Torsion and brations

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9Walter de Gruyter Berlin New York 1998

Torsion and brations

ByWolfgang L}ck,Thomas Schick andThomas Thielmann at M}nster

Abstract. We study the behaviour of analytic torsion under smooth brations.

Namely, letF1E__B^f Bbe a smooth ber bundle of connected closed oriented smooth manifolds and let V be a at vector bundle over E. Assume that E and B come with Riemannian metrics. Suppose that dim(E) is odd andV is unimodular and comes with an arbitrary Riemannian metric or that dim(E) is even andV comes with a unimodular (not necessarily at) Riemannian metric. Let oan(E;V) be the analytic torsion of Ewith coeicients in V, let oan(Fb;V) be the analytic torsion of the ber overb with coeicients in Vrestricted to Fb and let PfB be the Pfaian dim(B)-form. LetHdRq (F;V) be the at vector bundle over B whose ber over b`B is HdRq (Fb;V) with the Riemannian metric which comes from the Hodge-deRham decomposition and the Hilbert space structure on the space of harmonic forms induced by the Riemannian metrics. Let oan

(

B;HdRq (F;V)

)

be the analytic torsion of B with coeicients in this bundle. The Leray-Serre spectral sequence for deRham cohomology determines a certain correction term oLSdR(f). We prove

oan(E;V)^O B

oan(Fb;V) PfB^\ q

(^1)qoan

(

B;HdRq (F;V)

)

^oLSdR(f) .

This formula simplies in special cases such as bundles withSnas ber or base, in which case the correction term oLSdR(f) reduces to the torsion of the associated Gysin or Wang sequence, resp.

0. Introduction

LetMbe a connected closed smooth manifold with Riemannian metric andV be a at vector bundle with a not necessarily at Riemannian metric. The denition of analytic torsion due to Ray and Singer G20Hfor an orthogonal representationV, or, equivalently, for a at Riemannian metric onV, still makes sense in the setting above (G1H, page 35 and G18H, page 730). Namely, let fp(s) be the zeta-function of the Laplace operator

[p::p(E;V)1 :p(E;V)

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which is for Re(s)~~0 the holomorphic function \

J!0j+s wherejruns over the positive eigenvalues of[p. It has a meromorphic extension to the complex plane which is analytic in zero. Dene

(0.1) oan(E;V):1 2 \

q^~0(^1)qqfFq(0) `R.

We want to study it for smooth ber bundles. The main result of this paper is

Theorem 0.2. Let F1E__B^f B be a smooth ber bundle of connected closed oriented smooth manifolds and let V be a at vector bundle over E. Assume that E and B come with Riemannian metrics. Suppose that dim(E) is odd and V is unimodular and comes with an arbitrary Riemannian metric or that dim(E) is even and V comes with a unimodular Riemannian metric or that dim(E) is even and V comes with a unimodular Riemannian metric. Then

oan(E;V )^O

Boan(Fb;V) PfB^\

q (^1)qoan

(

B;HdRq (F;V)

)

^oLSdR(f) . ; Here are some explanations of the assumptions and the formula in Theorem 0.2.

For a path w in E the ber transport gives a linear isomorphism Vw:Vw(0)1Vw(1) which depends only on the homotopy class relative endpoints of wsinceV is at. We call V unimodular if for one (and hence all) e`E and all loops w with base point e we get det(Vw:Ve1Ve)^1. We call a Riemannian metric onV unimodular if for any pathwin E we get det(V<w Vw:Vw(0)1Vw(0))^1 where V<w is the adjoint ofVw with respect to the Hilbert space structure on the bers ofVgiven by the Riemannian metric. This is a weaker condition than being a at Riemannian metric what would mean that Vw is always an isometry. Notice thatV is unimodular if and only if it carries a unimodular Riemannian metric.

Of courseoan(E;V) is just the analytic torsion with respect to the given Riemannian metrics on E and V. These induce also a metric on HdRp (E,V). Each ber Fb^p+1(b) inherits a Riemannian metric fromE by restriction. We denote the restriction ofVto Fb again by V. Henceoan(Fb;V) is dened and is a smooth function in b`B.

Let PfB be the Pfaian dim(B)-form on the oriented Riemannian manifold B. It is a representative of the Euler class of B in Chern-Weil theory and satises by the Gauss- Bonnet theorem

O

BPfB^s(B)

wheres(B) is the Euler characteristic. If dim(B) is odd, then PfBis dened to be zero.

Let Hq(Fb;V) be the space of harmonic q-forms, i.e. the kernel of the Laplace operator [q::q(Fb;V)1 :q(Fb;V). It inherits an inner product from the Riemannian metrics onFbandV. Theharmonic Hilbert structureon the deRham cohomologyHdRq (Fb;V) is the Hilbert space structure for which the canonical Hodge isomorphism

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Hq(Fb;V) 1HdRq (Fb;V)

is isometric. Thus we get a Riemannian metric on the canonical at vector bundleHdRq (F;V) whose ber overb`BisHdRq (Fb;V). Hence the analytic torsionoan

(

B;HdRq (F;V)

)

is dened, andHdRp

(

B,HdRq (Fb;V)

)

inherits a natural Hilbert space structure.

There is the following natural descending ltration of the deRham complex:<(E;V).

DeneFp:n(E;V) to be thosen-forms with coeicients inVwhich can be written as nite sums of n-forms onEwith coeicients inVof the shapeu)f<gforu ` :n+k(E;V) and g ` :k(B) for some k~p. This ltration is compatible with the dierential since d(u)f<g)^d(u))f<g^u)f<d(g). The associated spectral cohomology sequence is the Leray-Serre spectral sequence for deRham cohomology, which we recall in Section 4.

Part of the Leray-Serre spectral sequence for deRham cohomology is the ltration of the cohomologyHn(E;V)

J0K^Fn;1,+1!!Fp;1,n+p+1!Fp,n+p!!F0,n^HdRn (E;V) , the natural identication of the E2-term

(0.3) V2p,q:HdRp

(

B;HdRq (F;V)

)

__B E2p,q,

the identication of the cohomology of the r-th term of the spectral sequence with the (r^1)-th term and the identication of the E'-term with the ltration quotients

p,qr :H0(Erp;r,q+(r+1)) __B Er;1p,q , tp,q:Fp,qFp;1,q+1 __B E'p,q. Forr suiciently large, the dierentials inEr<,<are trivial.

Next we explain the termoLSdR(f) appearing in Theorem 0.2.

For a linear isomorphism f:V1Wof nite-dimensional real Hilbert spaces, set

(0.4) f:1

2ln

(

det(f<f)

)

`R.

LetC^C< be an acyclic nite Hilbert cochain complex. Dene (0.5) o(C):(c<^c<) :Cev1Codd `R

wherec< is the dierential andc< a chain contraction. If f:C1D is a chain homotopy equivalence of nite Hilbert cochain complexes, cone(f) is the cochain complex withn-th dierential

^cn^fⁿ ^{^d}⁰ⁿ⁺¹

^:^Cn^Dn+1¹^Cn;1^Dn^.

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It is acyclic and we dene

(0.6) t(f):o

(

cone(f)

)

.

Let C be a nite Hilbert cochain complex such that H(C<) carries a Hilbert structure.

There is up to homotopy precisely one chain map i:H(C)1C whith H(i)^id, where we consider H(C) as a cochain complex with the trivial dierential. Dene

(0.7) o(C):^t(i)`R.

