On the higher homotopy groups of a finite CW-complex

l

is

ily

forma-t paper

Topology and its Applications 149 (2005) 273–284

www.elsevier.com/locate/topo

On the higher homotopy groupsof a finite CW-complex

Mihai Damian

Université Louis Pasteur, IRMA, 7, rue René Descartes, 67 084 Strasbourg, France

Received 23 May 2003; received in revised form 1 October 2004; accepted 4 October 2004

Abstract

Let Qq be a finite connected CW-complex of dimensionq � 2 whose fundamental groupAbelian. Denote byβ1(Q) its first Betti number.

We prove that ifβ1(Q) > q thenπi(Q) is not finitely generated for somei = 2, . . . , q. The sameconclusion holds ifχ(Q) �= 0 andπ1(Q) = Z × G whereG is finitely presented (but not necessarAbelian). If π1(Q) is Abelian andβ1(Q) equalsq or q − 1, we show thatπi(Q) is not finitelygenerated for somei = 2, . . . , q unless the case whenπ1(Q) ∼ Zβ1(Q) andQ has the homotopytype of theβ1(Q)-dimensional torus.

If Q is a closed connected manifold, we obtain in the same hypothesis(β1(Q) � q −1) thatπi(Q)

is not finitely generated for somei = 2, . . . ,max{[ q2 ],3} except for the case whenQ is homotopically

equivalent toTq . 2004 Elsevier B.V. All rights reserved.

MSC:57R70; 55P15; 57Q10

Keywords:CW complexes; Novikov homology

1. Introduction

There are not many tools which enable one to compute or at least to get some intion about the higher homotopy groups of a topological space. The aim of the presen

E-mail address:[email protected] (M. Damian).

0166-8641/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.topol.2004.10.002

274 M. Damian / Topology and its Applications 149 (2005) 273–284

unda-roups:d.oups

m

sible”

ticular

r

o-

and aamentalabove.

e

ement.

We

is to show that computable invariants of a finite CW complex, such as dimension, fmental group, or Euler characteristic have some influence on the higher homotopy gwe can infer under suitable hypothesis that not all these groups are finitely generate

One of the few tools available for the general study of the higher homotopy gris Hurewicz’s theorem. It asserts that forq � 2 the canonical (Hurewicz) morphisIq :πq(M) → Hq(M) is an isomorphism provided thatM is (q − 1)-connected.

In [17] Serre generalized this theorem (see also [19, p. 504]): For some “admisclasses of groupsC, he showed that, ifX is simply connected such thatπi(X) ∈ C fori = 1, . . . , q −1, whereq � 2, thenIq is an isomorphisme modC: This means that Ker(Iq)

and Coker(Iq) are inC.The class of finitely generated Abelian groups is such an admissible class. In par

we have:

Theorem 1.1. Let X be a simply connected space. Thenπi(X) is finitely generated foi � q iff Hi(X) is finitely generated fori � q.

In particular any closed, simply connectedCW-complex has finitely generated homtopy groups.

If π1(X) �= 0 the above theorem is false (take for instance the wedge of a circle2-sphere). In the present paper we prove that, under some hypothesis on the fundgroup, a closed nonsimply connected CW-complex does not satisfy the assertion 1.1The statement is:

Theorem 1.2. Letq � 2 and letQ be a finite connectedCW complex of dimensionq. Sup-pose thatπi(Q) are finitely generated for alli (or, equivalently, fori = 1, . . . , q, by 1.1).We have:

(a) Suppose thatπ1(Q) is Abelian. Then:(1) β1(Q) � q.(2) If β1(Q) ∈ {q − 1, q}, thenπ1(Q) ≈ Zβ1(Q) andQ has the homotopy type of th

torusTβ1(Q).(b) If π1(Q) = Z × G thenχ(Q) = 0.

Remark. We cannot drop the assumption on the fundamental group in the above statIf Q is homotopically equivalent to the wedge of a finite number of circles thenχ(Q) �= 0andπi(Q) vanish fori � 2.

If Q is a manifold, the hypothesis on the homotopy groups can be weakened:

Theorem 1.3.

(a) Let Mn be a closed connected manifold whose homotopy groupsπi(M) are finitelygenerated fori = 1, . . . ,max{[n

2],3} and whose fundamental group is Abelian.have then:(1) β1(M) � n andβ1(M) �= n − 1.

M. Damian / Topology and its Applications 149 (2005) 273–284 275

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(a)(1).

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r-hd.

