on conjectures of mathai and borel
TRANSCRIPT
On Conjectures of Mathai and Borel
STANLEY CHANG?
106 Central Street, Wellesley College, Wellesley, MA 02481, U.S.A.
e-mail: [email protected]
(Received: 2 February 2003; accepted in final form: 11 April 2003)
Abstract. Mathai has conjectured that the Cheeger–Gromov invariant rð2Þ ¼ Zð2Þ � Z is ahomotopy invariant of closed manifolds with torsion-free fundamental group. In this paperwe prove this statement for closed manifolds M when the rational Borel conjecture is known
for G ¼ p1ðM Þ, i.e. the assembly map a: H�ðBG;QÞ ! L�ðGÞ � Q is an isomorphism. Ourdiscussion evokes the theory of intersection homology and results related to the highersignature problem.
Mathematics Subject Classifications (2000). 57R67, 55N22.
Key words. signature, Borel conjecture, homotopy invariant, Witt spaces, algebraic Poincarecomplexes.
Let M be a closed, oriented Riemannian manifold of dimension 4k� 1, with k5 2. In
[Ma] Mathai proves that the Cheeger–Gromov invariant rð2Þ � Zð2Þ � Z is a homotopy
invariant of M if G ¼ p1ðM Þ is a Bieberbach group. In the same work, he conjectures
that rð2Þ will be a homotopy invariant for all such manifolds M whose fundamental
group G is torsion-free and discrete. This conjecture is verified by Keswani [K] when
G is torsion-free and the Baum–Connes assembly map mmax : K0 ðBGÞ ! K0ðC�GÞ is
an isomorphism. Yet it is now known that mmax fails to be an isomorphism for
groups satisfying Kazhdan’s property T. This paper improves on Keswani’s result
by showing that Mathai’s conjecture holds for torsion-free groups satisfying the
rational Borel conjecture, for which no counterexamples have been found.
As a consequence of a theorem of Hausmann [H], for every compact odd-dimen-
sional oriented manifold M with fundamental group G, there is a manifold W with
boundary such that G injects into G¼p1ðW Þ and @W¼ rM for some multiple rM of
M. Using this result, the author and Weinberger construct in [CW] a well-defined
Hirzebruch-type invariant for M4k�1 given by
tð2ÞðM Þ ¼1
rðsigG
ð2ÞðeWWÞ � sigðW ÞÞ;
where eWW is the universal cover of W. The map sigGð2Þ is a real-valued homomorphism
on the L-theory group L4kðGÞ given by sigGð2ÞðV Þ ¼ dimGðV
þÞ � dimGðV�Þ for any
?Research partially supported by NSF Grant DMS-9971657.
Geometriae Dedicata 106: 161–167, 2004. 161# 2004 Kluwer Academic Publishers. Printed in the Netherlands.
quadratic form V, considered as an ‘2ðGÞ-module. This invariant tð2Þ is in general a
diffeomorphism invariant, but is not a homotopy invariant when p1ðM Þ is not
torsion-free [CW]. It is now also known that tð2Þ coincides with rð2Þ by the work
of Luck and Schick [LS].
The purpose of this paper is to show that the diffeomorphism invariant tð2Þ, and
subsequently rð2Þ, is actually a homotopy invariant of M4k�1 if the fundamental
group G ¼ p1ðM Þ satisfies the rational Borel conjecture, which states that the
assembly map a : H�ðBG;QÞ ! L�ðGÞ � Q is an isomorphism if G is torsion-free.
We include in our discussion some background in L-theory and intersection
homology.
Introduction
The classical work on cobordism by Thom implies that every compact odd-dimen-
sional oriented manifold M has a multiple rM which is the boundary of an oriented
manifold W. Hausmann [H] showed furthermore that, for every such M with funda-
mental group G, there is a manifold W such that @W ¼ rM, for some multiple rM of
M, with the additional property that the inclusion M ,!W induces an injection
G ,!p1ðW Þ. For such a manifold M4k�1 with fundamental group G, a higher
‘Hirzebruch type’ real-valued function tð2Þ is defines in [CW] in the following
manner:
tGð2ÞðM Þ ¼1
rðsigG
ð2ÞðeWW Þ � sigðW ÞÞ;
where G ¼ p1ðW Þ and eWW is the universal cover of W.
To see that this quantity can be made independent of ðW;GÞ, we consider any
injection G ,!G0. Let WG0 be the G0-space induced from the G-action on eWW to G0.
