group Ln+3(Z[Fmicro]Σ) in the localization exact sequence

middot middot middot rarr Ln+3(Z[Fmicro]) rarr Ln+3(Σminus1Z[Fmicro]) rarr Ln+3(Z[Fmicro]Σ) rarr Ln+2(Z[Fmicro]) rarr

with Σ the set of Z-invertible square matrices over Z[Fmicro] The even-dimensionalboundary link cobordism groups are L2lowast+1(Z[Fmicro]Σ) = 0 The cobordismclass in L2k+2(Z[Fmicro]Σ) of a (2k minus 1)-dimensional micro-component boundarylink cupmicroS

2kminus1 sub S2k+1 was identified with the Witt class of a Σminus1Z[Fmicro]Z[Fmicro]-valued nonsingular (minus1)k+1-quadratic linking form on Hk(XZ[Fmicro]) gener-alizing the Blanchfield pairing on the homology of the infinite cyclic coverof a knot The localization Σminus1Z[Fmicro] was identified by Dicks and Sontag [7]and Farber and Vogel [11] with a ring of rational functions in micro noncommut-ing variables The high odd-dimensional boundary link cobordism groupsL2lowast+2(Z[Fmicro]Σ) have been computed by Sheiham [26]

17 Circle-valued Morse theory

Novikov [15] proposed the study of the critical points of Morse functionsf M rarr S1 on compact manifolds M The lsquoNovikov complexrsquo C(Mf)over Z((z)) = Z[[z]][zminus1] has one generator for each critical point of f andthe lsquoNovikov homologyrsquo

Hlowast(C(Mf)) = Hlowast(M Z((z)))

provides lower bounds on the number of critical points of Morse functions inthe homotopy class of f generalizing the inequalities of the classical Morsetheory of real-valued functions M rarr R Suppose given a Morse functionf M rarr S1 with M = flowastR such that π1(M) = π1(M ) times Z (for the sakeof simplicity) Let Σ be the set of square matrices over Z[π1(M )][z] which

6

become invertible over Z[π1(M)] under the augmentation z 7rarr 0 There is anatural morphism from the localization to the completion

Σminus1Z[π1(M)] rarr Z[π1(M)] = Z[π1(M)][[z]][zminus1]

which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also

features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]

18 3- and 4-dimensional manifolds

See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds

Part 2 The algebraic K- and L-theory of general-

ized free products via noncommutative localization

A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of

the Z[π1(M)]-module chain complex C(M) of the universal cover M is the

7

assembly of an A-module chain complex constructed from the chain com-

plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24

21 The algebraic K-theory of a noncommutative localiza-

tion

Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution

0 Ps

Q T 0

such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]

K1(A) rarr K1(Σminus1A) rarr K1(AΣ) rarr K0(A) rarr K0(Σ

minus1A)

was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo

TorAi (Σminus1AΣminus1A) = 0 (i gt 1)

then

(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B

(ii) the localization exact sequence extends to the higher K-groups

middot middot middot rarr Kn(A) rarr Kn(Σminus1A) rarr Kn(AΣ) rarr Knminus1(A) rarr middot middot middot rarr K0(Σ

minus1A)

with Kn(AΣ) = Knminus1(H(AΣ))

8

22 Matrix rings

The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings

Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring

A =

(A1 B0 A2

)

An A-module can be written as

M =

(M1

M2

)

with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection

A1 timesA2 rarr A (a1 a2) 7rarr

(a1 00 a2

)

induces isomorphisms of algebraic K-groups

Klowast(A1)oplusKlowast(A2) sim= Klowast(A)

The columns of A are fg projective A-modules

P1 =

(A1

0

) P2 =

(BA2

)

such that

P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)

HomA(P1 P2) = B HomA(P2 P1) = 0

The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring

Σminus1A = M2(C) =

(C CC C

)

with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1

sim= Σminus1P2 The Morita equivalence

Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L

9

induces isomorphisms in algebraic K-theory

Klowast(M2(C)) sim= Klowast(C)

The composite of the functor

A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M

and the Morita equivalence is the assembly functor

A-modules rarr C-modules

M =

(M1

M2

)7rarr (C C)otimesA M

= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)

inducing the morphisms

Klowast(A) = Klowast(A1)oplusKlowast(A2) rarr Klowast(Σminus1A) = Klowast(C)

in the algebraic K-theory localization exact sequence

There are evident generalizations to k times k matrix rings for any k gt 2

23 HNN extensions

The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with

α(s)z = zβ(s) isin R lowastαβ z (s isin S)

Define the triangular 2times 2 matrix ring

A =

(R Rα oplusRβ

0 S

)

with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with

σ1 =

((1 0)0

) σ2 =

((0 1)0

) P1 =

(R0

)rarr P2 =

(Rα oplusRβ

S

)

The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies

Σminus1A = M2(R lowastαβ z)

10

Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected

M = M1 cupNtimes01 N times [0 1]

N times [0 1]

M1

By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1

π1(M) = π1(M1) lowastαβ z

so thatZ[π1(M)] = Z[π1(M1)] lowastαβ z

As above define a triangular 2times 2 matrix ring

A =

(Z[π1(N)] Z[π1(M1)]α oplus Z[π1(M1)]β

0 Z[π1(M)]

)

with noncommutative localization

Σminus1A = M2(Z[π1(M1)] lowastαβ z) = M2(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are

injective and the universal cover M is a union

M =⋃

gisin[π1(M)π1(M1)]

gM1

11

of translates of the universal cover M1 of M1 and

g1M1 cap g2M1 =

hN if g1 cap g2z = h isin [π1(M) π1(N)]

g1M1 if g1 = g2

empty if g1 6= g2 and g1 cap g2z = empty

with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms

M zminus2M1 zminus1M1 M1 zM1 z2M1

zminus1N N zN z2N

The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps

iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)

iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)

defining a fg projective A-module chain complex

(C(M1)

C(N)

)with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)

)

= C(M)

by the Mayer-Vietoris theorem

Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms

R rarr R lowastαβ z S rarr R lowastαβ z

A =

(R Rα oplusRβ

0 S

)rarr Σminus1A = M2(R lowastαβ z)

12

are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C

0 R lowastαβ z otimesS Eiαminusziβ

R lowastαβ z otimesR D C 0

with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(RS α β)α β 01 1 0

Kn(A) = Kn(R)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R lowastαβ z) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(Rα β)(αminusβ)oplus0

Kn(R) Kn(R lowastαβ z)

In particular for α = β = 1 S = R rarr R the HNN extension is justthe Laurent polynomial extension

R lowastαβ z = R[z zminus1]

and the Mayer-Vietoris exact sequence splits to give the original splitting ofBass Heller and Swan [1]

K1(R[z zminus1]) = K1(R)oplusK0(R)oplus Nil0(R)oplus Nil0(R)

as well as its extension to the Quillen higher K-groups Klowast

24 Amalgamated free products

The amalgamated free product R1 lowastS R2 is defined for any ring morphismsi1 S rarr R1 i2 S rarr R2 with

r1i1(s) lowast r2 = r1 lowast i2(s)r2 isin R1 lowastS R2 (r1 isin R1 r2 isin R2 s isin S)

13

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

algebraic K-theory of modules and in the algebraic L-theory of quadraticforms The algebraic K-theory obstruction lives in the Whitehead groupWh(π) of the fundamental group π The L-theory obstruction lives in oneof the surgery groups Llowast(Z[π]) of Wall [32] and is defined when the topo-logical and algebraic K-theory obstructions vanish The groups Llowast(Λ) aredefined for any ring with involution Λ to be the generalized Witt groups ofnonsingular quadratic forms over Λ For manifolds of dimension gt 5 thevanishing of the algebraic obstructions is both a necessary and sufficientcondition for deforming a homotopy equivalence to a homeomorphism SeeRanicki [20] for the reduction of the Browder-Novikov-Sullivan-Wall surgerytheory to algebra

A homotopy equivalence of m-dimensional manifolds f M prime rarr M splits

along a submanifold Nn sub Mm if f is homotopic to a map (also denoted byf) such that N prime = fminus1(N) sub M prime is also a submanifold and the restrictionf | N prime rarr N is also a homotopy equivalence For codimension m minus n gt 3the splitting obstruction is just the ordinary surgery obstruction σlowast(f |) isinLm(Z[π1(N)]) For codimension m minus n = 1 2 the splitting obstructionsinvolve the interplay of the knotting properties of codimension (m minus n)submanifolds and Mayer-Vietoris-type decompositions of the algebraic K-and L-groups of Z[π1(M)] in terms of the groups of Z[π1(N)] Z[π1(MN)]

In the case m minus n = 1 π1(M) is a generalized free product ie eitheran amalgamated free product or an HNN extension by the Seifert-vanKampen theorem Codimension 1 splitting theorems and the algebraic K-and L-theory of generalized free products are a major ingredient of high-dimensional manifold topology featuring in the work of Stallings Brow-der Novikov Wall Siebenmann Farrell Hsiang Shaneson Casson Wald-hausen Cappell and the author Noncommutative localization providesa systematic development of this algebra using the intuition afforded by thetopological applications ndash see Part 2 below for a more detailed discussion

13 Homology surgery theory

For a morphism of rings with involution F Z[π] rarr Λ Cappell and Shaneson[3] considered the problem of whether a Λ-coefficient homology equivalenceof manifolds with fundamental group π is H-cobordant to a homeomor-phism Again the answer is no in general with obstructions in the topolog-ical K-theory of vector bundles and in the homology surgery groups Γlowast(F)which are generalized Witt groups of Λ-nonsingular quadratic forms over