In the Leray-Serre spectral sequence, equip E2p,q with the Hilbert space structure which makes Vp,q2 to an isometry. Equip inductively Erp;r,q+(r+1), H(Erp;r,q+(r+1)) (r~2) andEp,q' with the Hilbert sub- and quotient structures. In particular,p,qr become isometries. EquipFp,q!HdRp;q(E,V) andFp,qFp;1,q+1with the Hilbert sub- and quotient structures. Now dene

oLSdR(f): \ r^~2

r+1\ p:0 \

q (^1)p;qo(Erp;r,q+(r+1))^ \

p,q(^1)p;qtp,q. This number depends only on f and the Riemannian metrics onE,BandV.

In general the correction termoLSdR(f) is very involved and is as complicated as the Leray-Serre spectral cohomology sequence is. However, there are cases whereoLSdR(f) and the whole formula in Theorem 0.2 are easy to understand. Namely, we will prove under the assumptions of Theorem 0.2 the following three corollaries. The rst one generalizes a result of Fried G7Hfor orthogonalV.

Corollary 0.8. Suppose that HdRq (F;V) vanishes for all q. Then s(B)oan(Fb;V) is independent of b and

oan(E;V)^s(B) oan(Fb;V) . ;

Corollary 0.9. Suppose that F is Sn and V^f <W for a at vector bundle with Rie- mannian metric over B. Let G< be the acyclic cochain complex of nite-dimensional Hilbert spaces given by the Gysin sequence

^O B HdRp (B;W) ^)e(^f⁾B HdRp;n;1(B;W) ^f^< B HdRp;n;1(E;V)

O B HdRp;1(B;W) ^)e⁽^f⁾B

where)e( f ) is the product with the Euler class e( f )`HdRn;1(B) of the sphere bundle,O is integration over the ber and G1^HdR0 (E;V ). Then the torsion o(G<)`R is dened and we get:

oan(E;V)^s(Sn) oan(B;W)^o(G<) . ;

The conditionV^f<W is no loss of generality, provided thatn~2 or that finduces an isomorphismn1(E)1 n1(B).

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Corollary 0.10. Suppose that B^Sn. Let W< be the acyclic cochain complex of nite- dimensional Hilbert spaces given by the Wang sequence and the harmonic structures for some b`B:

1HdRq+1(E;V)1HdRq+1(Fb;V)1HdRq+n(Fb;V)1HdRq (E;V)1 where W1^HdR0 (E;V ). Then

oan(E;V)^s(Sn) oan(Fb;V)^o(W<) . ;

We make some remarks about the proof of Theorem 0.2. It will depend on the following deep results of Bismut-Zhang G1H and M}ller G18H. In the sequelM is a connected closed oriented Riemannian manifold and V is a at vector bundle over Mwith Riemannian metric. We have introduced the analytic torsion oan(M;V) above. Its topological counterpart

(0.11) otop(M;V)`R

is the Milnor torsion of M with respect to some triangulation and the harmonic Hilbert structure on cohomology which we will recall in Denition 3.3.

Theorem 0.12. If the Riemannian metric on V is unimodular, then oan(M;V)^otop(M;V) . ;

Let gMand gMbe two Riemannian metrics on M and gV and gV be two arbitrary Riemannian metrics onV. Letoan(M;V) andoan(M;V) be the analytic torsion with respect to (gM,gV) and (gM,gV). Analogously we denote the Hilbert spaces HdRp (M;V) and HdRp (M;V) equipped with the harmonic Hilbert structures with respect to (gM,gV) and (gM,gV) and the Hilbert spacesVxandVxequipped with the Hilbert structure with respect to gV and gV. Denote by PfM the Pfaian with respect to gM. Let Pf(M,^~ gM,gM) be the Chern-Simons n^1-form. Its image under the dierential is the dierence of the two Pfaians of Mwith respect to gM andgM. Let h(V,gV) be the closed 1-form dened in G1H, Denition 4.5. It measures the deviation of gV from being unimodular and vanishes if gV is unimodular.

Theorem 0.13. We get under the conditions and in the notations above:

1. If dim(M) is odd,then

oan(M;V)^oan(M;V)^^\

p (^1)pHdRp (M;V) ^id B HdRp (M;V). 2. Ifdim(M) is even, then

oan(M;V )^oan(M;V)^^\

p (^1)pHpdR(M;V) ^id BHdRp (M;V)

^ O

MVx ^id BVx PfM^ O

Mh(V,gV) Pf(M,^~ gM,gM) . ;

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Theorem 0.12 and Theorem 0.13 for odd-dimensionalMare Theorem 1 and Theorem 2.6 of M}llerG18H, who generalizes Cheeger's and his proofG3HandG17Hof the Ray-Singer Conjecture oan(M;V)^otop(M;V) for orthogonal representations V to the unimodular setting. Bismut and ZhangG1H, Theorem 0.1 and Theorem 0.2, have generalized M}ller's work to all dimensions and to a setting whereV is not necessarily unimodular. We say more about their version of Theorem 0.12 in Section 6.

Theorem 0.12 and Theorem 0.13 enable us to show that Theorem 0.2 follows from its topological version. This will be done in Section 6 where we also explain how Corollaries 0.8, 0.9 and 0.10 follow from Theorem 0.2 and its topological version. In the topological case we can treat a more general setting. Namely, we consider a bration f:E1Bsuch thatBis a connected nite CW-complex, the homotopy ber has the homotopy type of a niteCW-complex and a certain cohomology class hf`H1

(

B;Wh(E)

)

vanishes. Then we get for a local coeicient systemVwith unimodular Hilbert structure, and xed Hilbert structures on the relevant singular cohomology groups:

Theorem 5.4.

o(E;V)^s(B) o(F;V)^o

(

B;\

q (^1)qHsingq (F;V)

)

^oLSsing(f) . ;

FreedG6Hobtained results about torsion and spectral sequences similar to those of Section 4. The latter are used to deduce the topological bration formula. There is also work of Maumary G13Habout Whitehead torsion and spectral sequences.

The topological version includes manifolds with boundary and the question whether Theorem 0.2 generalizes to manifolds with boundary comes down to the question whether Theorem 0.12 and Theorem 0.13 generalize to manifolds with boundary. At least under the condition thatV is an orthogonal representation Cheeger and M}ller's results for closed manifolds have been extended to manifolds with boundary (seeG8H,G9HandG11H).

There are also interesting generalizations of the topological versions to other settings.

For example one can consider anL2-version. Or one can substituteVby a local coeicient system of nitely generated projective modules over a ringR. Then the torsion takes value in the algebraicK-theory ofR.

Instead of explicitely choosing Hilbert structures to dene torsion as a real number, one can consider it as an element in certain determinant spaces. The latter approach has the advantage not to depend on articial choices. However, it does not allow the generalizations we have mentioned. In particular, one is bound to the nite dimensional setting (i.e. even the singular cochain complex of a nite CW-complex is not allowed). Also, if one carries out explicit computations, in most cases it is preferable to deal with real numbers. Therefore we used the rst method. In our context, both languages are completely equivalent and we supply a dictionary to translate between them in Appendix A. One should also mention that there is a third equivalent approach which uses norms constructed on determinant lines (compare G1H).