Now define
tG0
ð2ÞðM Þ ¼1
rðsigG0
ð2ÞðWG0 Þ � sigðWG0=G0ÞÞ:
Since eWW is simply the G-cover WG of W and the quotient WG0=G0 is clearly diffeo-
morphic to W, we note that this definition of tG0
ð2Þ is consistent with the above
definition of tGð2Þ. However, by the G-induction property of Cheeger–Gromov
[CG, page 8, Equation (2.3)], we have
tG0
ð2ÞðM Þ ¼1
rðsigG
0
ð2ÞðWG0 Þ � sigðWG0=G0ÞÞ
¼1
rðsigGð2ÞðWGÞ � sigðW ÞÞ
¼ tGð2ÞðM Þ:
So from G one can pass to any large group G0 without changing the value of this
quantity. Given two manifolds W and W 0 with the required bounding properties,
we can use the larger group G0 ¼p1ðW Þ �G p1ðW0Þ, which contains both fundamental
162 STANLEY CHANG
groups p1ðW Þ and p1ðW 0Þ. The usual Novikov additivity argument? proves that
tGð2ÞðM Þ is independent of all choices [CW]. We will henceforth refer to it as tð2Þ.In [CW] we prove that tð2Þ is a differential invariant but not a homotopy invariant
of M4k�1 when p1ðM Þ is not torsion-free. We will use ideas given by Weinberger in
[W] which under the same conditions proves the homotopy invariance of the twisted
r-invariant raðM Þ ¼ ZaðM Þ � ZðM Þ, where a is any representation p1ðM Þ !UðnÞ.
We supply brief expository remarks about intersection homology and algebraic
Poincare complexes in Sections 1 and 2 before proving the theorem in Section 3.
1. Intersection Homology
Following Goresky and MacPherson, we say that a compact space X is a pseudo-
manifold of dimension n if there is a compact subspace S with dimðSÞ4n� 2 such that X� S is an n-dimensional oriented manifold which is dense in X.
We assume that our pseudomanifolds come equipped with a fixed stratification by
closed subspaces
X ¼ Xn � Xn�1 ¼ Xn�2 ¼ S � Xn�3 � � � � � X1 � X0
satisfying various neighborhood conditions (see [GM]). A perversity is a sequence of
integers �pp ¼ ð p2; p3; . . . ; pnÞ such that p2 ¼ 0 and pkþ1 ¼ pk or pk þ 1. A subspace
Y X is said to be ð �pp; iÞ-allowable if dimðY Þ4 i and dimðY \ Xn�kÞ4 i� kþ pk.
We denote by IC�ppi the subgroup of i-chains x 2 CiðX Þ for which jxj is ð �pp; iÞ-allowable
and j@xj is ð �pp; i� 1Þ-allowable. If X is a pseudomanifold of dimension n1 then we
define the i-th intersection homology group IH�ppi ðX Þ to be the i-th homology group
of the chain complex IC �pp� ðX Þ. For any perversity �pp, we have IH
�pp0ðX Þ ffi HnðX Þ
and IH �ppnðX Þ ffi HnðX Þ.
Whenever �ppþ �qq4 �rr, the intersection homology groups can be equipped with a
unique product \ : IH�ppi ðX Þ�IH
qj ðX Þ !IH �rr
iþj�nðX Þ that respects the intersection
homology classes of dimensionally transverse pairs ðC;DÞ of cycles. Let �mm ¼ ð0; 0; 1;
1; 2; 2; . . . ; 2k� 2; 2k� 2; 2k� 1Þ be the middle perversity. If X is stratified with only
strata of even codimension, and if dimðX Þ ¼ 4k, then the intersection pairing
\ : IH �mm2kðX Þ � IH �mm
2kðX Þ ! Z
is a symmetric and nonsingular form when tensored with Q. We can define the
signature sigðX Þ of X to be the signature of this quadratic form. If X is a manifold,
this definition coincides with the usual notion of signature.
?Novikov additivity, proved cohomologically in [AS], posits that signature is additive in the followingsense: if Y is an oriented manifold of dimension of 2n with boundary X and Y 0 is another such manifold
with boundary �X, then sigðY [X Y 0Þ ¼ sigðY Þ þ sigðY 0Þ. The additivity for sigGð2Þ is easy to argue on the
level of ‘2ðGÞ-modules V endowed with nonsingular bilinear form. To see that this additivity corresponds
to appropriate manifold glueings, we refer the reader to [F1] and [F2] of Farber, who puts L2 cohomology
groups into a suitable framework in which the same arguments can be repeated.