4

Z[π] Vogel [28] [29] identified the Λ-coefficient homology surgery groupswith the ordinary L-groups of the localization Σminus1Z[π] of Z[π] inverting theset Σ of Λ-invertible square matrices over Z[π]

Γlowast(F) = Llowast(Σminus1Z[π])

and identified the relative L-groups Llowast(Z[π] rarr Σminus1Z[π]) in the localizationexact sequence

middot middot middot rarr Ln(Z[π]) rarr Ln(Σminus1Z[π]) rarr Ln(Z[π] rarr Σminus1Z[π]) rarr Lnminus1(Z[π]) rarr

with generalized Witt groups Llowast(Z[π]Σ) of nonsingular Σminus1Z[π]Z[π])-valued quadratic linking forms on Σ-torsion Z[π]-modules of homologicaldimension 1

14 Codimension 2 embeddings

Suppose given a codimension 2 embedding Nn sub Mn+2 such as a knot orlink Let Σminus1A be the localization of A = Z[π1(MN)] inverting the set Σof matrices over A which become invertible over Z[π1(M)] By Alexanderduality the Σminus1A-coefficient homology modules

Hlowast(MN Σminus1A) sim= Hn+2minuslowast(MN Σminus1A) (lowast 6= 0 n + 2)

are determined by the homotopy class of the inclusion N sub M The A-coefficient homology groups Hlowast(MN A) and their Poincare duality prop-erties reflect more subtle invariants of N sub M such as knotting See Ranicki[21] for a general account of high-dimensional codimension 2 embedding the-ory including some of the applications of noncommutative localization

15 Open books

An (n + 2)-dimensional manifold Mn+2 is an open book if there exists acodimension 2 submanifold Nn sub Mn+2 such that the complement MN isa fibre bundle over S1 Every odd-dimensional manifold is an open bookQuinn [17] showed that for k gt 2 a (2k + 2)-dimensional manifold M isan open book if and only if an asymmetric form over Z[π1(M)] associatedto M represents 0 in the Witt group This obstruction was identified inRanicki [21] with an element in the L-group L2k+2(Ω

minus1Z[π1(M)][z zminus1]) ofthe Fredholm localization of Z[π1(M)][z zminus1] (cf section 11 above)

5

16 Boundary link cobordism

An n-dimensional micro-component boundary link is a codimension 2 embedding

Nn =⋃

micro

Sn sub Mn+2 = Sn+2

with a micro-component Seifert surface in which case the fundamental groupof the complement X = MN has a compatible surjection π1(X) rarr Fmicro

onto the free group on micro generators Duval [8] used the work of Cap-pell and Shaneson [4] and Vogel [29] to identify the cobordism group ofn-dimensional micro-component boundary links for n gt 2 with the relative L-group Ln+3(Z[Fmicro]Σ) in the localization exact sequence

middot middot middot rarr Ln+3(Z[Fmicro]) rarr Ln+3(Σminus1Z[Fmicro]) rarr Ln+3(Z[Fmicro]Σ) rarr Ln+2(Z[Fmicro]) rarr

with Σ the set of Z-invertible square matrices over Z[Fmicro] The even-dimensionalboundary link cobordism groups are L2lowast+1(Z[Fmicro]Σ) = 0 The cobordismclass in L2k+2(Z[Fmicro]Σ) of a (2k minus 1)-dimensional micro-component boundarylink cupmicroS

2kminus1 sub S2k+1 was identified with the Witt class of a Σminus1Z[Fmicro]Z[Fmicro]-valued nonsingular (minus1)k+1-quadratic linking form on Hk(XZ[Fmicro]) gener-alizing the Blanchfield pairing on the homology of the infinite cyclic coverof a knot The localization Σminus1Z[Fmicro] was identified by Dicks and Sontag [7]and Farber and Vogel [11] with a ring of rational functions in micro noncommut-ing variables The high odd-dimensional boundary link cobordism groupsL2lowast+2(Z[Fmicro]Σ) have been computed by Sheiham [26]

17 Circle-valued Morse theory

Novikov [15] proposed the study of the critical points of Morse functionsf M rarr S1 on compact manifolds M The lsquoNovikov complexrsquo C(Mf)over Z((z)) = Z[[z]][zminus1] has one generator for each critical point of f andthe lsquoNovikov homologyrsquo

Hlowast(C(Mf)) = Hlowast(M Z((z)))

provides lower bounds on the number of critical points of Morse functions inthe homotopy class of f generalizing the inequalities of the classical Morsetheory of real-valued functions M rarr R Suppose given a Morse functionf M rarr S1 with M = flowastR such that π1(M) = π1(M ) times Z (for the sakeof simplicity) Let Σ be the set of square matrices over Z[π1(M )][z] which

6

become invertible over Z[π1(M)] under the augmentation z 7rarr 0 There is anatural morphism from the localization to the completion

Σminus1Z[π1(M)] rarr Z[π1(M)] = Z[π1(M)][[z]][zminus1]

which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also

features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]

18 3- and 4-dimensional manifolds

See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds

Part 2 The algebraic K- and L-theory of general-

ized free products via noncommutative localization

A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of

the Z[π1(M)]-module chain complex C(M) of the universal cover M is the

7

assembly of an A-module chain complex constructed from the chain com-

plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24

21 The algebraic K-theory of a noncommutative localiza-

tion

Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution

0 Ps

Q T 0

such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]

K1(A) rarr K1(Σminus1A) rarr K1(AΣ) rarr K0(A) rarr K0(Σ

minus1A)

was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo

TorAi (Σminus1AΣminus1A) = 0 (i gt 1)

then

(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B

(ii) the localization exact sequence extends to the higher K-groups

middot middot middot rarr Kn(A) rarr Kn(Σminus1A) rarr Kn(AΣ) rarr Knminus1(A) rarr middot middot middot rarr K0(Σ

minus1A)

with Kn(AΣ) = Knminus1(H(AΣ))

8

22 Matrix rings

The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings

Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring

A =

(A1 B0 A2

)

An A-module can be written as

M =

(M1

M2

)

with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection

A1 timesA2 rarr A (a1 a2) 7rarr

(a1 00 a2

)

induces isomorphisms of algebraic K-groups

Klowast(A1)oplusKlowast(A2) sim= Klowast(A)

The columns of A are fg projective A-modules

P1 =

(A1

0

) P2 =

(BA2

)

such that

P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)

HomA(P1 P2) = B HomA(P2 P1) = 0

The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring

Σminus1A = M2(C) =

(C CC C

)

with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1

sim= Σminus1P2 The Morita equivalence

Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L

9

induces isomorphisms in algebraic K-theory

Klowast(M2(C)) sim= Klowast(C)

The composite of the functor

A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M

and the Morita equivalence is the assembly functor

A-modules rarr C-modules

M =

(M1

M2

)7rarr (C C)otimesA M

= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)

inducing the morphisms

Klowast(A) = Klowast(A1)oplusKlowast(A2) rarr Klowast(Σminus1A) = Klowast(C)

in the algebraic K-theory localization exact sequence

There are evident generalizations to k times k matrix rings for any k gt 2

23 HNN extensions

The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with

α(s)z = zβ(s) isin R lowastαβ z (s isin S)

Define the triangular 2times 2 matrix ring

A =

(R Rα oplusRβ

0 S

)

with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with

σ1 =

((1 0)0

) σ2 =

((0 1)0

) P1 =

(R0

)rarr P2 =

(Rα oplusRβ

S

)

The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies

Σminus1A = M2(R lowastαβ z)

10

Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected

M = M1 cupNtimes01 N times [0 1]

N times [0 1]

M1

By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1

π1(M) = π1(M1) lowastαβ z

so thatZ[π1(M)] = Z[π1(M1)] lowastαβ z

As above define a triangular 2times 2 matrix ring

A =

(Z[π1(N)] Z[π1(M1)]α oplus Z[π1(M1)]β

0 Z[π1(M)]

)

with noncommutative localization

Σminus1A = M2(Z[π1(M1)] lowastαβ z) = M2(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are

injective and the universal cover M is a union

M =⋃

gisin[π1(M)π1(M1)]

gM1

11

of translates of the universal cover M1 of M1 and

g1M1 cap g2M1 =

hN if g1 cap g2z = h isin [π1(M) π1(N)]

g1M1 if g1 = g2

empty if g1 6= g2 and g1 cap g2z = empty

with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms

M zminus2M1 zminus1M1 M1 zM1 z2M1

zminus1N N zN z2N

The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps

iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)

iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)

defining a fg projective A-module chain complex

(C(M1)

C(N)

)with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)

)

= C(M)

by the Mayer-Vietoris theorem

Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms

R rarr R lowastαβ z S rarr R lowastαβ z

A =

(R Rα oplusRβ

0 S

)rarr Σminus1A = M2(R lowastαβ z)

12

are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C

0 R lowastαβ z otimesS Eiαminusziβ

R lowastαβ z otimesR D C 0

with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(RS α β)α β 01 1 0

Kn(A) = Kn(R)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R lowastαβ z) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(Rα β)(αminusβ)oplus0

Kn(R) Kn(R lowastαβ z)

In particular for α = β = 1 S = R rarr R the HNN extension is justthe Laurent polynomial extension

R lowastαβ z = R[z zminus1]

and the Mayer-Vietoris exact sequence splits to give the original splitting ofBass Heller and Swan [1]

