higher franz-reidemeister torsion
TRANSCRIPT
Higher Franz-Reidemeister Torsion
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AMS/IP
Studies in Advanced Mathematics Volume 31
Higher Franz-Reidemeister Torsion
Kiyoshi Igusa
American Mathematical Society • International Press
https://doi.org/10.1090/amsip/031
Shing-Tung Yau , Genera l Edito r
2000 Mathematics Subject Classification. Primar y 19D10 ; Secondar y 55R40 , 57R45 , 19F27.
Library o f Congres s Cataloging-in-Publicatio n D a t a
Igusa, Kiyoshi , 1949 -Higher Franz-Reidemeiste r torsio n / Kiyosh i Igusa .
p. cm . — (AMS/I P studie s i n advance d mathematics , ISS N 1089-328 8 ; v. 31 ) Includes bibliographica l reference s an d index . ISBN 0-8218-3170- 4 (alk . paper ) 1. Reidemeiste r torsion . I . Title . II . Series .
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To the thre e me n wh o taugh t m e how to d o mathematics : Jun-Ichi Igusa , Alle n Hatche r an d Friedhel m Waldhausen .
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Contents List o f Figure s x i
Introduction xii i
1 Cocycle s i n Volodi n K-Theor y 1 1.1 Volodi n K-theor y 2
1.1.1 Th e smoot h Volodi n spac e o f C 5 1.2 Firs t formul a fo r th e cocycl e 7 1.3 Th e correctio n ter m 9
1.3.1 Cycli c homolog y 1 0 1.4 D2k gives the Bore l regulato r ma p 1 3 1.5 Th e involutio n i n V n(C) 1 4 1.6 Additivit y o f r ^ 1 6 1.7 Som e computations 1 7
2 Space s o f Matrice s an d Highe r FR-Torsio n 2 3 2.1 Th e spac e o f invertible matrice s 2 4
2.1.1 Th e spac e W.(# ,G,n ) 2 7 2.1.2 Th e Volodi n categor y V.(R,n) 2 8 2.1.3 Th e involutio n o n W.(R,G,n) 2 9 2.1.4 Functoria l propertie s o f W #(jR,G,n) 3 2
2.2 W. con*(Cm,n) axidWi iff{Cm,n) 3 3 2.3 Th e isovarian t subcomple x o f W #
con t(Cm ,G,n) 3 9 2.4 Highe r FR-torsio n fo r w2 iff(Cm,U{m),n) 4 2
2.4.1 Cycli c homolog y 4 4 2.4.2 Propertie s o f Tjt(m,n) 4 9
2.5 Stabilizatio n o f W.(R,G,n) 5 1 2.5.1 Propertie s o f Tfc(m) 5 5
2.6 Highe r FR-torsio n o f oriented S x-bundles 5 6 2.6.1 Mors e theory 5 7
2.7 Highe r FR-torsio n i n the isovarian t cas e 6 1 2.8 Transfe r an d polylogarithm s 6 3 2.9 Frame d function s o n S 1 6 9 2.10 Combinatoria l framing s 7 4
3 A Mode l fo r th e Whitehea d Spac e 8 3 3.1 Multipl e mappin g cylinder s 8 4 3.2 Morphism s o f multipl e mappin g cylinder s 9 3 3.3 Mappin g cone s 10 2 3.4 Morphism s o f bundles an d th e twiste d homolog y bundl e 10 7
vn
viii CONTENTS
3.5 Uniquenes s o f the fiberwise con e 11 9 3.6 A model fo r th e stabilize d Whitehea d spac e 12 6
4 Mors e Theor y an d Filtere d Chai n Complexe s 13 5 4.1 Compariso n with filtere d chai n complexe s 13 5 4.2 Mors e theory an d filtere d chai n complexe s 14 0 4.3 Th e G-monomia l functo r o f a family o f
Morse functions 14 7 4.4 G-expansio n functo r o f a family o f GMF's 15 4 4.5 Independenc e o f birth-death point s 15 9 4.6 Th e frame d functio n theore m 16 9
5 Homotop y Typ e o f th e Whitehea d Spac e 17 3 5.1 Filtere d endomorphism s 17 4 5.2 Th e G-monomia l categor y 17 7 5.3 Expansion s i n filtered chai n complexe s 18 1 5.4 Th e involutio n on Whitehead spac e 18 4 5.5 Poincar e duality an d conjugat e transpos e 18 7 5.6 Waldhause n if-theor y 19 9 5.7 Highe r FR-torsio n an d th e 2-inde x theorem 20 9
5.7.1 Th e suspensio n theore m 20 9 5.7.2 Propertie s o f higher FR-torsio n 21 2 5.7.3 Highe r FR-torsio n o f oriented spher e bundles 21 8
5.8 Th e involutio n agai n 22 4 5.9 Proo f o f the 2-inde x theorem 22 9
6 Th e Framin g Principl e an d Bokstedt' s Theore m 23 7 6.1 Th e framin g principl e 24 0 6.2 Pseudoisotopie s 24 4 6.3 Cerf' s approac h 24 6 6.4 Hatcher' s constructio n 24 7 6.5 Th e involutio n on C(M) 25 0 6.6 FR-torsio n fo r fibe r product s 25 4
7 Proo f o f Complexifie d Boksted t Theore m 25 7 7.1 System s of local sections fo r 5 1-bundles 25 8 7.2 Smoothl y varyin g incidence matrice s 25 9 7.3 Anothe r versio n of the integra l invarian t 26 3 7.4 Explici t loca l sections fo r th e canonica l 5 1-bundle ove r (S 2)k . . 26 4 7.5 Reductio n t o isolate d region s 26 8 7.6 A n explici t variabl e incidence matrix 27 3 7.7 Polylogarithm s a s functions o n S 1 . . ' 27 6 7.8 Simplificatio n o f the integra l 28 2 7.9 Partition s an d necklace s 28 8
7.9.1 Descendin g partition s 28 8 7.9.2 Necklace s 29 0
CONTENTS i x
7.9.3 Countin g necklace s 29 3 7.10 Statemen t o f the Theore m 29 5
8 Frame d Graph s 29 7 8.1 Graph s an d metri c graph s 29 8 8.2 Frame d graph s 30 8 8.3 Compariso n wit h frame d function s 31 4 8.4 Th e frame d grap h theore m 31 8
8.4.1 Trees , magma an d homotop y framing s 31 8 8.4.2 Homotop y frame d tree s admi t uniqu e framing s 32 2 8.4.3 Existence , uniquenes s o f homotopy framing s o f trees . . . 32 5 8.4.4 Framin g familie s o f graphs 33 0
8.5 Application s o f the frame d grap h theore m 33 2 8.5.1 Framin g o f graphs an d th e homolog y o f Out(F n) 33 2 8.5.2 Highe r FR-torsio n o f the Torell i grou p 34 1
8.6 Fa t graph s giv e framed function s 34 5 8.6.1 Fa t graph s 34 6 8.6.2 Integratin g vecto r fields 34 9 8.6.3 Proo f o f Lemmas 8.6. 5 an d 8.6. 6 35 3
Bibliography 35 9
Index 36 5
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List o f Figure s 1.1 E 2k = dW i n X^(C) an d £ 2/c bound s a con e i n H n 1 4