Finally we mention that in the case whereVis orthogonal Dai and MelroseG5Hhave proven the formula of Theorem 0.2 in the adiabatic limit by completely dierent methods,

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namely, by a careful analysis of the heat kernel in the adiabatic limit and the adiabatic version of the Leray-Serre spectral sequenceG14H. We remark that in the special case where V is an orthogonal representationoan(E;V) vanishes if dim(E) is even but this is not true in general for a non-at Riemannian metric onV.

The rst two authors decidate this paper to their friend and colleage Thomas Thiel- mann who died in a car accident in November 1994.

1. Simple structures on spaces

In this section we explain additional structures on arbitrary topological spaces and brations which allow the denition of torsion invariants on them. This extension from the category of nite CW-complexes serves two purposes: on the one hand there are interesting spaces which are not nite CW-complexes, f.i. classiying spaces of discrete groups. On the other hand, our approach singles out what exactly is used in the denition of torsion invariants, and this denitely claries the exposition.

We will always assume for a pair of spaces (X,A) that the inclusion ofAinto X is a cobration. This condition is satised if (X,A) is a pair ofCW-complexes or if X is a manifold with submanifold A. A map (F,f) : (X,A)1(Y,B) of pairs is a relative homotopy equivalenceifF4id :X4fB1Yis a homotopy equivalence. HereX4fBis obtained fromX]Bby identifyinga`Awithf(a)`B. Notice that (F,f) is a homotopy equivalence of pairs if and only if f is a homotopy equivalence and (F,f) is a relative homotopy equivalence. Given a cellular relative homotopy equivalence (F,f) : (X,A)1(Y,B) of pairs of nite CW-complexes, dene its Whitehead torsion as the Whitehead torsion of the homotopy equivalence F4id :X4fB1Yof nite CW-complexes

(1.1) q(F, f):q(F4id) `Wh(Y) .

We refer toG4H, * 6, * 21 and * 22, for the denition of the geometric Whitehead group and Whitehead torsion and their identications with the algebraic Whitehead group Wh

(

n1(Y)

)

and Whitehead torsion. Notice thatq(F,f) depends only on the homotopy class of (F,f) and satises the composition formula, the formula for pairs and the product formula as stated in 1.6. This follows from the special case A^.in G4H, * 22 and * 23. Given a pair (Y,B), we call two relative homotopy equivalences (Fi,fi) : (Xi,Ai)1(Y,B) with pairs of nite CW-complexes as source fori^0, 1 equivalent if

q

(

(G,g) (F0, f0)

)

^q

(

(G,g) (F1, f1)

)

holds for any relative homotopy equivalence (G,g) : (Y,B)1(YF,BF) into a pair of nite CW-complexes.

Denition 1.2. Arelative simple structureon a pair (Y,B) is an equivalence class of relative homotopy equivalence (F,f) : (X,A)1(Y,B) for a pair of nite CW-complexes as source. ;

1.3. Let (X,A) be a pair with a relative simple structure andg:AFF1Abe a homotopy equivalence with a nite CW-complex as source. Then we can extend gto a repre-

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sentative for the simple structure (G,g) : (XFF,AFF)1(X,A) which is a homotopy equivalence of pairs as follows.

Choose some representative (F,f) : (XF,AF)1(X,A) for the given simple structure.

Furthermore choose a homotopy inverse g+1:A1AFF, a homotopy fromg+1 f to a cellular map AF1AFFand a homotopy tfromg g+1 f to f. LetXFF be the nite CW- complex XF4y1AFF. Dene (G,g) : (XFF,AFF)1(X,A) by the following composition of (relative) homotopy equivalences or their homotopy inverses

XF4y1AFF1XF^G0,1H4yAFF2XF4g1 fAFF1XF4g g1 fA 1XF^G0,1H4TA2XF4fA1X.

The proof that (G,g) represents the given simple structure is done by the results of G4H,

* 5. ;

1.4. Given a (relative) simple structure on (X,A) and onA, we construct a preferred simple structure onXas follows. Because of 1.3 there is a homotopy equivalence of pairs (G,g) : (XFF,AFF)1(X,A) such that it represents the given relative simple structure on (X,A) andg represents the given simple structure on A. The preferred simple structure onX is then represented byG. This is independent of the choice of (G,g) by 1.6. ;

Given a relative homotopy equivalence (F,f) : (X,A)1(Y,B) of pairs with relative simple structures, we still can dene its (relative) Whitehead torsion

(F, f)^idY

)

^s(Y) i<q(F, f)

where i denotes the obvious inclusions. One easily checks that they remain true in the more general case that (X,A) is not necessarily a pair of niteCW-complexes, but carries a relative simple structure. ;

1.7. Let f:E1Bbe a bration such thatBis a nite CW-complex and the ber has the homotopy type of a niteCW-complex. Suppose that we are given acellular base point systemJbcc`IKforB, i.e. a choice of pointsbcin the interiorcEfor eachc`Iwhere

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here and elsewhere In is the set ofn-cells and I is the disjoint union of the In-s. Further- more suppose that we have specied a simple structure on each ber Fbc^f+1(bc). We want to dene a simple structure on Edepending only on these choices as follows. Recall that any homotopy class of pathswfromb0tob1denes a homotopy class of homotopy equivalences tw:Fb01Fb1by the ber transport G21H, 15.12.

Let En be f+1(Bn). As Bn+11Bn is a cobration, the same is true for En+11En G22H, I.7.14. Because of the construction 1.4 it suices to specify a relative simple structure for (En,En+1) for alln~0. This will be done by the next Lemma 1.8 taking into account 1.6 and that the Whitehead torsion of any homotopy equivalence (Dn,Sn+1)1(Dn,Sn+1) is trivial since Dn is simply-connected and hence Wh(Dn) is trivial. ;

Lemma 1.8. Suppose we have specied a cellular base point system for B and for each element in Inan orientation. Then there is a relative homotopy equivalence which is uniquely dened up to homotopy

]

c`InFbc^(Dn,Sn+1)1 (En,En+1) .

If we change the orientation of the cell c, then the map is changed by the selfhomotopy equivalenceid^s:Fbc^(Dn,Sn+1)1Fbc^(Dn,Sn+1) where s is a map of degree^1. If we change the base point bc of c to bFc, then the map is changed by the homotopy equivalence t^id :Fbc^(Dn,Sn+1)1FbDc^(Dn,Sn+1) where t:Fbc1FbDc is given by the ber transport along any path in cEconnecting bc and bFc.

Proof. The map will be constructed as the composition of the following four maps or their homotopy inverses.

There is up to homotopy one orientation preserving homotopy equivalence of pairs (Dn,Sn+1)1(cE,cE^bc). This gives the rst map

] c`In

Fbc^(Dn,Sn+1) 1 ] c`In

Fbc^(cE,cE^bc) .

Choose a homotopyc:cE^G0,1H1Bfrom the canonical inclusioncE1Bto the constant map with valuebc such that its evaluation atbcgives a path within cE. By the homotopy lifting property we obtain a strong ber homotopy equivalence which is unique up to ber homotopyG21H, Proposition 15.11

Fbc^(cE,cE^bc) 1(Ec^E,Ec^E+bc) .

The second map is the disjoint union of these maps over In. The third map is the relative homotopy equivalence given by the inclusion

]

cÌn(EcÊ,EcÊ+bc) 1(En,EBn+'cÌnp bc() .

The fourth map is the homotopy equivalence of pairs given by the inclusion (En,En+1) 1(En,EBn+'c`Inp bc() .