K1(R[z zminus1]) = K1(R)oplusK0(R)oplus Nil0(R)oplus Nil0(R)

as well as its extension to the Quillen higher K-groups Klowast

24 Amalgamated free products

The amalgamated free product R1 lowastS R2 is defined for any ring morphismsi1 S rarr R1 i2 S rarr R2 with

r1i1(s) lowast r2 = r1 lowast i2(s)r2 isin R1 lowastS R2 (r1 isin R1 r2 isin R2 s isin S)

13

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

Z[π] Vogel [28] [29] identified the Λ-coefficient homology surgery groupswith the ordinary L-groups of the localization Σminus1Z[π] of Z[π] inverting theset Σ of Λ-invertible square matrices over Z[π]

Γlowast(F) = Llowast(Σminus1Z[π])

and identified the relative L-groups Llowast(Z[π] rarr Σminus1Z[π]) in the localizationexact sequence

middot middot middot rarr Ln(Z[π]) rarr Ln(Σminus1Z[π]) rarr Ln(Z[π] rarr Σminus1Z[π]) rarr Lnminus1(Z[π]) rarr

with generalized Witt groups Llowast(Z[π]Σ) of nonsingular Σminus1Z[π]Z[π])-valued quadratic linking forms on Σ-torsion Z[π]-modules of homologicaldimension 1

14 Codimension 2 embeddings

Suppose given a codimension 2 embedding Nn sub Mn+2 such as a knot orlink Let Σminus1A be the localization of A = Z[π1(MN)] inverting the set Σof matrices over A which become invertible over Z[π1(M)] By Alexanderduality the Σminus1A-coefficient homology modules

Hlowast(MN Σminus1A) sim= Hn+2minuslowast(MN Σminus1A) (lowast 6= 0 n + 2)

are determined by the homotopy class of the inclusion N sub M The A-coefficient homology groups Hlowast(MN A) and their Poincare duality prop-erties reflect more subtle invariants of N sub M such as knotting See Ranicki[21] for a general account of high-dimensional codimension 2 embedding the-ory including some of the applications of noncommutative localization

15 Open books

An (n + 2)-dimensional manifold Mn+2 is an open book if there exists acodimension 2 submanifold Nn sub Mn+2 such that the complement MN isa fibre bundle over S1 Every odd-dimensional manifold is an open bookQuinn [17] showed that for k gt 2 a (2k + 2)-dimensional manifold M isan open book if and only if an asymmetric form over Z[π1(M)] associatedto M represents 0 in the Witt group This obstruction was identified inRanicki [21] with an element in the L-group L2k+2(Ω

minus1Z[π1(M)][z zminus1]) ofthe Fredholm localization of Z[π1(M)][z zminus1] (cf section 11 above)

5

16 Boundary link cobordism

An n-dimensional micro-component boundary link is a codimension 2 embedding

Nn =⋃

micro

Sn sub Mn+2 = Sn+2

with a micro-component Seifert surface in which case the fundamental groupof the complement X = MN has a compatible surjection π1(X) rarr Fmicro

onto the free group on micro generators Duval [8] used the work of Cap-pell and Shaneson [4] and Vogel [29] to identify the cobordism group ofn-dimensional micro-component boundary links for n gt 2 with the relative L-group Ln+3(Z[Fmicro]Σ) in the localization exact sequence

middot middot middot rarr Ln+3(Z[Fmicro]) rarr Ln+3(Σminus1Z[Fmicro]) rarr Ln+3(Z[Fmicro]Σ) rarr Ln+2(Z[Fmicro]) rarr

with Σ the set of Z-invertible square matrices over Z[Fmicro] The even-dimensionalboundary link cobordism groups are L2lowast+1(Z[Fmicro]Σ) = 0 The cobordismclass in L2k+2(Z[Fmicro]Σ) of a (2k minus 1)-dimensional micro-component boundarylink cupmicroS

2kminus1 sub S2k+1 was identified with the Witt class of a Σminus1Z[Fmicro]Z[Fmicro]-valued nonsingular (minus1)k+1-quadratic linking form on Hk(XZ[Fmicro]) gener-alizing the Blanchfield pairing on the homology of the infinite cyclic coverof a knot The localization Σminus1Z[Fmicro] was identified by Dicks and Sontag [7]and Farber and Vogel [11] with a ring of rational functions in micro noncommut-ing variables The high odd-dimensional boundary link cobordism groupsL2lowast+2(Z[Fmicro]Σ) have been computed by Sheiham [26]

17 Circle-valued Morse theory

Novikov [15] proposed the study of the critical points of Morse functionsf M rarr S1 on compact manifolds M The lsquoNovikov complexrsquo C(Mf)over Z((z)) = Z[[z]][zminus1] has one generator for each critical point of f andthe lsquoNovikov homologyrsquo

Hlowast(C(Mf)) = Hlowast(M Z((z)))

provides lower bounds on the number of critical points of Morse functions inthe homotopy class of f generalizing the inequalities of the classical Morsetheory of real-valued functions M rarr R Suppose given a Morse functionf M rarr S1 with M = flowastR such that π1(M) = π1(M ) times Z (for the sakeof simplicity) Let Σ be the set of square matrices over Z[π1(M )][z] which

6

become invertible over Z[π1(M)] under the augmentation z 7rarr 0 There is anatural morphism from the localization to the completion

Σminus1Z[π1(M)] rarr Z[π1(M)] = Z[π1(M)][[z]][zminus1]

which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also

features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]

18 3- and 4-dimensional manifolds

See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds

Part 2 The algebraic K- and L-theory of general-

ized free products via noncommutative localization

A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of

the Z[π1(M)]-module chain complex C(M) of the universal cover M is the

7

assembly of an A-module chain complex constructed from the chain com-

plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24

21 The algebraic K-theory of a noncommutative localiza-

tion

Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution

0 Ps

Q T 0

such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]

K1(A) rarr K1(Σminus1A) rarr K1(AΣ) rarr K0(A) rarr K0(Σ

minus1A)

was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo

TorAi (Σminus1AΣminus1A) = 0 (i gt 1)

then

(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B

(ii) the localization exact sequence extends to the higher K-groups

middot middot middot rarr Kn(A) rarr Kn(Σminus1A) rarr Kn(AΣ) rarr Knminus1(A) rarr middot middot middot rarr K0(Σ

minus1A)

with Kn(AΣ) = Knminus1(H(AΣ))

8

22 Matrix rings

The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings

Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring

A =

(A1 B0 A2

)

An A-module can be written as

M =

(M1

M2

)

with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection

A1 timesA2 rarr A (a1 a2) 7rarr

(a1 00 a2

)

induces isomorphisms of algebraic K-groups

Klowast(A1)oplusKlowast(A2) sim= Klowast(A)

The columns of A are fg projective A-modules

P1 =

(A1

0

) P2 =

(BA2

)

such that

P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)

HomA(P1 P2) = B HomA(P2 P1) = 0

The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring

Σminus1A = M2(C) =

(C CC C

)

with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1

sim= Σminus1P2 The Morita equivalence

Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L

9

induces isomorphisms in algebraic K-theory

Klowast(M2(C)) sim= Klowast(C)

The composite of the functor

A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M

and the Morita equivalence is the assembly functor

A-modules rarr C-modules

M =

(M1

M2

)7rarr (C C)otimesA M

= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)

inducing the morphisms

Klowast(A) = Klowast(A1)oplusKlowast(A2) rarr Klowast(Σminus1A) = Klowast(C)

in the algebraic K-theory localization exact sequence

There are evident generalizations to k times k matrix rings for any k gt 2

23 HNN extensions

The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with

α(s)z = zβ(s) isin R lowastαβ z (s isin S)

Define the triangular 2times 2 matrix ring

A =

(R Rα oplusRβ

0 S

)

with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with

σ1 =

((1 0)0

) σ2 =

((0 1)0

) P1 =

(R0

)rarr P2 =

(Rα oplusRβ

S

)

The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies

Σminus1A = M2(R lowastαβ z)

10

Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected

M = M1 cupNtimes01 N times [0 1]

N times [0 1]

M1

By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1

π1(M) = π1(M1) lowastαβ z

so thatZ[π1(M)] = Z[π1(M1)] lowastαβ z

As above define a triangular 2times 2 matrix ring

A =

(Z[π1(N)] Z[π1(M1)]α oplus Z[π1(M1)]β

0 Z[π1(M)]

)

with noncommutative localization

Σminus1A = M2(Z[π1(M1)] lowastαβ z) = M2(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are

injective and the universal cover M is a union

M =⋃

gisin[π1(M)π1(M1)]

gM1

11

of translates of the universal cover M1 of M1 and

g1M1 cap g2M1 =

hN if g1 cap g2z = h isin [π1(M) π1(N)]

g1M1 if g1 = g2

empty if g1 6= g2 and g1 cap g2z = empty

with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms

M zminus2M1 zminus1M1 M1 zM1 z2M1

zminus1N N zN z2N

The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps

iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)

iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)

defining a fg projective A-module chain complex

(C(M1)

C(N)

)with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)

)

= C(M)

by the Mayer-Vietoris theorem

Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms

R rarr R lowastαβ z S rarr R lowastαβ z

A =

(R Rα oplusRβ

0 S

)rarr Σminus1A = M2(R lowastαβ z)

12

are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C

0 R lowastαβ z otimesS Eiαminusziβ

R lowastαβ z otimesR D C 0

with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(RS α β)α β 01 1 0

Kn(A) = Kn(R)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R lowastαβ z) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(Rα β)(αminusβ)oplus0