2.1 Th e non-ascendin g se t o f f u 7 0 2.2 Pus h u p th e (- ) sheet s 7 1 2.3 Creat e cancellin g pair s unde r th e (- ) sheet s 7 1 2.4 Cance l th e (- ) sheet s wit h th e ne w (0 ) sheet s 7 2 2.5 W e ge t thi s shap e x ^ " 1 7 2 2.6 W e get thi s shap e xM f c _ m x D™ - 1 7 2 2.7 Eliminat e wha t i s left o f the (- ) se t 7 3 2.8 Th e crucia l ste p (f ) i n th e cas e m = 2 7 3
3.1 Drago n eatin g th e picture s fo r Chapte r II I 13 3
4.1 MUx^f-^c]) 14 2 4.2 Geometri c compariso n ma p 15 1 4.3 Simplice s nea r a birth-deat h lin e 15 7 4.4 Point s inciden t ove r x + an d x~ 15 8 4.5 Gradien t o f ± suspensio n o f / 16 1 4.6 Duplicatin g simplice s i n B 16 5 4.7 Eliminat e birth-deat h poin t o n a circl e 16 9
5.1 Dua l cell s in A 2 19 3 5.2 A 2 ha s 3 f c+1 - 2 k+1 = 1 9 dual chain s fo r eac h c e £(<J ) 19 7 5.3 Sw rallowrtail deformatio n , , , , , , , , 21 1 5.4 Resul t o f cancellation a t a n expansio n pai r 21 2 5.5 Slid e botto m o f swallowtai l t o lef t an d cance l 21 3 5.6 Preparin g th e expansio n pair s 23 1 5.7 Th e deformatio n H 23 1 5.8 f o n Y _ 23 2 5.9 Extendin g th e expansio n pai r t o Y 23 3 5.10 K(b) = St(SdB(b)) 23 4
6.1 Th e functio n 0(r ) 23 9 6.2 + / - suspensio n o f pseudoisotopies 24 5 6.3 cr +,cr_ ar e homotop y inverse s 24 6 6.4 Graphi c o f a 3-len s 24 8 6.5 L t i s a leve l surfac e o f f t 24 9
7.1 Pair s o f critica l point s x^,x~ o f a Mors e functio n o n a circl e . . 26 0 7.2 Imag e o f So o 26 5 7.3 Ste p 1 26 6
xi
X l l LIST OF FIGURES
7.4 Fou r loca l move s 26 6 7.5 Resul t o f Ste p 2 26 6 7.6 Ste p 3 at eac h ga p 26 7 7.7 Th e final ste p (4 ) 26 7 7.8 (U,Si) rectangl e i n Sf 27 1 7.9 Loca l sectio n domain s fo r k = 2 27 2
8.1 M(T ) = SIJ{{1,2,3},{4,5} } 31 9 8.2 Collapsin g tree s (circled ) wit h label s (dark ) 32 1 8.3 Cas e II : Collaps e S o to a poin t * 32 4 8.4 E i s the closur e o f al l simplices o f G" containin g r 32 8 8.5 77 2 on typ e 1 and typ e 2 subtrees 32 9 8.6 Cas e I I whe n e is a separatin g edg e 33 2 8.7 w\ = eie 2, w 2 = e 3ei, w 3 = e 2e3 34 6 8.8 G U (dS — D) i s the boundar y o f a surfac e S' no t containin g x . 35 2 8.9 Buildin g b l o c k s (dark ) withe-tab s T±(E) 35 3 8.10 g: [-4,4 ] - » [-§ , § ] i s an od d function , h(t) = 1 for t > 0 . . . . 35 4 8.11 Choos e a standar d deformatio n o f D{x~) U E(x+) int o th e e-ta b
T(x+) 35 6
Introduction The purpos e o f this boo k i s to develo p th e theor y o f higher Franz-Reidemeiste r torsion (FR-torsion ) fro m it s foundation s u p t o th e curren t results . W e giv e several description s o f th e highe r FR-torsion , sho w tha t the y ar e equivalent , use the m t o defin e th e highe r FR-torsio n i n a ver y genera l settin g an d als o show tha t i t ca n no w b e easil y compute d i n man y cases . Thi s invarian t i s no t the sam e a s highe r Whitehea d torsio n althoug h the y ar e closel y related . Also , higher FR-torsio n i s conceptually ver y differen t fro m th e highe r analyti c torsio n of Bismut an d Lot t althoug h th e numerica l result s appea r t o agree in many bu t not al l cases . On e o f the bi g challenge s i s to find som e theoretica l basi s fo r th e similarity i n the result s o f the tw o theories .
Classic Whitehea d torsion , Reidemeiste r torsio n and Ray-Singe r analyti c torsio n
There are three classic types of torsion: Whitehea d torsion , Reidemeiste r torsio n and Ray-Singe r analyti c torsion .
Whitehead torsion i s an invariant associate d t o a relative cel l complex (X , A) having a finit e numbe r o f cells :
X = A U ei U e2 U • • • U en
so tha t A i s a deformatio n retrac t o f X. B y takin g th e mappin g cylinde r (Z(f)1X) on e ca n als o associat e a Whitehea d torsio n invarian t t o a homotop y equivalence / : X — » Y betwee n finite cel l complexes . I f the torsio n o f / i s zer o then / i s called a simple homotopy equivalence.
By Mors e theor y on e ca n associat e t o an y compac t smoot h (i.e . C°°) man -ifold a finite cel l comple x whic h i s wel l define d u p t o simpl e homotop y equiv -alence. Consequently , a homotop y equivalenc e betwee n compac t smoot h man -ifolds ha s a wel l define d Whitehea d torsio n invariant . Similarly , a compac t smooth manifol d pai r (M , A) (wher e A i s a compac t smoot h submanifol d o f M) ha s a Whitehea d torsio n invarian t i f A i s a deformatio n retrac t o f M , fo r example i f M i s a smooth /i-cobordis m o f A.
The Whitehead torsio n invarian t lie s in the Whitehead grou p Wh\ (n) whic h is an abelia n quotien t o f the infinit e genera l linea r grou p o f Z[TT] wher e TT i s th e fundamental grou p o f the manifold . Typically , Whitehea d torsio n i s a complete invariant i n th e sens e tha t i t i s usuall y zer o i f an d onl y i f th e object s ar e topologically equivalent . Fo r example , th e s-cobordis m theore m say s that , i n the stabl e rang e dim M > 6 , a n h-cobordis m (M , 9QM) i s diffeomorphi c t o a
xm
XIV INTRODUCTION
product (doM x I,doM) i f an d onl y i f th e Whitehea d torsio n o f (M , <9oM) i s zero. Therefor e a computatio n o f the Whitehea d grou p give s a classificatio n o f /i-cobordisms u p t o diffeomorphism .