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(2) If β1(M) = n, thenπ1(M) ≈ Zn andM is homeomorphic toTn.(b) If the groupsπi(M) are finitely generated fori = 2, . . . , [n

2] and if π1(M) = Z × G,whereG is finitely presented, thenχ(M) = 0.

Remark on the proofs. In order to prove 1.2(a) and 1.3(a) we show that under the ghypothesis on the higher homotopy groups, ifβ1(M) � n − 1, thenM is necessarily anEilenberg–McLane spaceK(π1(M)). Its dimension is therefore bounded below byhomomological dimension ofπ1(M) which easily implies the two statements.

Example 1. A compact connected CW complex which is aH -space has finitely generatehomotopy groups and Abelian fundamental group (see [19, p. 510, Theorem 20]). Itsmology algebra with coefficients in a field of characteristic 0 is known to be isomorphthe cohomology algebra of a product of spheres [19, p. 269]: this easily implies 1.2Using 1.2(a)(2), we get the following

Corollary 1.4. Let Qq be a H-space withβ1(Q) ∈ {q − 1, q}. ThenQ is homotopicallyequivalent to theβ1(Q)-dimensional torus.

Example 2. The hypothesis of 1.3 are satisfied whenMn admits a metric with nonnegativscalar curvature. Our result is in this case a direct consequence of the “soul” theoGromoll and Cheeger [3]:

Theorem 1.5. If the closed manifoldMn admits a metric with nonnegative scalar cuvature, then there is a finite cover ofM which is isometrically diffeomorphic witM0 × Tβ1(M) endowed with a product metric, whereM0 is compact and simply connecte

The paper is organized as follows. In Section 2 we present the idea of the proof astate the main tool, Theorem 2.1. We show that 2.1 implies our main results 1.3 andSection 3 we recall the definition and the main properties of Novikov homology. FinaSection 4 we give the proof of 2.1, completing thus the proof of 1.3 and 1.2.

2. Idea of the proof

The proof of 1.2(a) and 1.3(a) is based on the existence of a fibration of a manifoM

over the circle when the homotopy groups ofM are supposed to be finitely generated. Tresult is:

Theorem 2.1. Let Mn be a closed connected manifold of dimensionn and G a fi-nitely presented group. Suppose thatπi(M) are finitely generated fori = 2, . . . , [n

2] andπ1(M) = Z × G. Suppose also thatWh(π1(M)) = 0.

Then, ifn � 6 there exists a fibrationf :M → S1, which induces the projection on thfirst factor at the level of fundamental groups. ForM of arbitrary dimensionn, there existsa fibrationf :M × Sp → S1 for p � 6− n.

In particular, the same conclusion holds forπ1(M) = Zm, m > 0.


.

y3.

lies

pose

t a

the

c-

up is

If the assumption onWh(π1(M)) is dropped, we only haveχ(M) = 0. The latter affir-mation is true for any dimensionn.

Remarks.

(1) HereWh(π1) denotes the Whitehead group associated toπ1 (see [10] for a definition)A theorem of Bass et al. [1] asserts thatWh(Zm) = 0 for anym.

(2) Forπ1(M) = Z andπi(M) finitely generated for alli the result above was proved bBrowder and Levine in [2]. We will discuss about other related results in Section

The proof of 2.1 will be given in Section 4. Let us show now how this theorem impthe statement (a) of our main results 1.3 and 1.2.

Proof of 2.1 ⇒ 1.3. Without restricting the generality of our statements, we may supthatn � 3. Suppose now thatπ1(M) is Abelian and denotem = β1(M). Let M0 be a finitecover ofM with π1(M0) = Zm.

(1) Suppose first thatn � 6. Using 2.1, we get a fibrationf :M0 → S1. Let F1 be afiber off . ThenF1 is connected (and closed) with fundamental groupπ1(F1) = Zm−1 andπi(F1) ∼ πi(M0) ∀i � 2. So, if m − 1 > 0 andn − 1 � 6, the fiberF1 still satisfies thehypothesis of 2.1. Suppose now thatm � n − 1. We apply successively 2.1 and we gesequence of closed connected manifolds

Fn−5 ↪→ Fn−6 ↪→ ·· · ↪→ F1 ↪→ M0 (1)

such that fork = 0, . . . , n − 1:

(1) π1(Fk) = Zm−k .(2) dim(Fk) = n − k.(3) The inclusionFk ↪→ M0 induces isomorphisms inπi for i � 2.