Kn(R) Kn(R lowastαβ z)

In particular for α = β = 1 S = R rarr R the HNN extension is justthe Laurent polynomial extension

R lowastαβ z = R[z zminus1]

and the Mayer-Vietoris exact sequence splits to give the original splitting ofBass Heller and Swan [1]

K1(R[z zminus1]) = K1(R)oplusK0(R)oplus Nil0(R)oplus Nil0(R)

as well as its extension to the Quillen higher K-groups Klowast

24 Amalgamated free products

The amalgamated free product R1 lowastS R2 is defined for any ring morphismsi1 S rarr R1 i2 S rarr R2 with

r1i1(s) lowast r2 = r1 lowast i2(s)r2 isin R1 lowastS R2 (r1 isin R1 r2 isin R2 s isin S)

13

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

16 Boundary link cobordism

An n-dimensional micro-component boundary link is a codimension 2 embedding

Nn =⋃

micro

Sn sub Mn+2 = Sn+2

with a micro-component Seifert surface in which case the fundamental groupof the complement X = MN has a compatible surjection π1(X) rarr Fmicro

onto the free group on micro generators Duval [8] used the work of Cap-pell and Shaneson [4] and Vogel [29] to identify the cobordism group ofn-dimensional micro-component boundary links for n gt 2 with the relative L-group Ln+3(Z[Fmicro]Σ) in the localization exact sequence

middot middot middot rarr Ln+3(Z[Fmicro]) rarr Ln+3(Σminus1Z[Fmicro]) rarr Ln+3(Z[Fmicro]Σ) rarr Ln+2(Z[Fmicro]) rarr

with Σ the set of Z-invertible square matrices over Z[Fmicro] The even-dimensionalboundary link cobordism groups are L2lowast+1(Z[Fmicro]Σ) = 0 The cobordismclass in L2k+2(Z[Fmicro]Σ) of a (2k minus 1)-dimensional micro-component boundarylink cupmicroS

2kminus1 sub S2k+1 was identified with the Witt class of a Σminus1Z[Fmicro]Z[Fmicro]-valued nonsingular (minus1)k+1-quadratic linking form on Hk(XZ[Fmicro]) gener-alizing the Blanchfield pairing on the homology of the infinite cyclic coverof a knot The localization Σminus1Z[Fmicro] was identified by Dicks and Sontag [7]and Farber and Vogel [11] with a ring of rational functions in micro noncommut-ing variables The high odd-dimensional boundary link cobordism groupsL2lowast+2(Z[Fmicro]Σ) have been computed by Sheiham [26]

17 Circle-valued Morse theory

Novikov [15] proposed the study of the critical points of Morse functionsf M rarr S1 on compact manifolds M The lsquoNovikov complexrsquo C(Mf)over Z((z)) = Z[[z]][zminus1] has one generator for each critical point of f andthe lsquoNovikov homologyrsquo

Hlowast(C(Mf)) = Hlowast(M Z((z)))

provides lower bounds on the number of critical points of Morse functions inthe homotopy class of f generalizing the inequalities of the classical Morsetheory of real-valued functions M rarr R Suppose given a Morse functionf M rarr S1 with M = flowastR such that π1(M) = π1(M ) times Z (for the sakeof simplicity) Let Σ be the set of square matrices over Z[π1(M )][z] which

6

become invertible over Z[π1(M)] under the augmentation z 7rarr 0 There is anatural morphism from the localization to the completion

Σminus1Z[π1(M)] rarr Z[π1(M)] = Z[π1(M)][[z]][zminus1]

which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also

features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]

18 3- and 4-dimensional manifolds

See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds

Part 2 The algebraic K- and L-theory of general-

ized free products via noncommutative localization

A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of

the Z[π1(M)]-module chain complex C(M) of the universal cover M is the

7

assembly of an A-module chain complex constructed from the chain com-

plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24

21 The algebraic K-theory of a noncommutative localiza-

tion

Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution

0 Ps

Q T 0

such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]

K1(A) rarr K1(Σminus1A) rarr K1(AΣ) rarr K0(A) rarr K0(Σ

minus1A)

was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo

TorAi (Σminus1AΣminus1A) = 0 (i gt 1)

then

(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B

(ii) the localization exact sequence extends to the higher K-groups

middot middot middot rarr Kn(A) rarr Kn(Σminus1A) rarr Kn(AΣ) rarr Knminus1(A) rarr middot middot middot rarr K0(Σ

minus1A)

with Kn(AΣ) = Knminus1(H(AΣ))

8

22 Matrix rings

The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings

Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring

A =

(A1 B0 A2

)

An A-module can be written as

M =

(M1

M2

)

with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection

A1 timesA2 rarr A (a1 a2) 7rarr

(a1 00 a2

)

induces isomorphisms of algebraic K-groups

Klowast(A1)oplusKlowast(A2) sim= Klowast(A)

The columns of A are fg projective A-modules

P1 =

(A1

0

) P2 =

(BA2

)

such that

P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)

HomA(P1 P2) = B HomA(P2 P1) = 0

The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring

Σminus1A = M2(C) =

(C CC C

)

with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1

sim= Σminus1P2 The Morita equivalence

Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L

9

induces isomorphisms in algebraic K-theory

Klowast(M2(C)) sim= Klowast(C)

The composite of the functor

A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M

and the Morita equivalence is the assembly functor

A-modules rarr C-modules

M =

(M1

M2

)7rarr (C C)otimesA M

= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)

inducing the morphisms

Klowast(A) = Klowast(A1)oplusKlowast(A2) rarr Klowast(Σminus1A) = Klowast(C)

in the algebraic K-theory localization exact sequence

There are evident generalizations to k times k matrix rings for any k gt 2

23 HNN extensions

The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with

α(s)z = zβ(s) isin R lowastαβ z (s isin S)

Define the triangular 2times 2 matrix ring

A =

(R Rα oplusRβ

0 S

)

with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with

σ1 =

((1 0)0

) σ2 =

((0 1)0

) P1 =

(R0

)rarr P2 =

(Rα oplusRβ

S

)

The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies

Σminus1A = M2(R lowastαβ z)

10

Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected

M = M1 cupNtimes01 N times [0 1]

N times [0 1]

M1

By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1

π1(M) = π1(M1) lowastαβ z

so thatZ[π1(M)] = Z[π1(M1)] lowastαβ z

As above define a triangular 2times 2 matrix ring

A =

(Z[π1(N)] Z[π1(M1)]α oplus Z[π1(M1)]β

0 Z[π1(M)]

)

with noncommutative localization

Σminus1A = M2(Z[π1(M1)] lowastαβ z) = M2(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are

injective and the universal cover M is a union

M =⋃

gisin[π1(M)π1(M1)]

gM1

11

of translates of the universal cover M1 of M1 and

g1M1 cap g2M1 =

hN if g1 cap g2z = h isin [π1(M) π1(N)]

g1M1 if g1 = g2

empty if g1 6= g2 and g1 cap g2z = empty

with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms

M zminus2M1 zminus1M1 M1 zM1 z2M1

zminus1N N zN z2N

The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps

iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)

iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)

defining a fg projective A-module chain complex

(C(M1)

C(N)

)with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)

)

= C(M)

by the Mayer-Vietoris theorem

Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms

R rarr R lowastαβ z S rarr R lowastαβ z

A =

(R Rα oplusRβ

0 S

)rarr Σminus1A = M2(R lowastαβ z)

12

are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C

0 R lowastαβ z otimesS Eiαminusziβ

R lowastαβ z otimesR D C 0

with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(RS α β)α β 01 1 0

Kn(A) = Kn(R)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R lowastαβ z) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(Rα β)(αminusβ)oplus0

Kn(R) Kn(R lowastαβ z)

In particular for α = β = 1 S = R rarr R the HNN extension is justthe Laurent polynomial extension

R lowastαβ z = R[z zminus1]

and the Mayer-Vietoris exact sequence splits to give the original splitting ofBass Heller and Swan [1]

K1(R[z zminus1]) = K1(R)oplusK0(R)oplus Nil0(R)oplus Nil0(R)

as well as its extension to the Quillen higher K-groups Klowast

24 Amalgamated free products

The amalgamated free product R1 lowastS R2 is defined for any ring morphismsi1 S rarr R1 i2 S rarr R2 with

r1i1(s) lowast r2 = r1 lowast i2(s)r2 isin R1 lowastS R2 (r1 isin R1 r2 isin R2 s isin S)

13

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

become invertible over Z[π1(M)] under the augmentation z 7rarr 0 There is anatural morphism from the localization to the completion

Σminus1Z[π1(M)] rarr Z[π1(M)] = Z[π1(M)][[z]][zminus1]

which is an injection if π1(M) is abelian or Fmicro (Dicks and Sontag [7] Farberand Vogel [11]) but may not be an injection in general (Sheiham [25]) SeePajitnov [16] Farber and Ranicki [10] Ranicki [22] and Cornea and Ran-icki [6] for the construction and properties of Novikov complexes of f overZ[π1(M)] and Σminus1Z[π1(M)] Naturally noncommutative localization also

features in the more general Morse theory of closed 1-forms ndash see Novikov[15] and Farber [9]

18 3- and 4-dimensional manifolds

See Garoufalidis and Kricker [12] Quinn [18] for applications of noncommu-tative localization in the topology of 3- and 4-dimensional manifolds