Reidemeister torsion i s a rea l numbe r associate d t o an y finite relativ e cel l complex (X, A) whic h is acyclic with respect t o a Hermitian locally trivial coeffi -cient shea f T. (Thi s means that T ha s stalks C n whic h ar e twisted b y a unitar y representation p : -K\X — > U(n) o f th e fundamenta l grou p o f X.) B y Mors e theory thi s als o applie s t o compac t smoot h manifol d pair s (M , A). Typically , we take A t o be empty. Fo r example, lens spaces have Reidemeister torsio n wit h respect t o severa l differen t unitar y representation s o f thei r fundamenta l grou p TTlL = Z/p.
Although Reidemeiste r torsio n i s define d i n mor e genera l situation s tha n Whitehead torsio n (sinc e any homotopy equivalenc e is a homology equivalence) , it require s a unitar y representatio n o f th e fundamenta l group . Whe n the y ar e both define d th e Reidemeiste r torsio n i s a functio n o f th e Whitehea d torsion : for eac h unitar y representatio n p of 7r , the Reidemeiste r torsio n i s the rea l par t of th e lo g o f th e determinan t o f th e comple x matri x induce d b y p fro m an y matrix representin g th e Whitehea d torsion .
The acyclicit y conditio n i s not alway s necessar y t o defin e Reidemeiste r tor -sion. I t i s sufficient t o hav e a volume for m o n th e homolog y £f*(M , A; J 7).
In the 1930' s Franz and DeRham classified len s spaces up to homeomorphis m using Reidemeiste r torsion . (Se e [Mil66]. ) Fo r thi s reason i t i s sometimes calle d Franz-Reidemeister-DeRham torsion.
In 1971 , Ra y an d Singe r [RS71 ] define d analytic torsion (als o calle d Ray-Singer analytic torsion) an d conjecture d tha t i t wa s th e sam e a s FR-torsion . This wa s prove d b y Cheege r [Che79 ] an d Miille r [Mul78] . (Se e als o [BZ92]. )
Higher Whitehea d torsio n
The histor y o f higher Whitehea d torsio n begin s i n 196 9 when Jea n Cer f showe d that pseudoisotop y implie s isotop y fo r smoot h simpl y connecte d manifold s o f dimension > 5 [Cer70] . I n othe r words , th e pseudoisotop y spac e C(M) i s con -nected. Alle n Hatcher an d Joh n (Jack ) Wagone r realized tha t th e algebraic par t of Cerf's argumen t wa s essentially equivalen t t o Matsumoto' s theorem ([Mat69] , also [Mil71] , pp 81-92 ) whic h says that th e algebrai c if-grou p K^L ha s only two elements. I n [HW73 ] Hatcher an d Wagone r define d Wh^is) a s a quotient o f the algebraic K-grou p l ^ Z ^ ] (s o that, e.g. , W7i2(l ) = 0 ) a n d obtaine d a surjectiv e mapping
&:7r0C(M)-»W%(7Ti(M)).
In [Igu79] , I extended thi s t o defin e a natura l mappin g
£ i :7 r iC(M)^W%(7r i (M))
where Whz(n\(M)) i s a quotient o f K^L[ni{M)\ an d i n [Igu84 ] I gave a formul a for th e cokerne l o f £i .
INTRODUCTION xv
The highe r homotop y group s o f C(M) ma p t o highe r Whitehea d torsio n groups o f 7Ti(M) . However , t o obtai n a complet e invarian t w e nee d a fanc y version o f these highe r Whitehea d torsio n group s which incorporate s th e highe r homotopy group s o f M a s wel l as 7Ti(M) . Friedhel m Waldhause n accomplishe d this aroun d 197 6 ([Wal85] , [Wal82] , [Wal87b] ) whe n h e constructed a if-theor y space of M whic h he called A(M). Rationally , thi s is the if-theory o f the intege r group rin g o f th e entir e loo p grou p o f M whic h incorporate s al l o f th e highe r homotopy group s o f M. (Integrally , i t i s the algebrai c If-theor y o f the ^^- r in g space ft 00S00(ftM+) i n th e languag e o f Pete r Ma y [May78]. ) Waldhause n als o constructed th e smooth Whitehea d spac e of M whic h he called Wh dl^(M) an d proved th e following .
• nWh diff{M) ~ BV{M) wher e V(M) = lim _ C{M x I n). I n particular , there i s a natura l mappin g BC(M) - > SlWh diff(M).
• A(M) ~ Wh diV(M) x ^ 0 0 5 0 0 ( M + ) .
Composing thes e w e get a mappin g
BC(M) - > ftA(M). (0.1 )
By naturalit y o f Waldhausen if-theor y ther e i s also a mappin g
A(M) - + Z x £GL(Z[7nM]) +. (0.2 )
(Recall tha t Ki{R) — TTi(BGL(R) +) fo r i > 1 and an y rin g R.) I t wa s shown i n [Igu88] that th e mappin g C(M) — > V(M) i s a homotopy equivalence s i n a stabl e range. Th e ma p (0.2 ) i s a rationa l homotop y equivalenc e i n the rang e i n whic h the highe r homotop y group s o f M ar e zero . Consequently , fo r example , i n th e special cas e whe n M i s a disk , Waldhausen' s theore m say s that , i n th e stabl e range:
^k-i{Diff{Dn)/0{n)) ® Q S* (K4fc+iZ) 0 Q .
By the wor k o f Armand Bore l ([Bor74b] , [Bor74a] ) w e know th e ran k o f th e higher X-theor y o f any number field, o r ring of integers of any number field, sa y F. Fo r th e case F = Z , I f^+ i Z ha s ran k 1 for al l k > 1 . Bore l detecte d al l of the rationa l highe r if-theor y o f F b y composin g th e map s K2k+\F — > if2fc+iC induced b y embedding s p : F — » C wit h a fixed mappin g i f ^+ i C — • R calle d the Borel regulator map.
It wil l tur n ou t tha t th e highe r FR-torsion , i n th e cas e o f pseudoisotopies , is equivalent t o th e compositio n o f the tw o know n mappings :
1. Waldhausen' s map :
7T,_!C(M) - 7T i+iA(M) - > K l+1Z{KXM]
2. Th e Bore l regulato r map :
/ W Z f a M ] - 7r 2fc+1BGL(Z[7r1M])+ Hurewic *>
H2fe+1(GL(Z[7nAf])) - ^ H 2k+1{GL{£)) - ^ R
where Ck G i/cfc+1(GL(C);R) i s a fixed continuou s cohomolog y class .