In particular we see for a bration p:E1B over a nite CW-complex B such that the homotopy ber has the homotopy type of a nite CW-complex and s(B)^0 andhf^0 holds thatE has a preferred simple structure. ;

Lemma 1.13. Let F1E__Bf B be a smooth bundle of compact smooth manifolds.

Equip B and each ber with the simple structure given by a smooth triangulation. This gives a coherent choice of simple structures on the bers. Then the simple structure on E given by a smooth triangulation agrees with the one given by 1.7. ;

Remark 1.14. The construction 1.7 can be extended to the case where B is not necessarily aCW-complex but carries a simple structure. Namely, choose a representative g:X1Bof the simple structure. The pull back construction yields a brationf:g<E1X

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and a ber homotopy equivalenceg:g<E1E. Equipfwith the coherent choice of simple structures on the bers induced by g and the given one of f. Construction 1.12 applies to fand gives a simple structure ong<E. Equip Ewith the simple structure for whichq(g) vanishes. It is not hard to check that this is independent of the choice of the representative g. The main step is to show in the case whereXandBare niteCW-complexes andg is an elementary expansion thatq(g) vanishes with respect to the simple structures onEand g<Egiven by construction 1.12. ;

2. Milnor torsion for cochain complexes

In this section we give a brief introduction to torsion invariants of cochain complexes as dened in the introduction. We give no proofs but refer to G16HandG12H.

Given linear isomorphismsf,gof nite-dimensional real Hilbert spaces, the number of 0.4 has the following properties:

(2.1) f^ln

(

det(A)

)

`R

whereAis the matrix describing f with respect to some choice of orthonormal basis for source and range.

f g^f^g,

⁰^f ^h^g

^{^}^f^{^}^g^,

f<^f.

Let C^C< be a nite Hilbert cochain complex, i.e. a cochain complex of nite-dimensional Hilbert spaces such that Ci^0 for i~N for some natural number N. If C is acyclic, we dened in (0.5)o(C):(c<^c<) :Cev1Codd `R, wherec< is the dierential andc< a chain contraction. This is independent of the choice ofc<. For f:C1Da chain homotopy equivalence of nite Hilbert cochain complexes, t(f) was dened in (0.6). It turns out that t(f) depends only on the homotopy class of f andt(f g)^t(f)^t(g).

Notice that with these conventions we get for an isomorphismf:C1Dof nite Hilbert cochain complexes

t(f)^\

n (^1)nfn.

LetCbe a nite Hilbert cochain complex such thatH(C<) carries a Hilbert structure, i.e.

Hn(C<) is equipped with a Hilbert space structure for eachn`Z. If i:H(C)1C is the chain map which induces on cohomology the indentity, we denedo(C):^t(i)`R. The minus sign ensures that this denition coincides with the one in 0.5 in the acyclic case. If we x an orthonormal basis for eachCnand eachHn(C), then the logarithm of the torsion dened by MilnorG16H, page 365, iso(C).

LetC be a nite Hilbert cochain complex and equip ker(cp) and im(cp+1) with the Hilbert substructures andHp(C)^ker(cp) im(cp+1) and Cp ker(cp) with the quotient

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Hilbert structures. If cp:Cp ker(cp)1im(cp) is the obvious isomorphism induced by cp and [p:Cp ker([p)1Cp ker([p) is the automorphism induced by the endomorphism [p^(cp)< cp^cp+1 (cp+1)< :Cp1Cp, then we get

(2.2) o(C)^\

p (^1)pcp^^\

p (^1)pp1

2 ln

(

det([p)

)

.

A simple structure on a (real) cochain complex C is an equivalence class of chain homotopy equivalences u:C 1Cwith a nite Hilbert cochain complex as source where u andv:C 1Care equivalent if t(v+1 u) vanishes. Let f:C1D be a chain homotopy equivalence of cochain complexes with simple structure. Dene

(2.3) t(f)^t(v+1 f u)`R

for any representativesu:C 1Candv:D 1Dof the simple structures. LetCbe a cochain complex with simple structure such that H(C) has a Hilbert structure. Dene

(2.4) o(C):o(C)`R

for any representative u:C 1C where we use the Hilbert structure onH(C) for which H(u) is an isometry.

Next we collect the basic properties of these invariants. We mention that one rstly veries them for nite Hilbert cochain complexes and uses this to show that the denitions and results extend to cochain complexes with simple structures.

If 01C1D1E10 is an exact sequence of nite Hilbert cochain complexes, we can view Cn1Dn1En as an acyclic nite Hilbert cochain complex concentrated in dimension 0, 1 and 2 and dene

o(C1D1E): \

n`Z(^1)no(Cn1Dn1En) .

This extends to an exact sequence01C1D1E10of cochain complexes with simple structures as follows. Construct a commutative diagram of cochain complexes

0 __B C __B D __B E __B 0

f g h

with the property that the rows are exact and the vertical arrows are homotopy equivalences which represent the given simple structures. In particular the lower row is an exact sequence of nite Hilbert cochain complexes and we put

(2.5) o(C1D1E):o(C 1D 1E).

2.6. In the following list of basic properties all cochain complexes come with simple structures.

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1. Homotopy invariance.

fg &t(f)^t(g) . 2. Composition formula.

t( f g)^t( f )^t(g) . 3. Exactness.

Given a commutative diagram of the shape above with exact rows and homotopy equivalences as vertical arrows, then

t( f )^t(g)^t(h)^o(C 1D 1E)^o(C1D1E) . 4. Sum formula.

Let 01C1D1E10 be exact and the cohomolgy of C, D and E come with Hilbert structures. Let LHS be the acyclic nite Hilbert cochain complex given by the long cohomology sequence whereLHS0^H0(C). Then:

o(D)ô(C)ô(E)ô(LHS)ô(C1D1E). 5. Transformation formula.

If f:C1Dis a homotopy equivalence and the cohomology ofCandDcome with Hilbert structures, then:

o(C)^o(D)^t( f )^ \

n`Z(^1)nHn( f ). 3. Milnor torsion for spaces and local coeicient systems

The fundamental groupoid %(X) of a space X has as objects points in X and a morphismw:x1yis a homotopy class relative to end points of paths inXfromytox.

Forx`X letX(x) be the set of all morphisms w:x1ywithxas source. The projection p(x):X(x)1X sendsw to y. Assume that X is locally path-connected and semi-locally simply connected. This condition is always satised if X is a CW-complex or manifold.

It ensures that there is precisely one topology on X(x) such that p(x) is a model for the universal covering of the path component ofXcontainingx. Thus we get a contravariant functor

X:%(X) 1SPACES .

Composing it with the covariant functor singular chain complex with real coeicients yields the contravariant functor

C< (sing X) :%(X) 1R^CHAIN .

A local coeicient system Von X is a contravariant functor V:%(X)1R^VECTOR into the category of nite-dimensional real vector spaces. For instance a at vector bundle

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over X denes a local coeicient system. Dene the singular cochain complex and the singular cohomology of X with coeicient in V by

(3.1) C<sing(X;V):hom

(

Csing< (X),V

)

, Hsing< (X;V):H<

(

C<sing(X;V)

)

where hom denotes the real vector space of natural transformations.