Part 2 The algebraic K- and L-theory of general-

ized free products via noncommutative localization

A generalized free product of groups (or rings) is either an amalgamated freeproduct or an HNN extension The expressions of Schofield [24] of gener-alized free products as noncommutative localizations of triangular matrixrings combine with the localization exact sequences of Neeman and Ranicki[14] to provide more systematic proofs of the Mayer-Vietoris decomposi-tions of Waldhausen [30] and Cappell [2] of the algebraic K- and L-theoryof generalized free products The topological motivation for these proofscomes from a noncommutative localization interpretation of the Seifert-vanKampen and Mayer-Vietoris theorems If (MN sube M) is a two-sided pairof connected CW complexes the fundamental group π1(M) is a general-ized free product an amalgamated free product if N separates M and anHNN extension otherwise The morphisms π1(N) rarr π1(MN) determinea triangular k times k matrix ring A with universal localization the full k times kmatrix ring Σminus1A = Mk(Z[π1(M)]) (k = 3 in the separating case k = 2in the non-separating case) such that the corresponding presentations of

the Z[π1(M)]-module chain complex C(M) of the universal cover M is the

7

assembly of an A-module chain complex constructed from the chain com-

plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24

21 The algebraic K-theory of a noncommutative localiza-

tion

Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution

0 Ps

Q T 0

such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]

K1(A) rarr K1(Σminus1A) rarr K1(AΣ) rarr K0(A) rarr K0(Σ

minus1A)

was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo

TorAi (Σminus1AΣminus1A) = 0 (i gt 1)

then

(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B

(ii) the localization exact sequence extends to the higher K-groups

middot middot middot rarr Kn(A) rarr Kn(Σminus1A) rarr Kn(AΣ) rarr Knminus1(A) rarr middot middot middot rarr K0(Σ

minus1A)

with Kn(AΣ) = Knminus1(H(AΣ))

8

22 Matrix rings

The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings

Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring

A =

(A1 B0 A2

)

An A-module can be written as

M =

(M1

M2

)

with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection

A1 timesA2 rarr A (a1 a2) 7rarr

(a1 00 a2

)

induces isomorphisms of algebraic K-groups

Klowast(A1)oplusKlowast(A2) sim= Klowast(A)

The columns of A are fg projective A-modules

P1 =

(A1

0

) P2 =

(BA2

)

such that

P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)

HomA(P1 P2) = B HomA(P2 P1) = 0

The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring

Σminus1A = M2(C) =

(C CC C

)

with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1

sim= Σminus1P2 The Morita equivalence

Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L

9

induces isomorphisms in algebraic K-theory

Klowast(M2(C)) sim= Klowast(C)

The composite of the functor

A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M

and the Morita equivalence is the assembly functor

A-modules rarr C-modules

M =

(M1

M2

)7rarr (C C)otimesA M

= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)

inducing the morphisms

Klowast(A) = Klowast(A1)oplusKlowast(A2) rarr Klowast(Σminus1A) = Klowast(C)

in the algebraic K-theory localization exact sequence

There are evident generalizations to k times k matrix rings for any k gt 2

23 HNN extensions

The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with

α(s)z = zβ(s) isin R lowastαβ z (s isin S)

Define the triangular 2times 2 matrix ring

A =

(R Rα oplusRβ

0 S

)

with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with

σ1 =

((1 0)0

) σ2 =

((0 1)0

) P1 =

(R0

)rarr P2 =

(Rα oplusRβ

S

)

The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies

Σminus1A = M2(R lowastαβ z)

10

Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected

M = M1 cupNtimes01 N times [0 1]

N times [0 1]

M1

By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1

π1(M) = π1(M1) lowastαβ z

so thatZ[π1(M)] = Z[π1(M1)] lowastαβ z

As above define a triangular 2times 2 matrix ring

A =

(Z[π1(N)] Z[π1(M1)]α oplus Z[π1(M1)]β

0 Z[π1(M)]

)

with noncommutative localization

Σminus1A = M2(Z[π1(M1)] lowastαβ z) = M2(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are

injective and the universal cover M is a union

M =⋃

gisin[π1(M)π1(M1)]

gM1

11

of translates of the universal cover M1 of M1 and

g1M1 cap g2M1 =

hN if g1 cap g2z = h isin [π1(M) π1(N)]

g1M1 if g1 = g2

empty if g1 6= g2 and g1 cap g2z = empty

with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms

M zminus2M1 zminus1M1 M1 zM1 z2M1

zminus1N N zN z2N

The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps

iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)

iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)

defining a fg projective A-module chain complex

(C(M1)

C(N)

)with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)

)

= C(M)

by the Mayer-Vietoris theorem

Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms

R rarr R lowastαβ z S rarr R lowastαβ z

A =

(R Rα oplusRβ

0 S

)rarr Σminus1A = M2(R lowastαβ z)

12

are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C

0 R lowastαβ z otimesS Eiαminusziβ

R lowastαβ z otimesR D C 0

with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(RS α β)α β 01 1 0

Kn(A) = Kn(R)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R lowastαβ z) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(Rα β)(αminusβ)oplus0

Kn(R) Kn(R lowastαβ z)

In particular for α = β = 1 S = R rarr R the HNN extension is justthe Laurent polynomial extension

R lowastαβ z = R[z zminus1]

and the Mayer-Vietoris exact sequence splits to give the original splitting ofBass Heller and Swan [1]

K1(R[z zminus1]) = K1(R)oplusK0(R)oplus Nil0(R)oplus Nil0(R)

as well as its extension to the Quillen higher K-groups Klowast

24 Amalgamated free products

The amalgamated free product R1 lowastS R2 is defined for any ring morphismsi1 S rarr R1 i2 S rarr R2 with

r1i1(s) lowast r2 = r1 lowast i2(s)r2 isin R1 lowastS R2 (r1 isin R1 r2 isin R2 s isin S)

13

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

assembly of an A-module chain complex constructed from the chain com-

plexes C(N) C(MN ) of the universal covers N MN of N MN Thetwo cases will be considered separately in sections 23 24

21 The algebraic K-theory of a noncommutative localiza-

tion

Given an injective noncommutative localization A rarr Σminus1A let H(AΣ) bethe exact category of homological dimension 1 A-modules T which admit afg projective A-module resolution

0 Ps

Q T 0

such that 1 otimes s Σminus1P rarr Σminus1Q is an Σminus1A-module isomorphism Thealgebraic K-theory localization exact sequence of Schofield [24 Theorem412]

K1(A) rarr K1(Σminus1A) rarr K1(AΣ) rarr K0(A) rarr K0(Σ

minus1A)

was obtained for any injective noncommutative localization A rarr Σminus1Awith K1(AΣ) = K0(H(AΣ)) Neeman and Ranicki [14] proved that ifA rarr Σminus1A is injective and lsquostably flatrsquo

TorAi (Σminus1AΣminus1A) = 0 (i gt 1)

then

(i) Σminus1A has the chain complex lifting property every finite fg freeΣminus1A-module chain complex C is chain equivalent to Σminus1B for a finitefg projective A-module chain complex B

(ii) the localization exact sequence extends to the higher K-groups

middot middot middot rarr Kn(A) rarr Kn(Σminus1A) rarr Kn(AΣ) rarr Knminus1(A) rarr middot middot middot rarr K0(Σ

minus1A)

with Kn(AΣ) = Knminus1(H(AΣ))

8

22 Matrix rings

The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings

Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring

A =

(A1 B0 A2

)

An A-module can be written as

M =

(M1

M2

)

with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection

A1 timesA2 rarr A (a1 a2) 7rarr

(a1 00 a2

)

induces isomorphisms of algebraic K-groups

Klowast(A1)oplusKlowast(A2) sim= Klowast(A)

The columns of A are fg projective A-modules

P1 =

(A1

0

) P2 =

(BA2

)

such that

P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)

HomA(P1 P2) = B HomA(P2 P1) = 0

The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring

Σminus1A = M2(C) =

(C CC C

)

with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1

sim= Σminus1P2 The Morita equivalence

Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L

9

induces isomorphisms in algebraic K-theory

Klowast(M2(C)) sim= Klowast(C)

The composite of the functor

A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M

and the Morita equivalence is the assembly functor

A-modules rarr C-modules

M =

(M1

M2

)7rarr (C C)otimesA M

= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)

inducing the morphisms

Klowast(A) = Klowast(A1)oplusKlowast(A2) rarr Klowast(Σminus1A) = Klowast(C)

in the algebraic K-theory localization exact sequence

There are evident generalizations to k times k matrix rings for any k gt 2

23 HNN extensions

The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with

α(s)z = zβ(s) isin R lowastαβ z (s isin S)

Define the triangular 2times 2 matrix ring

A =

(R Rα oplusRβ

0 S

)

with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with

σ1 =

((1 0)0

) σ2 =

((0 1)0

) P1 =

(R0

)rarr P2 =

(Rα oplusRβ

S

)

The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies

Σminus1A = M2(R lowastαβ z)

10

Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected

M = M1 cupNtimes01 N times [0 1]

N times [0 1]