XVI INTRODUCTION
By th e wor k o f Alexande r Beilinso n [BeT85 ] (se e als o [Esn89] , [JLDZOO] , [Ram88], [Rap88]) , i t wa s als o alread y know n tha t th e valu e o f th e Bore l reg -ulator o n a generato r o f i^2/e+iQ(C ) (generato r b y th e Galoi s action ) fo r ( a n n-th roo t o f unit y i s a rationa l multipl e o f the rea l par t o f
7 1 = 1
Thus, th e highe r FR-torsion fo r pseudoisotopies wa s already "known. " How -ever, pseudoisotopie s ar e a ver y specia l case , wherea s th e highe r FR-torsio n i s defined i n a very genera l settin g (fo r smoot h compac t manifol d bundle s ove r al l simply connecte d space s and mor e generally when the fundamenta l grou p of the base act s triviall y o n th e homolog y o f th e fiber). Also , despit e th e wealt h o f knowledge w e have abou t pseudoisotopies , Waldhause n i^-theory , algebrai c K-theory an d th e homotop y typ e o f diffeomorphism spaces , the highe r FR-torsio n for eve n th e simples t cas e o f linea r circl e bundle s wa s no t know n unti l ver y re -cently ([IK93 ] an d Chapte r VI I below) . I n thi s boo k w e will, for th e mos t part , avoid the pseudoisotopy cas e and concentrat e on the other cases . On e exceptio n will be Hatcher' s constructio n i n th e pseudoisotop y o f disks (§6.4) .
Higher analyti c torsio n
In 1995 , Jean-Michel Bismu t an d Joh n Lot t constructe d highe r analyti c torsio n forms
Tk(E;T)etfk(B)
for smoot h bundle s p : E — » B wit h Hermitia n locall y trivia l coefficien t system s T an d certain other additiona l structures. I n certain cases these forms are closed giving cohomolog y classe s
r f L (E ;^ ) eH 2k{B-R),
for example , whe n the fiber M i s odd dimensiona l an d acycli c (wit h coefficient s in JF) . Thi s cohomolog y clas s i s topological (define d fo r bundle s wit h structur e group Diff(M) an d independen t o f th e horizonta l distributio n an d vertica l metric) i n man y case s an d ha s bee n explicitl y compute d i n th e case o f circl e bundles b y Bismu t an d Lot t [BL95] , odd dimensiona l spher e bundle s b y Bunk e [Bun] an d i n othe r case s b y Bismu t an d Goett e [BGOO] .
Higher FR-torsio n
As note d earlier , highe r FR-torsio n invariant s i n th e cas e o f pseudoisotopie s can b e obtaine d a s a functio n o f highe r Whitehea d torsio n invariant s usin g the Waldhause n i^-theor y spac e A(X) an d it s relationshi p t o th e algebrai c K-theory o f discret e rings . I t wa s Joh n Wagone r [Wag78 ] wh o first propose d tha t higher FR-torsio n migh t b e define d mor e generally . Joh n Klei n [Kle89 ] was th e
INTRODUCTION xvii
first t o accomplis h this . Klei n use d a variatio n A P(X) o f Waldhausen' s A(X) which replaced homotop y equivalence s with p-homolog y equivalence s where p is a comple x representatio n o f TT\X. Late r Klei n an d I developed tw o othe r meth -ods to describe the highe r FR-torsion . (W e take the Platoni c poin t o f view tha t the highe r FR-torsio n alread y exist s an d w e ar e lookin g fo r way s t o describ e and comput e it. ) Th e first wa s th e spac e o f acycli c filtered chai n complexe s FCh{X,£) whic h w e use d t o defin e highe r FR-torsio n i n a genera l settin g i n ([Igu91], [IK95] ) an d th e secon d wa s the computatio n i n th e specia l cas e o f cir -cle bundles ove r S 2 usin g "pictures " an d th e Bloch-Wigne r dilogarith m [IK93] . In th e meantim e Dwyer , Weis s an d William s obtaine d a differen t topologica l construction fo r highe r FR-torsio n [DWW] . Thes e differen t topologica l con -structions ar e probabl y equivalent .
In thi s boo k w e develo p wha t i s essentiall y a n algebrai c refinemen t o f th e filtered chai n comple x approac h t o highe r FR-torsion .
Contents o f thi s boo k
In this book we show how higher Franz-Reidemeiste r torsio n ca n be defined an d how i t ca n b e computed . Highe r FR-torsio n i s defined fo r al l smoot h manifol d bundles p : E — > B wit h compac t manifol d fiber M s o that TTI (B) act s triviall y on the homology groups of the fiber H* (M, J7) where J7 is a Hermitian coefficien t system fo r E. The y ar e cohomolog y classes :
Tk(E;F)€H2k{B;R).
The construction o f these higher FR-torsio n classe s is analogous to the pseu -doisotopy cas e explaine d above . First , w e us e Mors e theor y i n th e for m o f the frame d functio n theore m ([Igu87] , explaine d i n §4. 6 below ) t o g o fro m th e smooth bundl e E — » B t o a mapping fro m B int o a Whitehead space . The n w e compose wit h th e Bore i regulato r ma p o n Voiodi n i^-theory .
Mapping t o th e Whitehea d spac e
The frame d functio n theore m tell s u s tha t ther e exist s a smoot h ma p
f :E-+R
which i s a "framed " generalize d Mors e functio n o n eac h fiber provide d tha t dimM > dim B. (Whe n thi s i s not satisfie d w e stabilize E b y replacin g i t wit h a produc t E x D n.) Takin g th e resultin g famil y o f cellula r chai n complexe s {C(Ma)} fo r al l generi c smal l simplice s a i n B w e obtai n a functo r £ # fro m the categor y o f generic smal l smoot h simplice s i n the bas e B t o th e (simplicial ) Whitehead categor y Wh 9(Z[7TiE},7TiE) o f base d fre e chai n complexe s ove r th e ring Z[TTIE]:
£E • simp B - • Wh.(1[ii lE}1iT1E). (0.3 )
XV111 INTRODUCTION
Furthermore, fo r ever y unitar y representatio n p : ix\E — > U(n) w e obtai n a simplicial functo r
p* : Wh.(2,[7r1E\,ir1E) - Wh.{M n{C),U (">))•
In th e specia l cas e whe n th e fiber M i s acycli c wit h respec t t o th e repre -sentation p th e compositio n p * o £E lift s t o th e simplicia l ful l subcategor y W/^(Mn(C), U(n)) o f acycli c base d fre e chai n complexe s ove r th e rin g M n(C).
By a theore m o f Joh n Klei n an d mysel f ([IK95], also Theore m 0.0. 2 below ) we know tha t ther e ar e universa l classe s fo r highe r FR-torsion :
rk G H2k{Whh.(Mn(C),U(n));R) (0.4 )
which w e ca n no w pul l bac k t o giv e highe r FR-torsio n classe s fo r E (sinc e B ~ |simpB|) :
Tk(E,p) = e EP*(Tk)eH2k(B;R).