Suppose that X is a CW-complex. Then X becomes a functor from %(X) into the category of CW-complexes and we can dene the cellular versions Ccell< (X;V) and Hcell< (X;V) of the denitions above by using the cellular chain complex instead of the singular one. There is a homotopy equivalence

(3.2) Ccell< (X;V)1 C<sing(X;V)

which is unique up to homotopy and natural in X andV (G10H, page 263). It induces a natural isomorphismHcell< (X;V)1Hsing< (X;V).

LetX be a space and V0,V1, ...,Vrlocal coeicient systems. We say \r q:0

(^1)qVq isunimodularif we have for each automorphism w:x1xin %(X)

]r

q:0det(Vqw :Vqx 1Vqx)(+1)q^1 . AHilbert structureon \r

q:0(^1)qVqis a choice of Hilbert space structure onVqx for each q`J1, 2, ...,rKandx`X. It is calledunimodularif we have for each morphismw:y1x in %(X)

\r

q:0(^1)qVqw :Vqx1Vqy^0 in the notation of 0.4. Notice that \r

q:0(^1)qVqis unimodular if and only if it admits a unimodular Hilbert structure.

Denition 3.3. Let X be a space andV0,V1, ...,Vr be local coeicient systems.

Assume that Xcomes with a simple structure, r

\

q:0(^1)qVq with a unimodular Hilbert structure andHsing< (X;Vq) with a Hilbert structure. Dene theMilnor torsion

o

^X;q:0^\^r (^1)qVq

(3.5) '

q:0^\^r

(^1)qVq

^{: Wh(X}⁾ ¹^R

as follows. An element GuH`Wh(X) can be represented by an automorphismu:M1M of a nitely generated freeZ%(X)-moduleM. Composition withudenes an automorphism uq: hom(M,Vp)1hom(M,Vq) of a nite-dimensional real vector space. Put

'

q:0^\^r (^1)qVq

^(GuH)^:q:0^\^r (^1)q lndet(uq). The unimodularity condition on r

\

q:0(^1)qVqensures that trivial units are send to 0. The following lemma is a consequence of the denitions and the formulas 2.6.

Lemma 3.6. Let f:X1Y be a homotopy equivalence of spaces with simple structures.

Let V0,V1, ...,Vr be local coeicient systems on Y. Assume that V: \r

q:0(^1)qVq comes with a unimodular Hilbert structure and H<sing(X; f<Vq) andHsing< (Y;Vq)come with Hilbert structures. Then:

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o(Y;V)^o(X; f<V)^'(V)

(

q(f)

)

^ \r

q:0(^1)q \

p^~0(^1)pfp:Hsingp (Y;Vq)1Hsingp (X;f<Vq). ;

4. Torsion and spectral sequences

LetCbe a cochain complex with nite descending ltration by cochain complexesFpC C^F0C"F1C"F2C""FpC"Fp;1C" .

, Ep,qr :Zp,qr Bp,qr ,

Ep,q' :Zp,q' B'p,q,

Fp,q:im

(

Hp;q(Fp)1Hp;q(C)

)

. We have the inclusions:

J0K^Bp,q1 !!Bp,qr !Bp,q' !Zp,q' !Zrp,q!!Zp,q1 ^Hp;q(FpFp;1) . The mapHp;q(FpFp;r)1Hp;q(FpFp;1) resp.Hp;q(FpFp;r)1Hp;q;1(Fp;rFp;r;1) is induced by the inclusion resp. is a boundary operator. We get epimorphisms

Hp;q(FpFp;r) 1Zp,qr Zp,qr;1 and

Hp;q(FpFp;r)1 Bp;r,q+r;1r;1 Brp;r,q+r;1.

Now the standard diagram chase shows that these maps have the same kernel. Hence we obtain canonical isomorphisms

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cp,qr :Zp,qr Zp,qr;11 Bp;r,q+r;1r;1 Bp;r,q+r;1r for r~1 . Analogously one gets a natural isomorphism

tp,q:Fp,qFp;1,q+11 Ep,q' . We dene the dierential

drp,q:Ep,qr ^Zp,qr Bp,qr 1 Ep;r,q+r;1r ^Zp;r,q+r;1r Bp;r,q+r;1r by the composition:

Zrp,qBp,qr 1Zrp,qZp,qr;1

cp,qr B Bp;r,q+r;1r;1 Bp;r,q+r;1r 1Zp;r,q+r;1r Bp;r,q+r;1r . We obtain cochain complexes

(Erp;r,q+(r+1),drp;r,q+(r+1)) , r~ 0

if we use for r~1 the denition above and dene E0p, to be the (^p)-th suspension

&+pFpFq;1of FpFp;1. Since ker(drp,q) isZp,qr;1Bp,qr and im(drp,q) is Bp;r,q+r;1r;1 Bp;r,q+r;1r ,

we obtain a canonical isomorphism

rp,q:H0(Erp;r,q+(r+1))1 Ep,qr;1

in the caser~1, in the caser^0 we use the identicationH0(E0p,q;)^Hp;q(FpFp;1).

4.1. Now suppose we have xed the following data:

1. Simple structures on FpFp;1; 2. a Hilbert structure on H(C). ;

EquipFpinductively with the simple structure for which the number dened in (2.5) satises

o(Fp;11Fp1FpFp;1)^0 .

In particular we get a preferred simple structure on C^F0 and o(C) is dened. Equip Fp,q!Hp;q(C) with the Hilbert substructure and Fp,qFp;1,q+1 with the Hilbert quotient structure. Do iteratively the same forEp,qr andH(Erp;r,q+(r+1)). Observe thatrp,q become isometries. Notice that 4.1.1 implies thatErp,q is nite-dimensional forr~1.

Denition 4.2. Dene for the nite descending ltration F<C with respect to the data 4.1 and choices above:

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o^~fil2(F<C): \ r^~2

r+1\ p:0 \

q (^1)p;qo(Erp;r,q+(r+1))^\

p,q(^1)p;qtp,q, ofil(F<C):\

p o(FpFp;1)^ \ r^~1

r+1\ p:0 \

q (^1)p;qo(Erp;r,q+(r+1))

^ \

p,q(^1)p;qtp,q; whereowas dened in (2.4) and in (0.4). ;

Remark 4.3. ofil(F<C) depends on the choices 4.1.

For the denition ofo^~fil2(F<C) one can x Hilbert structures onE2p,qinstead of data 4.1.1 and equip Erp,q and H(Erp;r,q+(r+1)) (r~2) with the corresponding Hilbert sub- and quotient structures. Then, o^~fil2(F<C) depends not on 4.1.1 but on these choices as follows: if U2p,q:E p,q2 1E2p,q is the identity on E2 equipped with two dierent Hilbert structures, then

o^~fil2(F<C)^o^~fil2(F<C)^\

p,q(^1)p;qU2p,q

(torsion computed using these two Hilbert structures). This follows from the transformation formula 2.6.

The main result of this section is:

Theorem 4.4. We get with respect to the data 4.1 and the conventions above o(C)^ofil(F<C) .

It will follow from the next three lemmas.

Lemma 4.5. It suices to treat the following special case: C itself is a nite Hilbert cochain complex with the simple structure represented by id :C1C and the Hilbert structures on Fpand FpFp;1,are obtained by the given one on C by taking Hilbert sub- and quotient structures.

Proof. One easily constructs a nite Hilbert cochain complexDwith nite coltra- tionF<D together with a chain homotopy equivalence f:D1Cwith the following property:f induces chain homotopy equivalencesFpf:FpD1FpCfor allpsuch that the given simple structure onFpCFp;1C is represented byFpfFp;1f:FpDFp;1D1FpCFp;1C.