M1

By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1

π1(M) = π1(M1) lowastαβ z

so thatZ[π1(M)] = Z[π1(M1)] lowastαβ z

As above define a triangular 2times 2 matrix ring

A =

(Z[π1(N)] Z[π1(M1)]α oplus Z[π1(M1)]β

0 Z[π1(M)]

)

with noncommutative localization

Σminus1A = M2(Z[π1(M1)] lowastαβ z) = M2(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are

injective and the universal cover M is a union

M =⋃

gisin[π1(M)π1(M1)]

gM1

11

of translates of the universal cover M1 of M1 and

g1M1 cap g2M1 =

hN if g1 cap g2z = h isin [π1(M) π1(N)]

g1M1 if g1 = g2

empty if g1 6= g2 and g1 cap g2z = empty

with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms

M zminus2M1 zminus1M1 M1 zM1 z2M1

zminus1N N zN z2N

The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps

iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)

iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)

defining a fg projective A-module chain complex

(C(M1)

C(N)

)with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)

)

= C(M)

by the Mayer-Vietoris theorem

Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms

R rarr R lowastαβ z S rarr R lowastαβ z

A =

(R Rα oplusRβ

0 S

)rarr Σminus1A = M2(R lowastαβ z)

12

are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C

0 R lowastαβ z otimesS Eiαminusziβ

R lowastαβ z otimesR D C 0

with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(RS α β)α β 01 1 0

Kn(A) = Kn(R)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R lowastαβ z) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(Rα β)(αminusβ)oplus0

Kn(R) Kn(R lowastαβ z)

In particular for α = β = 1 S = R rarr R the HNN extension is justthe Laurent polynomial extension

R lowastαβ z = R[z zminus1]

and the Mayer-Vietoris exact sequence splits to give the original splitting ofBass Heller and Swan [1]

K1(R[z zminus1]) = K1(R)oplusK0(R)oplus Nil0(R)oplus Nil0(R)

as well as its extension to the Quillen higher K-groups Klowast

24 Amalgamated free products

The amalgamated free product R1 lowastS R2 is defined for any ring morphismsi1 S rarr R1 i2 S rarr R2 with

r1i1(s) lowast r2 = r1 lowast i2(s)r2 isin R1 lowastS R2 (r1 isin R1 r2 isin R2 s isin S)

13

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

22 Matrix rings

The amalgamated free product of rings and the HNN construction are spe-cial cases of the following type of noncommutative localization of triangularmatrix rings

Given rings A1 A2 and an (A1 A2)-bimodule B define the triangular2times 2 matrix ring

A =

(A1 B0 A2

)

An A-module can be written as

M =

(M1

M2

)

with M1 an A1-module M2 an A2-module together with an A1-modulemorphism B otimesA2 M2 rarr M1 The injection

A1 timesA2 rarr A (a1 a2) 7rarr

(a1 00 a2

)

induces isomorphisms of algebraic K-groups

Klowast(A1)oplusKlowast(A2) sim= Klowast(A)

The columns of A are fg projective A-modules

P1 =

(A1

0

) P2 =

(BA2

)

such that

P1 oplus P2 = A HomA(Pi Pi) = Ai (i = 1 2)

HomA(P1 P2) = B HomA(P2 P1) = 0

The noncommutative localization of A inverting a non-empty subset Σ subeHomA(P1 P2) = B is the 2times 2 matrix ring

Σminus1A = M2(C) =

(C CC C

)

with C the endomorphism ring of the induced fg projective Σminus1A-moduleΣminus1P1

sim= Σminus1P2 The Morita equivalence

Σminus1A-modules rarr C-modules L 7rarr (C C)otimesΣminus1A L

9

induces isomorphisms in algebraic K-theory

Klowast(M2(C)) sim= Klowast(C)

The composite of the functor

A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M

and the Morita equivalence is the assembly functor

A-modules rarr C-modules

M =

(M1

M2

)7rarr (C C)otimesA M

= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)

inducing the morphisms

Klowast(A) = Klowast(A1)oplusKlowast(A2) rarr Klowast(Σminus1A) = Klowast(C)

in the algebraic K-theory localization exact sequence

There are evident generalizations to k times k matrix rings for any k gt 2

23 HNN extensions

The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with

α(s)z = zβ(s) isin R lowastαβ z (s isin S)

Define the triangular 2times 2 matrix ring

A =

(R Rα oplusRβ

0 S

)

with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with

σ1 =

((1 0)0

) σ2 =

((0 1)0

) P1 =

(R0

)rarr P2 =

(Rα oplusRβ

S

)

The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies

Σminus1A = M2(R lowastαβ z)

10

Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected

M = M1 cupNtimes01 N times [0 1]

N times [0 1]

M1

By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1

π1(M) = π1(M1) lowastαβ z

so thatZ[π1(M)] = Z[π1(M1)] lowastαβ z

As above define a triangular 2times 2 matrix ring

A =

(Z[π1(N)] Z[π1(M1)]α oplus Z[π1(M1)]β

0 Z[π1(M)]

)

with noncommutative localization

Σminus1A = M2(Z[π1(M1)] lowastαβ z) = M2(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are

injective and the universal cover M is a union

M =⋃

gisin[π1(M)π1(M1)]

gM1

11

of translates of the universal cover M1 of M1 and

g1M1 cap g2M1 =

hN if g1 cap g2z = h isin [π1(M) π1(N)]

g1M1 if g1 = g2

empty if g1 6= g2 and g1 cap g2z = empty

with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms

M zminus2M1 zminus1M1 M1 zM1 z2M1

zminus1N N zN z2N

The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps

iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)

iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)

defining a fg projective A-module chain complex

(C(M1)

C(N)

)with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)

)

= C(M)

by the Mayer-Vietoris theorem

Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms

R rarr R lowastαβ z S rarr R lowastαβ z

A =

(R Rα oplusRβ

0 S

)rarr Σminus1A = M2(R lowastαβ z)

12

are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C

0 R lowastαβ z otimesS Eiαminusziβ

R lowastαβ z otimesR D C 0

with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(RS α β)α β 01 1 0

Kn(A) = Kn(R)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R lowastαβ z) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(Rα β)(αminusβ)oplus0

Kn(R) Kn(R lowastαβ z)

In particular for α = β = 1 S = R rarr R the HNN extension is justthe Laurent polynomial extension

R lowastαβ z = R[z zminus1]

and the Mayer-Vietoris exact sequence splits to give the original splitting ofBass Heller and Swan [1]

K1(R[z zminus1]) = K1(R)oplusK0(R)oplus Nil0(R)oplus Nil0(R)

as well as its extension to the Quillen higher K-groups Klowast

24 Amalgamated free products

The amalgamated free product R1 lowastS R2 is defined for any ring morphismsi1 S rarr R1 i2 S rarr R2 with

r1i1(s) lowast r2 = r1 lowast i2(s)r2 isin R1 lowastS R2 (r1 isin R1 r2 isin R2 s isin S)

13

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

induces isomorphisms in algebraic K-theory

Klowast(M2(C)) sim= Klowast(C)

The composite of the functor

A-modules rarr Σminus1A-modules M 7rarr Σminus1M = Σminus1AotimesA M

and the Morita equivalence is the assembly functor

A-modules rarr C-modules

M =

(M1

M2

)7rarr (C C)otimesA M

= coker(C otimesA1 B otimesA2 M2 rarr C otimesA1 M1 oplus C otimesA2 M2)

inducing the morphisms

Klowast(A) = Klowast(A1)oplusKlowast(A2) rarr Klowast(Σminus1A) = Klowast(C)

in the algebraic K-theory localization exact sequence

There are evident generalizations to k times k matrix rings for any k gt 2

23 HNN extensions

The HNN extension R lowastαβ z is defined for any ring morphisms α β S rarr R with

α(s)z = zβ(s) isin R lowastαβ z (s isin S)

Define the triangular 2times 2 matrix ring

A =

(R Rα oplusRβ

0 S

)

with Rα the (RS)-bimodule R with S acting on R via α and similarly forRβ Let Σ = σ1 σ2 sub HomA(P1 P2) with

σ1 =

((1 0)0

) σ2 =

((0 1)0

) P1 =

(R0

)rarr P2 =

(Rα oplusRβ

S

)

The A-modules P1 P2 are fg projective since P1 oplus P2 = A Theorem 131of [24] identifies

Σminus1A = M2(R lowastαβ z)

10

Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected

M = M1 cupNtimes01 N times [0 1]

N times [0 1]

M1

By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1

π1(M) = π1(M1) lowastαβ z

so thatZ[π1(M)] = Z[π1(M1)] lowastαβ z

As above define a triangular 2times 2 matrix ring

A =

(Z[π1(N)] Z[π1(M1)]α oplus Z[π1(M1)]β

0 Z[π1(M)]

)

with noncommutative localization

Σminus1A = M2(Z[π1(M1)] lowastαβ z) = M2(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are

injective and the universal cover M is a union

M =⋃

gisin[π1(M)π1(M1)]

gM1

11

of translates of the universal cover M1 of M1 and

g1M1 cap g2M1 =

hN if g1 cap g2z = h isin [π1(M) π1(N)]

g1M1 if g1 = g2

empty if g1 6= g2 and g1 cap g2z = empty

with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms

M zminus2M1 zminus1M1 M1 zM1 z2M1

zminus1N N zN z2N

The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps

iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)

iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)

defining a fg projective A-module chain complex

(C(M1)

C(N)

)with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)

)

= C(M)

by the Mayer-Vietoris theorem

Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms

R rarr R lowastαβ z S rarr R lowastαβ z

A =

(R Rα oplusRβ

0 S

)rarr Σminus1A = M2(R lowastαβ z)

12

are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C

0 R lowastαβ z otimesS Eiαminusziβ

R lowastαβ z otimesR D C 0

with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(RS α β)α β 01 1 0

Kn(A) = Kn(R)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R lowastαβ z) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(Rα β)(αminusβ)oplus0