ON CONJECTURES OF MATHAI AND BOREL 163
If X is a pseudomanifold, we say that X is a Witt space if IH �mmk ðL;QÞ ¼ 0 whenever
L2k is the link of an odd-codimensional stratum of X. If Xq is a Witt space, there is a
nondegenerate rational pairing
IH �mmi ðX;QÞ � IH �mm
j ðX;QÞ ! Q
whenever iþ j¼ q. If q¼ 4k> 0, then IH �mm2kðX;QÞ is a symmetric inner product space.
In this case, we define the Witt class wðX Þ of X to be the equivalence class of
IH �mm2kðX;QÞ in the Witt ring WðQÞ of classes of symmetric rational inner product
spaces. If q ¼ 0, set wðX Þ to be rank ðH0ðX;QÞÞ.h1i, and set it to zero if q 6� 0
mod 4. If ðX; @X Þ is a Witt space with boundary, set wðX Þ ¼ wðXX Þ, where
XX ¼ X [ coneð@X Þ. Let sigQ: WðQÞ!Z be the signature homomorphism developed
by Milnor and Husemoller [MH]. We define the signature sigðX Þ to the integer
sigQðwðX ÞÞ. See [S].
Denote by OWitt� the bordism theory based on Witt spaces; i.e. if Y is a Witt space,
define OWittn ðY Þ to be the classes ½X; f �, where X is an n-dimensional Witt space and
f : X!Y a continuous map, such that ½X1; f1� � ½X2; f2� iff there is an ðnþ 1Þ-dimen-
sional Witt space W with @W¼X1
‘X2 and a map W!Y that restricts to f1 and f2
on the boundary. Witt bordism enjoys the important properties that (1) there is a
signature invariant defined on the cycle level which is a cobordism invariant, and
(2) the signature can be extended to relative cycles ðX; @X Þ so that it is additive.
LEMMA 1. Let M1 and M2 be homotopy equivalent manifolds of the same dimension.
Suppose that their fundamental group G is torsion-free and satisfies the rational Borel
conjecture. Then there is rationally a Witt cobordism between M and M0 over BG.Proof. It is well known that, if G is torsion-free and satisfies the Borel conjecture,
then it satisfies the Novikov conjecture, which asserts that, if f : M!BG is a map,
then the generalized Pontrjagin number (or ‘higher signature’)
f�ðLðM Þ \ ½M�Þ 2 H�ðBG;QÞ
is an oriented homotopy invariant. Here LðM Þ is the Hirzebruch L-polynomial in
terms of the Pontrjagin classes, and ½M� is the fundamental class of M. Let
fi : Mi !BG be the map classifying the universal cover of Mi. We would like to show
that ½M1; f1� and ½M2; f2� rationally define the same class in OWitt� ðBGÞ.
For a Witt space X, Siegel [S] constructs an L-class LðX Þ 2 H�ðX;QÞ that gene-
ralizes the L-class of Goresky and MacPherson [GM], the latter of which is defined
only on Whitney stratified pseudomanifolds with even-codimensional strata. If
f : X!BG is the universal cover of X, then f�ðLðX ÞÞ 2 H�ðBG;QÞ coincides with
the higher signature given above. In addition, it is a bordism invariant in
OWitt� ðBGÞ. Rationally the Atiyah–Hirzebruch spectral sequence for Witt cobordism
collapses, so we obtain an isomorphism
i: OWittn ðBGÞ � Q �!
Mk50
Hn�4kðBG;QÞ
164 STANLEY CHANG
such that ið½M; f �Þ is the higher signature f�ðLðM Þ \ ½M�Þ. But these signatures are
homotopy invariants by assumption. &
2. Algebraic Poincare Complexes
An n-dimensional Poincare complex over a ring A with involution is an A-module
chain complex C with a collection of A-module morphisms fs : Cn�rþs !Cr such
that the chain map f0 is a chain equivalence inducing abstract Poincare duality
isomorphisms f0 : Hn�rðCÞ!HrðCÞ. As defined by Wall [Wa], an n-dimensional
geometric Poincare complex X with fundamental group G is a finitely dominated
CW-complex together with a fundamental class ½X � 2 Hnð eXX;ZÞ such that cap
product with ½X � gives a family of ZG-module isomorphisms
\½X�: HrðeXXÞ �!Hn�rðeXXÞfor 04 r4 n (there is additional business about an orientation group morphism
w: p1ðX Þ ! Z2 for which one should consult [Ran2]). An n-dimensional geometric
Poincare complex X with fundamental group G naturally determines an n-dimen-
sional symmetric Poincare complex Cð eXX;fXÞ over ZG. Such symmetric complexes
(as opposed to quadratic complexes, which define lower L-theory) over a ring A
can be assembled into an Abelian group LnðAÞ under a cobordism relation defined
by abstract Poincare-Lefschetz [M,Ran1], with addition given by
ðC;fÞ þ ðC 0;f0Þ ¼ ðC� C 0;f� f0
Þ 2 LnðAÞ
For a more detailed discussion on algebraic Poincare complexes and in particular the
manner in which algebraic Poincare complexes are glued together along a common
boundary, see [Ran3].