This wa s th e procedur e whic h wa s outline d i n [Igu91] . I t wa s als o announce d that highe r FR-torsio n coul d b e define d i n th e cas e whe n th e homolog y o f th e fiber inject s int o the homolog y o f the tota l space . I n thi s boo k thi s i s strength -ened t o th e following . (Se e Corollar y 3.6.8. )
Theorem 0.0.1 . Let p : E — > B be a smooth bundle with compact manifold fiber M and suppose that G is a subgroup of the group of units of a ring R and p : TTIE — » G is a homomorphism with the property that the fiberwise homology H*(M]R) of M with twisted R coefficients is a projective R-module with trivial action of TT\B. Then the natural mapping
B^Wh.(R,G)
lifts canonically to Wh 9(R,G). In particular when (it , G) — (Mn(C),[/(n)) we obtain well-defined higher FR-torsion invariants for E.
Since Mn(C) i s semisimple, al l modules are projective an d the only conditio n for th e existenc e o f higher FR-torsio n i s the trivia l actio n o f 7Ti B a s we claime d earlier.
In th e specia l cas e whe n th e coefficien t rin g R i s a field w e hav e a fur -ther extensio n o f thi s resul t t o th e cas e whe n ix\B act s o n H*(M; R) b y uppe r triangular transformation s wit h respec t t o som e basis . (Se e Theore m 3.5.11 , Corollary 3.6.12. )
Computation o f highe r FR-torsio n
In orde r t o explicitl y comput e thi s highe r FR-torsio n clas s w e nee d a n explici t formula fo r th e universa l clas s r k. I n [IK93 ] Joh n Klei n an d I use d th e Block -Wigner dilogarithm to find a formula fo r T\ and compute it in the case of oriented linear circl e bundle s ove r s 2.
INTRODUCTION xix
In the present wor k I use a rational multipl e of the Kamber-Tondeur-Dupon t formula [Dup76 ] fo r th e continuou s cohomolog y clas s
CfcG#f+ 1(£G£(C);M)
which give s th e Bore l regulato r ma p i^fc+i C — > R . I us e thi s t o comput e th e higher FR-torsio n invariant s fo r oriente d linea r circl e bundles , oriente d linea r sphere bundles, lens-space bundles , nonlinear dis k bundles (Hatcher' s example) , the mappin g clas s grou p an d th e Torell i group .
The first ste p i s to conver t thi s (2k + l)-cocycl e fo r GL(C) int o a 2/c-cocycl e for th e Volodi n spac e V(C ) ~ OJ3GL(C) +. Thi s i s carrie d ou t i n Chapte r I . The Volodi n spac e o f a rin g R i s a spac e o f invertibl e matrice s ove r R whic h locally var y b y elementar y colum n operations . I n Chapte r I I w e embe d thi s space int o th e large r spac e o f invertibl e matrice s ove r R whic h locall y var y b y elementary ro w an d colum n operation s an d b y G-monomia l basi s change . W e call thi s W 9(R,G) an d w e extend th e Volodi n if-theor y cohomolog y clas s t o a class
rk e H 2k(W.(Mn(C),U(n));R). (0.5 )
This allow s u s t o defin e an d comput e th e highe r FR-torsio n fo r al l oriente d circle bundles . Th e computatio n i s reduce d t o on e specia l cas e usin g transfe r arguments i n Chapter I I and thi s special case is worked ou t i n detail in Chapte r VII.
When th e cellula r chai n comple x o f the fiber i s bot h acycli c an d i n tw o in -dices, i t i s give n b y a singl e invertibl e matri x an d therefor e w e ge t a mappin g from B int o som e W*(R,G). I n orde r t o defin e th e highe r FR-torsio n classe s more generally we need a chain complex version o f W9(R, G). I n Chapte r II I we construct th e simplicia l categor y Wh 9(R,G) o f base d fre e i?-complexe s whic h locally var y b y G-monomia l an d elementar y basi s chang e i n each degree . Sinc e invertible matrice s ar e the same as acyclic based fre e chai n complexe s which ar e nonzero i n onl y tw o consecutiv e degrees , W.(R,G) i s i n som e sens e a ful l sub -category o f Wh 9(R, G) wher e ( ) h mean s w e take onl y acycli c chai n complexes . The two-inde x theore m (prove d i n §5.9 ) say s tha t
W.(R,G) ~Whi(R,G). (0.6 )
In particula r the y hav e th e sam e cohomolog y fo r an y (R, G) s o the cohomolog y classes r ^ i n (0.5 ) uniquel y determine d th e universa l classe s i n (0.4) .
Chapter II I als o contain s th e basi c resul t b y whic h th e highe r FR-torsio n can b e extende d t o th e non-acycli c cas e whe n on e o f th e followin g condition s holds (Theore m 3.6.7) .
1. Th e homolog y H*(M\R) o f th e fiber M i s a projectiv e R- module wit h trivial 7T i inaction o r
2. R i s a field an d H*{M\ R) ha s a filtration b y i2[7riB]-submodule s s o tha t TTIB act s triviall y o n th e associate d grade d module .
X X INTRODUCTION
Chapters I V an d V explain s th e relationshi p betwee n Wh m(R,G) an d th e space o f filtered chai n complexes . Th e resul t is :
\Whh.(R,G)\ ~ \FC h{BG,R)\ (0.7 )
and th e sam e wit h "/z " deleted . W e nee d thi s resul t fo r th e followin g reason . When we use the frame d functio n theore m w e get a canonical famil y o f oriente d fiberwise generalize d Mors e function s (GMF's ) o n th e fibers. Thi s give s us , i n a canonica l way , a mappin g
simpB - > FC{Bn 1E,rL[KlE])
by sendin g eac h simple x t o th e tota l singula r comple x o f it s invers e imag e i n E together wit h a filtratio n give n b y th e fiberwis e GMF . Th e homotop y equiva -lence (0.7 ) tell s u s tha t w e ge t a mappin g £ # a s i n (0.3 ) whic h i s wel l define d up t o homotopy .
In Chapte r V , w e prove th e two-inde x theore m (0.6 ) an d explai n th e mai n theorem o f [IK95] :
Theorem 0.0.2 . There is a homotopy fiber sequence
\FCh(X.,R)\ - > fi 00S00(|X.|+) - • Z x BGL{R) +
where Z is the image of KQZ = Z in KQR.
This theore m i s importan t fo r severa l reasons . First , i t say s tha t th e spac e of filtered chai n complexe s (an d als o W.(R,G), Wh h(R,G) b y (0.6 ) an d (0.7) ) has th e "correct " homotop y typ e an d therefor e w e hav e th e correc t definitio n of highe r FR-torsion . Secondly , thi s implie s tha t th e involutio n o f reversin g the sig n o f the fiberwise GM F change s th e sig n o f the highe r FR-torsio n i n th e expected way . Thi s i s proved i n §5.8 .