EquipH(D) with the Hilbert structure for which H(f) :H(D)1H(C) becomes an isometry. We have by assumption and construction for allpthat

o(Fp;1D1FpD1FpDFp;1D)^0^o(Fp;1C1FpC1FpCFp;1C) . Hence we conclude from 2.6

o(C)^o(D) .

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The mapsFpfinduce isometries between theErp;r,q+(r+1)forr~1 andH(Erp;r,q+(r+1)) for r~0 associated to F<DandF<C. Now one easily veries

ofil(F<C)^ofil(F<D) . This nishes the proof of Lemma 4.5. ;

From now on we will only consider the special case described in Lemma 4.5. The proof of the next lemma is similiar to the proof in G19H, Theorem 2.2.

Lemma 4.6.

o(C) ^\

p o(FpFp;1)^ \ r^~1 \

p,q(^1)p;qcrp,q^\

p,q(^1)p;qtp,q.

Proof. We do induction overlwithFlC^J0K. The begin of inductionl^0 is trivial, the induction step from l^ 1 tol~1 done as follows.

LetC beF1^F1Cwith the coltration

Fp^FpC:Fp;1C.

We will denote the various data coming from the spectral sequence associated toC as the ones forCdecorated with an additional bar. LetLHSbe the acyclic nite Hilbert cochain complex given by the long cohomology sequence associated to 01C 1C1CC 10.

It induces forn~0 an acyclic nite Hilbert cochain complexD(n) concentrated in dimensions 0, 1, 2 and 3 by

Hn(C)F1,n+11Hn(CC)1Hn;1(C)1F1,n

where we equipF1,n+1^im

(

Hn(C)1Hn(C)

)

respectivelyHn;1(C)F1,nwith the Hilbert substructure respectively quotient structure. Dene an acyclic Hilbert subcochain complex D(n,r) forr~1 of D(n) by

Hn(C)F1,n+11Z0,nr 1F r+1,n;2+r1Fr,n;1+r.

Notice that D(n,1)^D(n). The following diagram of acyclic nite Hilbert cochain complexes concentrated in dimensions 1, 2 and 3 commutes:

Z0,nr Zr;10,n __B F r+1,n;2+rF r,n;1+r __B Fr,n;1+rFr;1,n+r

Br,n;1+rr;1 Br,n;1+rr __B Z r+1,n;2+r' B r+1,n;2+r' __B Zr,n;1+rBr,n;1+r'

c0,nr t r+1,n;2+r tr,n;1+r

where the upper row is the quotient D(n,r)D(n,r^1) and the lower row is induced by the obvious inclusions and projections if one takes the following identities into account:

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B r+1,n;2+r' ^Br,n;1+rr ; Br,n;1+rr;1 ^Br,n;1+r' ; Z r+1,n;2+r' ^Zr,n;1+r' .

We conclude from the sum formula and transformation formula 2.6 o(C)ô(C)ô(CC)ô(LHS) ,

o(LHS)^ \

^t0,n. We compute using the fact

c p,qr ^ cp;1,q+1r for p~1 and the induction hypothesis applied to C:

o(C)^o(C)^o(F0F1)

^ \ n^~+1

(^1)n;1

^t0,n^{^}^l+1_r:1^\ ^{(^}^c0,n^r ^{^}^t^r+1,n;2+r^{^}^tr,n;1+r)

^ \ p^~1

o(FpFp;1)^ \ r^~1

\ p^~1,q

(^1)p;q+1crp,q+1^\ p,q

(^1)p;qt p,q

^o(F0F1)

^ \ n^~+1

(^1)n;1

^t0,n^{^}^l+1r:1^\

^c0,nr ^t r+1,n;2+r^tr,n;1+r

^\

p o(FpFp;1)^ \ r^~1 \

p,q(^1)p;qcp,qr ^\

p,q(^1)p;qtp,q. This nishes the proof of Lemma 4.6. ;

Lemma 4.7.

ofil(F<C)^\

p o(FpFp;1)^ \ r^~1 \

p,q(^1)p;qcp,qr ^\

p,q(^1)p;qtp,q. Proof. We get from 2.2

o(Erp;r,q+(r+1))^\

k (^1)kcp;rk,q+(r+1)kr .

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This implies

\ r^~1

\ p,q

(^1)p;qcrp,q^ \ r^~1

r+1\ p:0

^0 where we think of hfas homomorphism n1(B,b)1Wh(E) and'(V) was introduced in 3.5. ;

Notice that\

q (^1)qHsingq (F;V) is unimodular but not necessarily eachHsingq (F;V).

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5.3. We will consider the following real numbers:

1. o(E;V).

We get by construction 1.7 a simple structure onEfrom the data 5.1.3 if we specify a cellular base point system onB. However, by 1.9 the choice of cellular base point system does not matter. Now we use this simple structure and data 5.1.1 and 5.1.2 to dene the Milnor torsion o(E;V) according to Denition 3.3. Notice that it depends only on 5.1.1, 5.1.2 and 5.1.3. This is the invariant we want to compute.

2. o(F;V).

Chooseb`B. Then we get from Denition 3.3 applied to data 5.1.1, 5.1.3 and 5.1.4 the Milnor torsiono(Fb;V). Here and elsewhere we supress in the notation that we view Vas a local coeicient system overFbby the inclusion ofFbinto E. We get from Lemma 3.6 that o(Fb:V) is independent ofbsince B is path connected. We abbreviate

o(F;V)^o(Fb;V) . This number depends only on 5.1.1, 5.1.3 and 5.1.4.

3. o

(

B;\ q

(^1)qHsingq (F;V)

)

.

This is the Milnor torsion and depends on the data 5.1.4 and 5.1.5.

4. oLSsing(f).

We have the Leray-Serre spectral sequence for singular cohomology associated to the brationf:E1B. Namely, the skeletal ltration ofBinduces a ltrationEp^f+1(Bp).

It yields a coltration onC<sing(E;V) by putting

B;\

q (^1)qHsingq (F;V)

)

^oLSsing(f) .

Proof. Fix a cellular base point systemJbcc`IKforBand an orientation for each cellc`I. The homotopy equivalence of Lemma 1.8 together with the suspension cochain homotopy equivalence and the obvious identication of FpFp;1 withC<sing(Ep,Ep+1;V) yields a cochain homotopy equivalence unique up to homotopy

U0p,: c`Ip

&pC<sing(Fbc;V)1FpFp;1. We have dened an isomorphism in (3.4)

kp,q:Cpcell

(

B;Hsingq (F;V)

)

1

c`IpHsingq (Fbc;V) .

Recall that we equipCpcell

(

B;Hsingq (F;V)

)

dened in 2.3 vanishes.

EquipFp inductively with the simple structure for whicho(Fp;11Fp1FpFp;1) dened in (2.5) vanishes. Then this simple structure on F0^C<sing(E;V) agrees with the one we get from the simple structure on E which we have dened in 1.7 by an inductive proce- dure over the En-s and Lemma 1.8. We conclude from Theorem 4.4:

o(E;V)^ofil

(

F<C<sing(E;V)

)

.