LEMMA 2. Let M and M0 be homotopy equivalent manifolds of the same dimension
4k� 1 with torsion-free fundamental group G. Suppose that Y is a rational Witt
cobordism of M and M0 over BG. If the Borel conjecture holds for G, then
sigGð2ÞðYGÞ ¼ sigðY Þ, where YG is the induced G-cover of Y.Proof. By the preceding lemma, there is a Witt cobordism between n copies of M
and n copies of M 0. Let Ch be the homotopy cylinder given by the homotopy
equivalence h : M!M 0. Attach n copies of Ch to the rational Witt cobordism Y to
form a space X. This space is usually not a pseudomanifold because the singular
space is of codimension one, but it is an algebraic Poincare space. Notice that all of
these spaces come equipped with a map to BG. Since upper and lower L-theory
coincide rationally, we may then consider X as an element of L�ðBGÞ � Q. Consider
then the composition of maps
OSO� ðBGÞ � Q�!H�ðBG;QÞ �!L�ðGÞ � Q:
ON CONJECTURES OF MATHAI AND BOREL 165
The first map is well known to be surjective, and the second is surjective by our
assumption that the Borel conjecture holds for G. Therefore there is a closed mani-
fold X 0 without boundary bordant to X as algebraic Poincare complexes. Atiyah’s
G-index theorem [A] shows that sigGð2ÞðX
0GÞ ¼ sigðX 0Þ, so that the cobordism invariance
of signature implies that sigGð2ÞðXGÞ ¼ sigðX Þ. For a topological proof of the G-index
theorem for signature, see the appendix of [CW].
By Novikov additivity, we may decompose the complex X to arrive at the equation
sigGð2ÞðYGÞ þ n � sigG
ð2ÞðChGÞ ¼ sigðY Þ þ n � sigðChÞ:
By Novikov additivity, the homotopy cylinder Ch enjoys the convenient property
that sigGð2ÞðC
hGÞ ¼ sigðChÞ ¼ 0, so sigG
ð2ÞðYGÞ ¼ sigðY Þ. &
3. The Main Theorem
THEOREM. Let M be an oriented compact manifold of dimension 4k� 1. If p1ðM Þ is
torsion-free, then the real-valued quantity tð2ÞðM Þ is a homotopy invariant.
Proof. Let M and M0 be homotopy equivalent closed manifolds of dimension
n ¼ 4k� 1. There are closed manifolds W and W 0 such that @W ¼ rM and
@W 0 ¼ sM0 with injecting fundamental groups. Let G ¼ p1ðW Þ and G0 ¼ p1ðW0Þ:
Construct s copies of W and r copies of W 0 and attach ts Witt cobordisms Y between
the boundary components M and M0. Call this space X with H ¼ p1ðX Þ. Note that
both G and G 0 inject into H. Since the map OSOðBHÞ � Q ! OWitt
ðBHÞ � Q is
surjective (see [Cu]), it follows that X is Witt cobordant to a smooth manifold X 0.
Hence, they share the same signatures. Note that X 0 has the property that
sigHð2ÞðX
0HÞ ¼ sigðX 0Þ and, hence, by Novikov additivity, we can partition X along the
original attachments to obtain
s � sigGð2ÞðWGÞþ rs � sigrð2ÞðYrÞ� r � sigG 0
ð2ÞðW0G0 Þ ¼ s � sigðW Þþ rs � sigðY Þ� r � sigðW 0Þ:
By Lemma 2, the cylindrical terms cancel, leaving
1
rðsigG
ð2ÞðWGÞ � sigðW ÞÞ ¼1
sðsigG
0
ð2ÞðW0G0 Þ � sigðW 0ÞÞ:
Equivalently tð2ÞðM Þ ¼ tð2ÞðM0Þ, so tð2Þ is a homotopy equivalence. &
Remark. Given the identification of tð2Þ and rð2Þ, one can conjecture that rð2Þ is
homotopy invariant iff the fundamental group of M is torsion-free (see [CW]).
However, recent work of [NW] makes use of secondary invariants for groups with
unsolvable word problems, which, a fortiori, are not residually finite. Potentially,
the results of this paper can have applications to the geometry of certain moduli
spaces.
166 STANLEY CHANG
Acknowledgement
I would like to thank Shmuel Weinberger for some useful conversations.
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ON CONJECTURES OF MATHAI AND BOREL 167