In Sectio n 5. 7 w e use th e two-inde x theore m t o mak e calculation s o f highe r FR-torsion fo r linear and sphere bundles and their associated lens space bundles . These agre e with Ulric h Bunke' s computation s whe n th e dimensio n o f the fiber is odd . Fo r eve n dimensiona l fibers th e computation s d o no t agree . However , the discrepanc y seem s t o vanis h i f we replace th e highe r FR-torsio n define d b y a frame d functio n / wit h th e averag e o f th e highe r FR-torsion s define d b y / and —/ . (Thi s i s an applicatio n o f Theore m 0.0. 2 explaine d i n Sectio n 5.8. )
In Chapter V I we derive a general rule which we call the "framin g principle " by which we may determine th e highe r FR-torsio n invarian t o f a bundle E — • B from a n oriented fiberwise GM F o n E whic h is not framed . Sectio n 6.1 gives the precise statemen t o f the framin g principl e an d prove s i t a s a n eas y consequenc e of th e highe r FR-torsio n computatio n fo r linea r spher e bundles . Section s 6. 2 and 6. 3 contai n a revie w o f pseudoisotop y theor y an d result s an d method s o f Jean Cerf . I n §6. 4 w e giv e Hatcher' s constructio n fo r a rationall y nontrivia l element o f 7^ Ak-1(Diff{Dn)/0(n)). Th e well-know n argumen t ([FH78] ) fo r th e existence o f suc h exoti c element s i s explaine d i n §6.5 . Hatcher' s exampl e i n particular wa s show n t o b e equivalen t t o th e know n exoti c classe s b y Marce l
INTRODUCTION xxi
Bokstedt usin g homotop y theoreti c method s ([B6k84]) . W e giv e anothe r proo f of this by computing the higher FR-torsion invarian t fo r Hatcher's example using the framin g principle . Finally , i n §6.6 we give a product formul a fo r highe r FR -torsion an d prov e i t fo r fibe r product s o f oriente d linea r spher e bundle s an d i n two othe r cases .
In Chapte r VII I we look at th e modul i spac e of curves and the Torell i group . It i s a resul t o f Richar d Hain , Rober t Penne r an d myself , base d o n wor k o f John Klei n [Kle93] , that th e highe r FR-torsio n T2k of th e mappin g clas s grou p is a nonzer o multipl e o f th e Miller-Morita-Mumfor d classe s K,2k plu s a linea r combination o f product s o f lower terms . B y restrictio n th e sam e i s true fo r th e Torelli group . Usin g th e framin g principl e w e refin e thi s t o obtai n a n explici t computation (Theore m 8.5.19) :
T2k{M°g) = (_i)* C(2* + l ) ^ L .
Combinatorial construction s o n th e categor y o f graph s whic h ar e neede d to fil l i n th e detail s o f thi s argumen t ar e given . W e defin e a structur e calle d a "frame d graph " whic h i s th e additiona l structur e o n th e grap h neede d t o construct a frame d functio n o n a tubula r neighborhoo d o f the grap h whe n i t i s immersed i n R n. Sinc e th e categor y o f connecte d graph s wit h H\ = Z n i s th e classifying spac e of the oute r automorphis m grou p o f the fre e grou p o n n letter s by Culle r an d Vogtman n [CV86 ] , we obtain highe r FR-torsio n classe s
r2k(Outh(Fn)) € H ik{Outh{Fn);R)
which restric t t o th e highe r FR-torsio n classe s o n th e Torell i grou p (Corol -lary 8.5.17) .
The mapping BOut h(Fn) - • W/£(Z , 1 ) used to define th e higher FR-torsio n on Out h(Fn) fits int o th e followin g commutin g diagra m wher e th e row s ar e homotopy fiber sequences .
BOuth(Fn)
|W7£(Z,1)|
Since |W/i.(Z , 1)| ~ QS° i s rationall y trivial , thi s implie s tha t th e mappin g Out{Fn) — » GL(1i) i s trivial i n rationa l homology . (Se e Theore m 8.5.3. )
The logica l dependenc y o f th e chapter s i s give n a s follows . However , fro m Chapter VI I onl y Theore m 7.10. 2 i s needed. Th e proo f an d th e res t o f Chapte r VII ca n b e skipped .
Chap I Chap II — Chap VII
I Chap III — Chap IV — - Chap V — - Chap VI — Chap VIII
- BOut{F n) BGL n{Z)
I I * \Wh.{Z, 1) | Z x BGL{Z) +
XXII INTRODUCTION
I a m gratefu l fo r th e hel p an d encouragemen t I receive d fro m man y peopl e during the many years that I worked on this project: Me l Rothenberg, Donggen g Gong, Igo r Najfeld , Richar d Palais , Joh n Klein , To m Goodwillie , Rober t Pen -ner, Richar d Hai n an d Sebastia n Goette . I als o ow e a grea t deb t t o Jea n Cer f whose pioneerin g work s o n diffeomorphis m group s ([Cer68] , [Cer70] ) inspire d me to stud y pseudoisotop y theory . Also , both Joh n Klei n an d I were motivate d by Joh n Wagoner' s origina l articl e o n highe r torsio n [Wag78] . I shoul d than k the Nationa l Scienc e Foundatio n fo r thei r generou s support . I woul d als o lik e to than k m y wif e Gordan a Todoro v an d he r mothe r Rakil a Todoro v fo r thei r support an d understandin g whil e I concentrated al l m y effort s o n finishing thi s book. Als o Gordan a deserve s credi t fo r revisin g thi s introductio n s o tha t i t i s readable.
November 2001 , Kiyoshi Igus a
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[Wal87a] Friedhel m Waldhausen , Algebraic K-theory of spaces, concordance and stable homotopy theory, Algebrai c Topolog y an d Algebrai c K-theory (Willia m Browder, ed.) , Annals of Math. Studies , vol. 113, 1987, pp. 392-417 .