Kn(R) Kn(R lowastαβ z)

In particular for α = β = 1 S = R rarr R the HNN extension is justthe Laurent polynomial extension

R lowastαβ z = R[z zminus1]

and the Mayer-Vietoris exact sequence splits to give the original splitting ofBass Heller and Swan [1]

K1(R[z zminus1]) = K1(R)oplusK0(R)oplus Nil0(R)oplus Nil0(R)

as well as its extension to the Quillen higher K-groups Klowast

24 Amalgamated free products

The amalgamated free product R1 lowastS R2 is defined for any ring morphismsi1 S rarr R1 i2 S rarr R2 with

r1i1(s) lowast r2 = r1 lowast i2(s)r2 isin R1 lowastS R2 (r1 isin R1 r2 isin R2 s isin S)

13

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

Example Let (MN sube M) be a non-separating pair of connected CW com-plexes such that N is two-sided in M (ie has a neighbourhood N times [0 1] subeM) with MN = M1 connected

M = M1 cupNtimes01 N times [0 1]

N times [0 1]

M1

By the Seifert-van Kampen theorem the fundamental group π1(M) is theHNN extension determined by the morphisms α β π1(N) rarr π1(M1)induced by the inclusions N times 0 rarr M1 N times 1 rarr M1

π1(M) = π1(M1) lowastαβ z

so thatZ[π1(M)] = Z[π1(M1)] lowastαβ z

As above define a triangular 2times 2 matrix ring

A =

(Z[π1(N)] Z[π1(M1)]α oplus Z[π1(M1)]β

0 Z[π1(M)]

)

with noncommutative localization

Σminus1A = M2(Z[π1(M1)] lowastαβ z) = M2(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms α β are

injective and the universal cover M is a union

M =⋃

gisin[π1(M)π1(M1)]

gM1

11

of translates of the universal cover M1 of M1 and

g1M1 cap g2M1 =

hN if g1 cap g2z = h isin [π1(M) π1(N)]

g1M1 if g1 = g2

empty if g1 6= g2 and g1 cap g2z = empty

with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms

M zminus2M1 zminus1M1 M1 zM1 z2M1

zminus1N N zN z2N

The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps

iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)

iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)

defining a fg projective A-module chain complex

(C(M1)

C(N)

)with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)

)

= C(M)

by the Mayer-Vietoris theorem

Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms

R rarr R lowastαβ z S rarr R lowastαβ z

A =

(R Rα oplusRβ

0 S

)rarr Σminus1A = M2(R lowastαβ z)

12

are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C

0 R lowastαβ z otimesS Eiαminusziβ

R lowastαβ z otimesR D C 0

with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(RS α β)α β 01 1 0

Kn(A) = Kn(R)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R lowastαβ z) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(Rα β)(αminusβ)oplus0

Kn(R) Kn(R lowastαβ z)

In particular for α = β = 1 S = R rarr R the HNN extension is justthe Laurent polynomial extension

R lowastαβ z = R[z zminus1]

and the Mayer-Vietoris exact sequence splits to give the original splitting ofBass Heller and Swan [1]

K1(R[z zminus1]) = K1(R)oplusK0(R)oplus Nil0(R)oplus Nil0(R)

as well as its extension to the Quillen higher K-groups Klowast

24 Amalgamated free products

The amalgamated free product R1 lowastS R2 is defined for any ring morphismsi1 S rarr R1 i2 S rarr R2 with

r1i1(s) lowast r2 = r1 lowast i2(s)r2 isin R1 lowastS R2 (r1 isin R1 r2 isin R2 s isin S)

13

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

of translates of the universal cover M1 of M1 and

g1M1 cap g2M1 =

hN if g1 cap g2z = h isin [π1(M) π1(N)]

g1M1 if g1 = g2

empty if g1 6= g2 and g1 cap g2z = empty

with hN the translates of the universal cover N of N In the diagram it isassumed that α β are isomorphisms

M zminus2M1 zminus1M1 M1 zM1 z2M1

zminus1N N zN z2N

The cellular fg free chain complexes C(M1) C(N) are related by Z[π1(M1)]-module chain maps

iα Z[π1(M1)]α otimesZ[π1(N)] C(N) rarr C(M1)

iβ Z[π1(M1)]β otimesZ[π1(N)] C(N) rarr C(M1)

defining a fg projective A-module chain complex

(C(M1)

C(N)

)with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

(iα minus ziβ Z[π1(M)]otimesZ[π1(N)] C(N) rarr Z[π1(M)] otimesZ[π1(M1)] C(M1)

)

= C(M)

by the Mayer-Vietoris theorem

Let R lowastαβ z be an HNN extension of rings in which the morphismsα β S rarr R are both injections of (S S)-bimodule direct summands andRα Rβ are flat S-modules (This is the case in the above example if π1(N) rarrπ1(M) is injective) Then the natural ring morphisms

R rarr R lowastαβ z S rarr R lowastαβ z

A =

(R Rα oplusRβ

0 S

)rarr Σminus1A = M2(R lowastαβ z)

12

are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C

0 R lowastαβ z otimesS Eiαminusziβ

R lowastαβ z otimesR D C 0

with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(RS α β)α β 01 1 0

Kn(A) = Kn(R)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R lowastαβ z) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(Rα β)(αminusβ)oplus0

Kn(R) Kn(R lowastαβ z)

In particular for α = β = 1 S = R rarr R the HNN extension is justthe Laurent polynomial extension

R lowastαβ z = R[z zminus1]

and the Mayer-Vietoris exact sequence splits to give the original splitting ofBass Heller and Swan [1]

K1(R[z zminus1]) = K1(R)oplusK0(R)oplus Nil0(R)oplus Nil0(R)

as well as its extension to the Quillen higher K-groups Klowast

24 Amalgamated free products

The amalgamated free product R1 lowastS R2 is defined for any ring morphismsi1 S rarr R1 i2 S rarr R2 with

r1i1(s) lowast r2 = r1 lowast i2(s)r2 isin R1 lowastS R2 (r1 isin R1 r2 isin R2 s isin S)

13

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

are injective and Σminus1A is a stably flat universal localization withH(AΣ) =Nil(RS α β) the nilpotent category of Waldhausen [30] The chain com-plex lifting property of Σminus1A gives a noncommutative localization proof ofthe existence of Mayer-Vietoris presentations for finite fg free R lowastαβ z-module chain complexes C

0 R lowastαβ z otimesS Eiαminusziβ

R lowastαβ z otimesR D C 0

with D (resp E) a finite fg free R- (resp S-) module chain complex ([30]Ranicki [23]) The algebraic K-theory localization exact sequence of [14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(RS α β)α β 01 1 0

Kn(A) = Kn(R)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R lowastαβ z) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(Rα β)(αminusβ)oplus0

Kn(R) Kn(R lowastαβ z)

In particular for α = β = 1 S = R rarr R the HNN extension is justthe Laurent polynomial extension

R lowastαβ z = R[z zminus1]

and the Mayer-Vietoris exact sequence splits to give the original splitting ofBass Heller and Swan [1]

K1(R[z zminus1]) = K1(R)oplusK0(R)oplus Nil0(R)oplus Nil0(R)

as well as its extension to the Quillen higher K-groups Klowast

24 Amalgamated free products

The amalgamated free product R1 lowastS R2 is defined for any ring morphismsi1 S rarr R1 i2 S rarr R2 with

r1i1(s) lowast r2 = r1 lowast i2(s)r2 isin R1 lowastS R2 (r1 isin R1 r2 isin R2 s isin S)

13

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

Define the triangular 3times 3 matrix ring

A =

R1 0 R1

0 R2 R2

0 0 S

and the A-module morphisms

σ1 =

100

P1 =

R1

00

rarr P3 =

R1

R2

S

σ2 =

010

P2 =

0R2

0

rarr P3 =

R1

R2

S

The A-modules P1 P2 P3 are fg projective since P1 oplus P2 oplus P3 = A Thenoncommutative localization of A inverting Σ = σ1 σ2 is the full 3 times 3matrix ring

Σminus1A = M3(R1 lowastS R2)

(a modification of Theorem 410 of [24])

Example Let (MN sube M) be a separating pair of CW complexes such thatN has a neighbourhood N times [0 1] sube M and

M = M1 cupNtimes0 N times [0 1] cupNtimes1 M2

with M1M2 N connected

M1 M2N times [0 1]

By the Seifert-van Kampen theorem the fundamental group of M is theamalgamated free product

π1(M) = π1(M1) lowastπ1(N) π1(M2)

14

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

so thatZ[π1(M)] = Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]

As above define a triangular matrix ring

A =

Z[π1(M1)] 0 Z[π1(M1)]

0 Z[π1(M2)] Z[π1(M2)]0 0 Z[π1(N)]

with noncommutative localization

Σminus1A = M3(Z[π1(M1)] lowastZ[π1(N)] Z[π1(M2)]) = M3(Z[π1(M)])

Assume that π1(N) rarr π1(M) is injective so that the morphisms

i1 π1(N) rarr π1(M1) i2 π1(N) rarr π1(M2)

π1(M1) rarr π1(M) π1(M2) rarr π1(M)

are all injective and the universal cover M of M is a union

M =⋃

g1isin[π1(M)π1(M1)]

g1M1 cup ⋃hisin[π1(M)π1(N)]

hN

⋃

g2isin[π1(M)π1(M2)]

g2M2

of [π1(M) π1(M1)] translates of the universal cover M1 of M1 and [π1(M)