Becausep,qr are isometries, we get from the transformation formula 2.6:

\

p o(FpFp;1)^\ p \

c`Ico

(

&pC<sing(Fbc;V)

)

^\ p \

q (^1)qHq(U0p,)

^\

p (^1)p \

c`Ipo(F;V)^\

p,q(^1)p;qHp;q(U0p,)

^s(B) o(F;V)^\

p,q(^1)p;qU1p,q,

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\

q (^1)qo(E<,q1 )^\

q (^1)qo

(

C<cell

(

B;Hsingq (F;V)

))

^\

q (^1)q\

p (^1)pU1p,q

^\

q (^1)q\

p (^1)pHp(U1,q)

^o

(

B;\

q (^1)qHq(F;V)

)

^\

p,q(^1)p;qU1p,q

^\

p,q(^1)p;qU2p,q. We get from 5.3.4 and Denition 4.2:

ofil

(

F<C<sing(E;V)

)

^\

p o(FpFp;1)^\

q (^1)qo(E1<,q)

^oLSsing(f)^\ p,q

(^1)p;qU2p,q. Now Theorem 5.4 follows. ;

This is independent of the choice of hand Theorem 5.4 remains true.

6. The ber bundle formula for analytic torsion

In this section we give the proof of Theorem 0.2 and Corollaries 0.8, 0.9 and 0.10.

We begin with the proof of Theorem 0.2. In the rst step we reduce the claim to the case where the Riemannian metric on V is unimodular. By assumption we only have to treat the case where dim(E) is odd. We will vary the Riemannian metric onV, but x the

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Riemannian metrics onEandB. Denote byVthe at bundleVwith the given Riemannian metric and by V\ the at vector bundleVwith some unimodular Riemannian metric. Let oLSdR(f) and⁾oLSdR(f) be thetwo values of the correction term for the choice of these two Riemannian metrics on V. Notice that either PfB is zero or dim(Fb) is odd. We get from Theorem 0.13 and Denition A.7

oan(E;V)^oan(E;V\)^^\

p (^1)pHdRp (E;V) __B^id HdRp (E;V\), O

Boan(Fb;V) PfB^O

Boan(Fb;V\) PfB

^^O B

(

\

(

B;HdRq (F;V)

)

__B^id HdRp

(

B;HdRq (F;V\ )

)

^O

BHdRq (Fb;V) __B^id HdRq (Fb;V\) PfB, oLSdR(f)^⁾oLSdR(f)^\

p,q

. Hence we can assume in the sequel without loss of generality that the Riemannian metric onV is unimodular.

Given a closed smooth manifold M and a at vector bundle W over M with Riemannian metrics on M and W, the harmonic Hilbert structure on the singular cohomology Hsing< (M;W) is given by the deRham isomorphism

HdR<(M;W) 1Hsing< (M;W)

and the harmonic Hilbert structure onHdR< (M;W) coming from the Hodge decomposition and the Hilbert space structure on the space of harmonic forms as explained in the introduction. If we have choosen a smooth triangulation onM, the harmonic Hilbert structure onHcell< (M;W) is induced by the canonical isomorphism

H<cell(M;W)1 Hsing< (M;W)

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induced from (3.2). Denote by otop(M;W) the Milnor torsion of M dened in 3.3 with respect to the simple structure represented by a smooth triangulation and the harmonic Hilbert structure on H<sing(M;W).

Denote byHsingq (F;V) the at bundle overBwith bers Hsingq (Fb;V) equipped with some Riemannian metric such that the induced Hilbert structure on\

q

(^1)qHsingq (F;V) is unimodular. Recall that this can be done by Lemma 5.2. EquipBandFbwith theCW- structure given by some smooth triangulation. Let o(Fb;V) be the Milnor torsion of Fb with respect toHsingq (F;V) (see Denition 3.3). We have seen in 5.3.2 that it is independent ofb`Band abbreviate it byo(F;V). LetoLSsing(f) be the correction term of 5.3 with respect to the harmonic Hilbert structures onHsing< (E;V) andHsingp

(

B;Hsingq (F;V)

)

. We conclude from Lemma 1.13 and Theorem 5.4:

otop(E;V)^s(B) o(F;V)^o

(

B;\

q (^1)qHsingq (F;V)

)

^oLSsing(f) . (6.1)

Next we want to show:

(6.2) oan(E;V)^otop(E;V) , (6.3) oan(Fb;V )^otop(Fb;V) , (6.4) \

q (^1)qoan

(

B;Hsingq (F;V)

)

^o

(

B;\

q (^1)qHsingq (F;V)

)

.

The rst two equations follow directly from Theorem 0.12. For the proof of the last equation we must take a closer look at the result of Bismut and ZhangG1H, Theorem 0.2.

The problem is that the Riemannian metric on\

q (^1)qHsingq (F;V) is unimodular but it is not true in general that each of the at bundlesHsingq (F;V) is unimodular or, equivalently, admits some unimodular Riemannian metric.

Choose a Morse function f onB. LetX be its gradient vector eld with respect to some Riemannian metric on Bwhich can be dierent from the given Riemannian metric such thatXsatises the Smale transversality conditions. We obtain aCW-structure onB, denoted by BF, and a cellular base point system Jbcc`IK given by the critical points of f. The simple structure represented by id :BF1Bis the same as the one represented by any smooth triangulation. Let C<cell

q (^1)qHsingq (F;V)

)

^\

q (^1)qo

(

C<cell

(

))

.

But C<cell

(

)

is isometrically isomorphic to the Thom-Smale complex.

Let volharmbe the volume form associated to the harmonic Hilbert structure. Hence we get from the denitions (cf. G1H, Remark 2.3 and Remark 1.10)

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oan

(

B;Hsingq (F;V)

)

^^ln

(

volharmLSdetH.

))

^O B

B,\

q (^1)qHsingq (F;V)

)

^\

q (^1)qoan

(

B;Hsingq (F;V)

)

^\

q (^1)qo

(

Ccell<

(

))

^O B \

q (^1)qh

(

Hsingq (F;V)

)

X<t(B) . Now one easily checks that \

q (^1)qh

(

Hsingq (F;V)

)

vanishes as the Riemannian metric on\

q (^1)qHsingq (F;V) is unimodular. This nishes the proof of (6.4).

Denote byHdRq (F;V) andHsingq (F;V) the at bundles with the harmonic Riemannian metrics (in contrast toHsingq (F;V)). We get from the transformation formula 2.6 and (6.3)

(6.5) s(B) o(F;V)^O

Boan(Fb;V) PfB

^O

B

(

o(Fb;V)^otop(Fb;V)

)

PfB

^^O B \

q (^1)qHsingq (Fb;V) __B^id Hsingq (Fb;V) PfB. We conclude from (6.4) and Theorem 0.13:

(6.6) o

(

B;\

q (^1)qHsingq (F;V)

)

^\

q (^1)qoan

(

B;HdRq (F;V)

)

^\

(

^\

p (^1)pHdRp

(

B,Hsingq (F;V)

)

__B^id HdRp

(

B,Hsingq (F;V)

)

^O

BHsingq (F;V) __B^id Hsingq (F;V) PfB

)

^^\

p,q(^1)p;qHdRp

(

B,Hsingq (F;V)

)

__B^id HdRp

(

B,Hsingq (F;V)

)

^\

q (^1)qO

BHsingq (Fb;V) __B^id Hsingq (Fb;V) PfB.