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Index OS(n), 30 7 Out(Fn), 30 7 AP = lattic e o f closed subset s o f P ,
88, 9 7 7Ti infixed submodule , 123 , 215
additivity, 3 0 for W. , 5 0 for t%, 16
additivity lemma , 55 , 64, 21 3 additivity theorem , 33 6 admissible
morphisms, 5 9 simplex, 5 9
admissible grap h epimorphism , 29 9 antisimplicial map , 18 5
6, boundary i n HC, 11 , 45, 49 6*, boundary i n H C(A, *) , 46 balanced product , 4 barycentric subdivision, 149 , 153, 162,
230, 234 , 306 , 326 relative subdivision , 32 7
basis poset , 11 4 B2k, fc-th Bernoulli number, 249 , 252 Beilinson, Alexander , xv i bifurcation set , 57 , 59 Bismut, Jean-Michel , xiii , xvi , 21 8 Bokstedt, Marcel , xxi, 237, 249, 250,
257, 34 3 Borel regulator map , xv, xvi , xix, 1 ,
7, 9 , 13 , 14, 23 , 24, 42, 48, 241
Borel, Armand , xv , 15 , 253, 342 Bott, Raoul , 25 0 Bunke, Ulrich , xvi , xx , 21 8
canonical cone , 13 0 Cfc, k-th. Catalan number , 28 9 cellular chai n complex , 14 4 Cerf, Jean , xxii , 237 , 244 , 246 , 252,
297 Chern character , 220 , 25 5
0/1(7), Chern character of 7, 219 c/i2fc> the degre e 2k ter m o f c/i ,
219 of a rea l bundle , 224 , 239 , 240
C2/C+1, 7 , 1 0
closed bijection (wit h order preserv -ing inverse), 27, 28, 34, 99, 107, 149, 178, 184, 201, 203 , 340
closed bijection, graded , 99 , 107, 126, 127
closed subse t o f P , 8 8 closure wit h respec t t o expansion ,
55 coindex, 16 0 collapsing morphism , 30 9 collapsing pair , 30 8 collapsing tree , 29 9 color weight , 4 5 comparison case , 140-142 , 151 , 153 comparison map , 144 , 147, 217, 237,
238, 240, 241, 249, 250, 343-345
reverse, 23 7 comparison torsio n class , 23 8 complete invariant , xii i concordance space , 24 4 cone, mappin g cone , 10 3
C(a), 10 8 C( / ) , 10 2 alternating mappin g cone , 132 ,
140, 142, 147, 151, 153, 159, 210,211,217,218,238,241, 253
cellular mappin g cone , 14 1 fiberwise cone , 11 2 fiberwise mappin g cone, 109, 115,
125, 12 8 induced morphism , 10 3 of a pose t morphism , 9 7 uniqueness of fiberwise mapping
cone, 128 , 130
365
366
configurations GF(J), linearl y ordere d config -
urations, 7 4 CF{S1
1 cyclicall y ordere d con -figurations, 7 5
EF(I), configuration s embed -ded i n / , 7 6
EF{Sl), configuration s embed -ded i n S\ 7 7
conjugate, 3 4 conjugate morphisms , 2 8 conjugate transpose , 7 , 15 , 29 conjugation, 15 , 29 conjugation lemma , 52 , 64 , 21 4 corank, 16 0 correction term , 20 , 21 , 43 Culler, Marc , xxi, 297, 307, 346, 347 cyclic homology , 4 4 cylinder functor , 202 , 20 3
degeneration, 32 1 Dennis, R . Keith , 22 6 descending partition , 28 8 diagonal automorphisms , 4 0 diagonal category , 53 , 59 D2k, 1 0 E>2k — F2k — G2ki 4 8 D. (R, P) , A-se t of acyclic chain com-
plexes wit h basi s P , 2 5
extension sequence , 10 0
f.dim (finit e dimensional ) algebra , 5 fat graphs , 297 , 298 , 345-35 8 fiber homotop y (o f morphisms), 10 9 fiberwise frame d function , 62 , 65 ,
170, 176, 219, 220, 222, 223, 237-239, 242, 254, 342, 345
filtered Hor n complex , 17 4 filtered morphism , 13 6 F2fc, a cochai n o n V* iff(C), 8 framed functions , xxi , 23, 62, 69-74,
79, 80 , 212 , 218 , 219 , 252 , 297, 298 , 345
F(I) i s contractible , 7 1 Fis1) i s contractible , 7 3
INDEX
F+ (S1), positivel y framed func -tions o n S 1, 7 3
comparison with framed graphs , 314, 31 8
definition, 16 9 framed functio n theorem , xvii ,
131, 135 , 169-171, 345, 358 space of , 7 6 spaces of , 29 8
framed graph , xxi , 297, 308-314, 318 metric frame d graph , 31 7 theorem, 24 , 297, 297 , 298, 318-
332, 335 , 349 framed structure , 30 9 framed tre e lemma , 33 0 framing principle , 23 7 framing principle , xxi, 240, 250, 257,
342-344 for dis k bundles , 23 9 for dis k bundle s reversed , 23 9 reverse framin g principle , 24 2
free expansio n pair , 121 , 182 free G-expansio n pair , 12 7
general pose t morphism , 9 8 geometric realization , 3 G-expansion, 5 1 G-expansion functor , 84 , 127 , 12 8 G-expansion morphism , 12 6 G-expansion pair , 5 1 ghosts, 58 , 79 , 31 3 GMF, generalize d Mors e function ,
62, 15 4 G-monomial chai n isomorphism , 2 7 G-monomial functor , 10 9 G-monomial monomorphis m
/isomorphism, 2 7 G-monomial morphism , 9 9 G-monomial simplicia l homotopy, 10 9 Goette, Sebastian , xvi , xxii , 21 8 graded poset , 24 , 85 gradient-like vecto r field, 15 9 graphic, 24 8
Hain, Richard , xxi , xxii , 343 , 345 handle exchang e lemma , 22 9
INDEX 367
Hatcher, Allen , vi , xiv, xvi, xx, 229, 237, 247-253 , 297, 333
Hattori-Stallings trace , 22 6 Hermitian coefficients , xiv , xvi , 61,
64, 212 , 254 higher analytic torsion, xiii , xvi, 218,
225 higher FR-torsio n
7*(ra,n) fo r W.(C m ,n) , 4 9 r ^ ( l , n) fo r Volodi n space , 1 3 T^{m) fo r F(M m(C)), 1 3 tk{m) fo r Wi iff, 5 3 for Out h(Fn), 33 5 for Out^) aP(F2g), 33 9
higher FR-torsio n computation s CPn~^bundles, 24 3 ^ - i ^ / / P 2 n + 1 r e l 9 ) , 2 5 2 circle bundles , 66 , 296 circle bundles-exceptional case ,
220 complex linea r spher e bundle ,
220 fiber produc t o f sphere s bun -
dles, 25 6 fiber produc t o f lens space an d
Morse bundles , 25 5 Hatcher's example , 24 9 lens-space bundle , 21 9 mapping clas s group , 34 5 Morse bundles , 25 5 real linea r spher e bundle , 22 4 Torelli group , 34 3
higher homotopies , 8 5 Hochschild homology , 22 6 homology bundle , 90 , 11 0
twisted homolog y bundle , 11 2 homology markin g o f a graph , 33 5 homotopy betwee n morphisms , 93 ,
99 homotopy framing , 318-33 0