π1(M2)] translates of the universal cover M2 of M2 with intersection the[π1(M) π1(N)] translates of the universal cover N of N

N M2M1

15

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

The cellular fg free chain complexes C(M1) C(N) of the universal covers

M1 N are related by Z[π1(M1)]-module chain maps

i1 Z[π1(M1)]otimesZ[π1(N)] C(N) rarr C(M1)

i2 Z[π1(M2)]otimesZ[π1(N)] C(N) rarr C(M2)

defining a fg projective A-module chain complex

C(M1)

C(M2)

C(N)

with assembly

the cellular fg free Z[π1(M)]-module chain complex of M

coker

((1otimes i1

1otimes i2

) Z[π1(M)]otimesZ[π1(N)] C(N) rarr

Z[π1(M)]otimesZ[π1(M1)] C(M1)oplus Z[π1(M)]otimesZ[π1(M1)] C(M2)

)

= C(M)

by the Mayer-Vietoris theorem

Let R1 lowastS R2 be an amalgamated free product of rings in which the mor-phisms i1 S rarr R1 i2 S rarr R2 are both injections of (S S)-bimoduledirect summands and R1 R2 are flat S-modules (This is the case in theabove example if π1(N) rarr π1(M) is injective) Then the natural ring mor-phisms

R1 rarr R1 lowastS R2 R2 rarr R1 lowastS R2 S rarr R1 lowastS R2

A =

R1 0 R1

0 R2 R2

0 0 S

rarr Σminus1A = M3(R1 lowastS R2)

are injective and Σminus1A is a stably flat noncommutative localization withH(AΣ) = Nil(R1 R2 S) the nilpotent category of Waldhausen [30] Thechain complex lifting property of Σminus1A gives a noncommutative localizationproof of the existence of Mayer-Vietoris presentations for finite fg freeR1 lowastS R2-module chain complexes C

0 R1 lowastS R2 otimesS E R1 lowastS R2 otimesR1 D1 oplusR1 lowastS R2 otimesR2 D2 C 0

with Di (resp E) a finite fg free Ri- (resp S-) module chain complex([30] Ranicki [23]) The algebraic K-theory localization exact sequence of

16

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

[14]

middot middot middot rarr Kn+1(AΣ) = Kn(S)oplusKn(S)oplus Niln(R1 R2 S)

i1 0 00 i2 01 1 0

Kn(A) = Kn(R1)oplusKn(R2)oplusKn(S)

rarr Kn(Σminus1A) = Kn(R1 lowastS R2) rarr

is just the stabilization by 1 Klowast(S) rarr Klowast(S) of the Mayer-Vietoris exactsequence of [30]

Kn(S)oplus Niln(R1 R2 S)i1 0i2 0

Kn(R1)oplusKn(R2) Kn(R1 lowastS R2)

25 The algebraic L-theory of a noncommutative localization

See Chapter 3 of Ranicki [19] for the algebraic L-theory of a commutativelocalization

The algebraic L-theory of a ring A depends on an involution that is afunction A rarr A a 7rarr a such that

a+ b = a+ b ab = b a a = a 1 = 1 (a b isin A)

For an injective noncommutative localization A rarr Σminus1A of a ring A with aninvolution which extends to Σminus1A Vogel [29] obtained a localization exactsequence in quadratic L-theory

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

with Ln(AΣ) = Lnminus1(H(AΣ)) (See [14] for the symmetric L-theory lo-calization exact sequence in the stably flat case) At first sight it does notappear possible to apply this sequence to the triangular matrix rings of sec-tions 22 23 24 How does one define an involution on a triangular matrixring

A =

(A1 B0 A2

)

17

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

The trick is to observe that if A1 A2 are rings with involution and (B β) isa nonsingular symmetric form over A1 such that B is an (A1 A2)-bimodulethen A has a chain duality in the sense of Definition 11 of Ranicki [20]

sending an A-module M =

(M1

M2

)to the 1-dimensional A-module chain

complex

TM TM1 =

(Mlowast

1

0

)rarr TM0 =

(B otimesA2 M

lowast2

Mlowast2

)

The quadratic L-groups of A are just the relative L-groups in the exactsequence

middot middot middot rarr Ln(A) rarr Ln(A2)(Bβ)otimesA2

minus Ln(A1) rarr Lnminus1(A) rarr

In particular for generalized free products of rings with involution the tri-angular matrix rings A of section 23 24 have such chain dualities and inthe injective case the torsion L-groups Llowast(AΣ) = Llowastminus1(H(AΣ)) in thelocalization exact sequence

middot middot middot rarr Ln(A) rarr Ln(Σminus1A) rarr Ln(AΣ) rarr Lnminus1(A) rarr

are just the unitary nilpotent L-groups UNillowast of Cappell [2]

References

[1] H Bass A Heller and R Swan The Whitehead group of a polynomial

extension Publ Math IHES 22 61ndash80 (1964)

[2] S Cappell Unitary nilpotent groups and hermitian K-theory BullAMS 80 1117ndash1122 (1974)

[3] and J Shaneson The codimension two placement problem and

homology equivalent manifolds Ann of Maths 99 277ndash348 (1974)

[4] and Link cobordism Comm Math Helv 55 20ndash49(1980)

[5] P M Cohn Free rings and their relations Academic Press (1971)

[6] O Cornea and A Ranicki Rigidity and glueing for the Morse and

Novikov complexes httparXivmathAT0107221 to appear in J EurMath Soc

18

[7] W Dicks and E Sontag Sylvester domains J Pure Appl Algebra 13243ndash275 (1978)

[8] J Duval Forme de Blanchfield et cobordisme drsquoentrelacs bords CommMath Helv 61 617ndash635 (1986)

[9] M Farber Morse-Novikov critical point theory Cohn localization and

Dirichlet units Commun Contemp Math 1 467ndash495 (1999)

[10] and A Ranicki The Morse-Novikov theory of circle-valued func-

tions and noncommutative localization httparXivmathDG9812122Proc 1998 Moscow Conference for SPNovikovrsquos 60th Birthday ProcSteklov Inst 225 381ndash388 (1999)

[11] and P Vogel The Cohn localization of the free group ring MathProc Camb Phil Soc 111 433ndash443 (1992)

[12] S Garoufalidis and A Kricker A rational noncommutative invariant

of boundary links httparXivmathGT0105028 (2001)

[13] B Hughes and A Ranicki Ends of complexes Cambridge Tracts inMathematics 123 CUP (1996)

[14] A Neeman and A Ranicki Noncommutative localization and chain

complexes I Algebraic K- and L-theory httparXivmathRA0109118(2001)

[15] S P Novikov The hamiltonian formalism and a multi-valued analogue

of Morse theory Russian Math Surveys 375 1ndash56 (1982)

[16] A Pajitnov Incidence coefficients in the Novikov complex for

Morse forms rationality and exponential growth propertieshttparXivmathdg-ga9604004 St Petersburg Math J 9 969ndash 1006 (1998)

[17] F Quinn Open book decompositions and the bordism of automor-

phisms Topology 18 55ndash73 (1979)

[18] Dual decompositions of 4-manifolds II Linear link invariantshttparXivmathGT0109148 (2001)

[19] A Ranicki Exact sequences in the algebraic theory of surgery Mathe-matical Notes 26 Princeton (1981)

19

[20] Algebraic L-theory and topological manifolds CambridgeTracts in Mathematics 102 CUP (1992)

[21] High dimensional knot theory Springer Mathematical Mono-graph Springer (1998)

[22] The algebraic construction of the Novikov complex of a circle-

valued Morse function httparXivmathAT9903090 Math Ann322 745ndash785 (2002)

[23] Algebraic and combinatorial codimension 1 transversalityhttparXivmathAT0308111

[24] A Schofield Representations of rings over skew fields LMS LectureNotes 92 Cambridge (1985)

[25] D Sheiham Noncommutative characteristic polynomials and Cohn lo-

calization J London Math Soc 64 13ndash28 (2001)

[26] Invariants of boundary link cobordism Edinburgh Ph D thesis(2001) httparXivmathAT0110249 AMS Memoir (to appear)

[27] L Siebenmann The obstruction to finding the boundary of an open

manifold of dimension greater than five Princeton PhD thesis (1965)httpwwwmathsedacuk aarsurgerysiebenpdf

[28] PVogel On the obstruction group in homology surgery Publ MathIHES 55 165ndash206 (1982)

[29] Localisation non commutative de formes quadratiques SpringerLecture Notes 967 376ndash389 (1982)

[30] F Waldhausen Algebraic K-theory of generalized free products Annof Maths 108 135ndash256 (1978)

[31] CTC Wall Finiteness conditions for CW complexes Ann of Maths81 55ndash69 (1965)

[32] Surgery on compact manifolds Academic Press (1970) 2ndedition AMS (1999)

School of MathematicsUniversity of EdinburghJames Clerk Maxwell Building

20

Kingrsquos BuildingsMayfield RoadEdinburgh EH9 3JZSCOTLAND UK

e-mail aarmathsedacuk

21

• Finite domination
• Codimension 1 splitting
• Homology surgery theory
• Codimension 2 embeddings
• Open books
• Boundary link cobordism
• Circle-valued Morse theory
• 3- and 4-dimensional manifolds
• The algebraic K-theory of a noncommutative localization
• Matrix rings
• HNN extensions
• Amalgamated free products
• The algebraic L-theory of a noncommutative localization