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The deRham isomorphismHdRn (E;V)1Hsingn (E;V) is compatible with the two l- trations and the identications of the E2-term in the Leray-Serre spectral sequence for deRham and singular cohomology (0.3) and (5.5). It yields isomorphisms onErp,qforr~2.

This implies using Remark 4.3

(6.7) oLSsing(f)^oLSdR(f)

^\

p,q(^1)p;qHdRp

(

B,Hsingq (F;V)

)

__B^id HdRp

(

B,Hsingq (F;V)

)

. Now Theorem 0.2 follows from (6.1), (6.2), (6.5), (6.6) and (6.7). ;

Next we prove Corollary 0.8. Recall that HdRq (F;V) vanishes by assumption. In particular the E2-term of the Leray-Serre spectral sequences for cohomology vanishes.

Hence Theorem 0.2 implies

oan(E;V)^O

Boan(Fb;V) PfB.

If dim(B) is odd, the right hand side of the equation above and s(B) oan(Fb;V) vanish and the claim follows. Suppose that dim(B) is even. Then dim(Fb) is even and the Riemannian metric on V is unimodular by assumption or dim(Fb) is odd. In both cases oan(Fb;V) is independent of b`B by Theorem 0.13 and the claim follows. This nishes the proof of Corollary 0.8. ;

Next we outline the proof of Corollary 0.9. Analogously to the rst step in the proof of Theorem 0.2 we can show that we can assume without loss of generality that the Riemannian metric onW is unimodular. PutV^f<W. There is a canonical isomorphism of local coeicient systems overB

Hsingq (F;Z)ZW1Hsingq (F;V) .

Recall that we assume that EandB are oriented. (One can drop this assumption if one allows an additional twist forW.) This implies thatn1(B) acts orientation preserving on the homology of the ber. Hence we can choose an isomorphism of local coeicient systems of Z-modules from Hsingq (F;Z) to the trivial coeicient system with value Z. Hence we obtain identications

Hsingq (F;V)^

^W,^{0 ,} ^q^q^{^}^@^0,^0,ⁿⁿ^,^.

We conclude from Theorem 5.4

otop(E;V)^s(B) o(Fb;V)^o

(

B;\

q (^1)qHsingq (F;V)

)

^oLSsing(f)

where we use onHsingq (Fb;V) the unimodular Hilbert structure given by the one onWand the identication above and onHsingq

(

B;\

q (^1)qHsingq (F;V)

)

the harmonic one. We get

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o(F;V)^0 , o

(

B;\

q (^1)qHsingq (F;V)

)

^s(Sn) otop(B;W) , oLSsing(f)^oLSdR(f) ,

oan(E;V)^otop(E;V) , oan(B;W)^otop(B;W) since there is a canonical isomorphism

Csing< (Fb;Z)ZW1 C<sing(Fb;V)

and Hsing< (Fb;Z) is free as Z-module, the deRham isomorphism is compatible with the coltrations on the deRham complex and the singular cochain complex and with the identications of the E2-terms of the associated Leray-Serre spectral sequences (0.3) and (5.5) and we have Theorem 0.12. Hence it remains to show

oLSdR(f)^o(G<) .

Notice that the E2-term of the Leray-Serre spectral sequence is trivial except for the 0-th andn-th row. Hence we can splice the spectral sequence together to one exact sequence

1Enp,n

dp,nn;1B Ep;n;1,0n 1Hp;n;1(E;V) 1En;1p;1,n

dp;1,nn;1 B .

If one takes the identication of theE2-term (0.3) and the obvious identicationEp,q2 ^Enp,q into account, one gets an exact sequence whose torsion is preciselyoLSdR(f). Moreover, it can be identied with the Gysin sequence up to sign. This nishes the proof of Corollary 0.9. The proof of Corollary 0.10 is similar. ;

A. Torsion and determinants

We start with a description of the determinant of a nite dimensional vector space.

Let V be an n-dimension (real) vector space. Dene its determinantdet(V) to be the 1- dimensional vector space nVgiven by then-th exterior power. There are canonical isomorphisms

det(VW) __B det(V)det(W) , det(V<) __B det(V)< ,

det (V)<det(W) __B homR

(

det(V), det(W)

)

, det(U)<det (V< )det(V)det(W) __B det(U<)det (W) ,

det(V) 1

(

det(V)<

)

<.

Given a homomorphism f:V1W, we obtain a well-dened element

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Hp(C)

)

_p

)

.

The topological torsion o(X) of a nite CW-complex X is dened as the torsion of the cellular cochain complex.

The denition of the torsion of a ltrationspectral sequence is a little bit more elaborate.

LetF<W be a bration of a nite-dimensional vector spaceW J0K^F+1!!Fp!Fp;1!!Fn^W.

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This determines an isomorphism (A.4) ofil(F<W) :

p^~0det (FpFp+1) __B det (W)

which is dened inductively over n. The begin of induction n^1 is given by the Milnor torsion of the acyclic 2-dimensional cochain complexF01W1WF0. For the step n^1 to nobserve that

ofil(F<Fn+1) :n+1

p:0det (FpFp+1) __B det (Fn+1)

is dened by induction, and we simply compose (after tensoring with det(WFn+1)) with the Milnor torsion of Fn+11W1WFn+1.

In particular, suppose f:E1B is a bration of closed manifolds as in the introduction. This yields a ltration of HdRn (E;V) for n~0. Take the tensor product of the corresponding torsion isomorphisms resp. their inverses depending on the parity of n to obtain an isomorphism

o(Erp;r,q+(r+1)) :

n det

(

Hn(Erp;r,q+(r+1))(+1)n

)

1

n det(Erp;rn,q+(r+1)n)(+1)n. Taking r+1

p:0

(

B,HdHq (F,V)

)

1E2p,q and rp,q between the homology of ErandEr;1this denes an isomorphism

(A.5) odRLS(f) : det

((

HdRp

(

B;HdRq (F;V)

)

_p,q

))

__B det

((

HdRn (E;V)

)

_n

)

as the composition of isomorphisms (use the fact that Er and E' are equal for r large enough)

det

1det

(

(E'p,q)p,q

)

^((tp,q)+1)^p,qBdet

(

(Fp,qFp;1,q+1)p,q

)

^o^fil Bdet

((

HdRn (E;V)

)

_n

))

. For the spectral sequence associated to an arbitrary ltrationF<Wof a nite-dimensional vector space Wone can deneofil(F<W) and o^~2

fil (F<W) in the obvious analogous way.

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Now, we have to relate the invariantso to the previously dened real numbers o_ using given inner products.

If the nite-dimensional vector space V has a Hilbert structure, it induces in a canonical way a Hilbert space structure on det(V). In particular the norm v` R^~0 is dened for any elementv`V. IfVandWcome with Hilbert space structures, for any element

u`homR

(

det(U), det(V)

)

^det(V)<det (W) we get its norm

(A.6) u`R^~0.

.

Of course, with similar denitions a similar result can be obtained foroLSsing(f). Also, the corresponding equation for ofilof a ltration is true.

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G18HW. M}ller, Analytic torsion andR-torsion for unimodular representations, J. AMS6(1993), 721_753.

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Alg.20(1981), 195_225.

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Fachbereich f}r Mathematik und Informatik, Westf{lische Wilhelms-Universit{t M}nster, Einsteinstr. 62, D-48149 M}nster

e-mail: lueckWmath.uni-muenster.de e-mail: thomas.schickWmath.uni-muenster.de

Eingegangen 16. Dezember 1996