definition, 32 0 horizontal
arrows, 52 , 53 isomorphism, 6 3 morphisms, 5 4
transformation, 5 2
incidence matri x continuous, 25 7 smooth, 259 , 26 1
independent, 15 5 inversion functor , 30 , 52 involution, xx , 29 , 30 , 50 , 174 , 184 ,
186, 187, 224, 237, 250-253 g-* = (g*)-\8 onW.(i? ,G), 5 2 on W.CR,G,n) , 2 9 on W/4°' m](#,G), 18 6 on y n(C), 14-1 6
isovariant subcomplex , 4 0
Ki(R), 2 K2(R), 2 K2k, a cochai n o n W # , 43 Klein, John , xvi , xviii, xxi, xxii, 135,
173, 34 3
label, 32 0 last simple x functor , 21 0 linear interpolation , 5 , 5 3 Z/2fc, another cochai n onlV , , 43 Lott, John , xiii , xv i
magma, 31 9 mapping clas s group , xxi , 343-345 ,
347 marking o n a graph , 30 7 Matsumoto, Hideya , xi v maximal morphism , 97 , 11 1 May, Peter , x v metric graphs , 297 , 298 , 301 , 302 ,
309, 31 1 MQ., 302 , 30 3
W . + , 303
families, 306 , 311 , 320, 33 0 marked, 33 5 metric frame d graphs , 30 9
Miller-Morita-Mumford classes , xxi, 297, 298 , 342, 34 3
368 INDEX
minimal orderin g o f E CT, 149 minimal subcomplex , 138 , 141, 142,
144, 146, 147, 151, 152,159, 162, 163, 165, 166,174,184, 208, 23 7
is the cellula r complex , 14 1 monomial matrices , 15 , 23
Mon(G), G-monomia l matrices, 24
monomials, 1 1 morphism o f arrows , 9 8 Morse bundles , 25 4 multiple incidence , 144 , 148 , 155
nd mean s nondegenerate , 9 1 necklaces, 290-29 5 NA(f), non-ascendin g se t o f / , 7 0
ordering o f critica l se t minimal ordering, 142 , 145, 149-
151, 156 , 16 5 modified minima l ordering, 156,
176 standard ordering , 142 , 149, 156
oriented fiberwise GMF , 15 8 oriented fiberwise Morse function, 14 7 oriented Mors e function , 14 3
parametrized Mors e lemma , 14 7 Penner, Robert , xxi , xxii , 297 , 343,
345-347 perfect group , 2 permutation morphism , 9 9 PGMF, positivel y frame d GMF , 5 7 Poincare duality , 187 , 18 8 polylogarithms, 21 , 23 , 63-69 , 258 ,
262, 270, 272, 276-282, 287 component, 264 , 268 , 273, 275,
288 polylogarithm invariant , 273 , 274
poset, 2 4 poset morphism , 9 6 projective homolog y case , 12 5 pseudoisotopy, xiv , xvi , 24 4
Ray-Singer analyti c torsion , xi v
realization lemma , 23 0 Reidemeister torsion , xiv , 6 6 relative Eule r characteristic , 25 4 rescaling operation , 30 2
Serre spectra l sequence , 9 0 Serre subcategory , 21 5 £*(£), 23 0 E2( /) , critica l point s o f / o f inde x
z, 240 E l ( / ) , critica l point s o f / o f inde x
i, 24 0 E2( /) , critica l point s o f / o f inde x
i, 24 0 E2( /) , critica l point s o f / o f inde x
i, 24 0 S(i,j, fc), uppe r triangula r chai n ho-
motopies, 2 5 simple homotop y equivalence , xii i simple morphis m o f pairs , 10 5 simple pose t morphism , 9 8 simplicial forgetfu l map , 5 simplicial homotopy , 5 simplicial map , 5 simplicial set , 3 singular nod e o r 0-cell , 308 , 355 small simple x i n B, 15 5 small subset s o f £? , 148 smooth nod e (whic h i s no t a ver -
tex), 308 , 312, 320 smooth, i.e . C°° , xii i Splitting Lemma , 21 4 St(R), Steinber g group , 2 stability theorem , 24 5 stable disk , 18 9 stable equivalence , 12 1 stable pseudoisotop y space , 24 5 stable rang e condition , 5 stable rank , 5 standard bal l neighborhood, 142 , 148 standard orderin g o f E a , 14 9 Steinberg relations , 2 strongly acyclic , 13 6 strongly exact , 13 6 strongly free , 13 6 strongly projective , 13 6
INDEX
subordinate to a homotop y framing , 32 0 to a pose t morphism , 93 , 98 to the functo r P , 10 9
support, 25 1 suspension
alternating, 13 2 anticommutes wit h involution ,
251 functor, 10 0 generalized, 24 5 maps, 244 , 245 multiple, 16 1 theorem, 20 9 twisted, 171 , 219
swallowing lemma , 180 , 18 1 symmetric simplicia l set , 18 6 symplectic group , 33 7 system o f endomorphisms, 2 6 system o f higher homotopies , 8 5
constant system , 8 6 system o f loca l sections , 25 9
Torelli group, xxi, 131 , 298, 308, 341-345, 349 , 358
total complex , 9 0 trace o f a deformation , 21 0 transfer, 6 3
N, th e transfe r functor , 33 , 55, 56
for Volodi n space , 6 on W/*.(P,G) , 21 4 smooth transfe r N d ^ ' , 6 transfer lemma , 56 , 65, 213 transfer lemm a fo r W #, 4 9
T°(R), a-uppe r triangular matrix group, 3
two-index theorem , 173 , 209 , 229 -236
unique refinemen t property , 38 , 40 unstable con e G(x) , 14 6 unstable disk , 18 9 unstable manifold , 35 1 upper triangula r
P-upper triangular , 2 4
369
cr-upper triangular , 2 4 strictly a-uppe r triangular , 2 4
upper triangula r homolog y bundle , 114
upper triangula r morphism , 10 5
vertical arrows , 5 2 vertical gradient , 148 , 15 5 vertical tangen t bundle , 147 , 18 7 vertically oriented , 18 7 Vogtmann, Karen , xxi , 297, 307, 346,
347 Volodin, 23 , 24, 83 , 298
if-theory, xvii , xix , 1-5 , 1 3 K-theory KY(R), 3 category V #(P, n), 2 8 space V n(C), 1 2 space V(C) , xix , 1 space V n(Z), 34 0 space V n(R), 3 , 28 space V* iff(C), 5 , 7 , 1 0 space o f P , xi x stable categor y V.(P) , 5 2
V.2(P,G,n), 3 0
Wagoner, John , xiv , xvi , xxii , 4 Waldhausen, Priedhelm , vi , xv, xvi ,
180, 199-208, 245, 253, 333, 334, 33 6
W™nt{B{n), p) , a A-subset o f ObW.coni(Cm, G 34
Wiiff(Cm,n) = Vtf iff(Cm,U{m),n\ 34
u>, weight o f a monomial , 1 1 Whi (R,G), Whitehea d group , 12 2 Whitehead torsion , xiii , 122 , 123 ,
125, 130 , 147 , 184 , 20 3 fiberwise, 129 , 13 0
Wi*°-c{B(n),p), a A-subset o f 06Wi 5 0~c(Cm , 40
Wj*°- c(Cm ,G,n), 4 0 Wi s ° - d (C m ,G ,n) ,40 Wi s o- c i(C r n ,n) ,40 W.(M,G,n) = W.(End R(MR),G,n),
33 W.(P,G), 5 1
370
W.(R,G,n), 2 7 W,(R,G,n)/cmj,28
zpk+i), 46 , 48