In particular the manifoldFn−5 is of dimension 5 and its fundamental group isZm−n+5

and its higher homotopy groupsπi are finitely generated fori = 1, . . . ,max{[n2],3}.

In order to apply 2.1 to this manifold, we consider the productF = Fn−5 × S3.By 2.1, we get a fibration of the product manifoldF over S1. Denote byG0 its fiberwhich is a closed connected manifold of dimension 7 with finitely generatedπi fori = 1, . . . ,max{[n

2],3} (sinceS3 has the same property by 1.1). Since[7/2] = 3, we mayapply 2.1 toG0 and then again to its fiberG1 to obtain a sequence as above, wheremaps are inclusions of fibrations over the circle:

G2 ↪→ G1 ↪→ G0 ↪→ F.

It follows that the universal covers ofG2 andF are homotopically equivalent, in partiular

H∗(G2

) ∼ H∗(Fn−5 × S3). (2)

Now G2 is a closed connected 5-dimensional manifold whose fundamental groZm−n+2. SinceG2 is not simply connected (as we supposed thatm � n − 1), we havethatHi(G2) vanishes fori > 4. Indeed, we have


of

e cir-up is

-

folds:

th-

we

H 0(G2) = HomZ[π1(G2)](H0

(G2

),Z

[π1(G2)

])≈ HomZ[π1(G2)](Z,Z

[π1(G2)

]) = 0,

and we infer thatH5(G2) = 0 by Poincaré duality.Using the Kunneth formula we infer from (2) thatHi(Fn−5) vanishes fori > 0. There-

fore Fn−5 is contractible and, using the sequence (1)M0 = M is contractible, too. SoM is an Eilenberg–McLane spaceK(π1(M)) ; In particular the homological dimensionπ1(M) equalsn. This is clearly impossible (and contradicts thus our assumptionm � n−1)unlessm = n andπ1(M) = Zn. In the latter caseM has the homotopy type of then-torus.By a theorem of Farell and Jones [6] it follows forn � 5 thatMn is homeomorphic toTn.

(2) We treat now the casesn = 3,4,5. Consider the productF = M0 × S3. As above,we can successively apply 2.1 to get a sequence

Gn−2 ↪→ Gn−3 ↪→ ·· · ↪→ G1 ↪→ F,

where the maps are inclusions of the fiber into the total space of fibrations over thcle. The manifoldGn−2 is closed connected of dimension 5 and its fundamental groZm−n+2. We use it to finish the proof using the same argument as above.�

Using again Theorem 2.1 (whose proof is postponed in Section 4), we give the:

Proof of Theorem 1.2. Suppose first thatπ1(Q) is Abelian andβ1(Q) = q + p withp > 0. LetQ0 be the finite cover ofQ whose fundamental group isZq+p .

EmbedQ0 in an Euclidean spaceR2q+r+1 for somer � 0. Let W be a tubular neighbourhood ofQ and denote byM2q+r the smooth manifold∂W . SinceM is a deformationretract ofW \ Q, using a general position argument we get isomorphisms betweenπi(W)

andπi(M) for i � q + r − 1. In particular, forr > 1 the inclusionM ↪→ W induces anisomorphism at the level of fundamental groups and, sinceQ is a retract ofW , the higherhomotopy groupsπi(M) are finitely generated fori � q + r − 1. If r is large enough (tohaveq + r − 1� [2q+r

2 ], then the hypothesis of 2.1 is satisfied. We obtain then forr largeenough (r � p − q + 5), as in the proof of 1.3, a sequence of closed connected mani

Fq+p ↪→ Fq+r−p−1 ↪→ ·· · ↪→ F1 ↪→ M (1)

such that fork = 0, . . . , q + r − p:

(1) π1(Fk) = Zq+p−k .(2) dim(Fk) = 2q + r − k.(3) The inclusionFk ↪→ M induces isomorphisms inπi for i � 2.

The constantr � 1 must be chosen large enough to have:

(a) dim(Fk) � 5∀k, which meansq + r − p � 5 (this is needed for the dimension hypoesis in 2.1).

(b) q + r − 1 � [2q+r2 ], for the hypothesis on the higher homotopy groups in 2.1, as

explained above.(c) r > p.


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poth-

It follows that the first manifoldFq+p in the sequence (1) is closed and simply cnected of dimensionq + r − p. Using the condition (3) we find that the composapplication

Fq+p ↪→ M ↪→ W → Q0 (2)

induces isomorphisms at the level of the higher homotopy groupsπi for i = 2, . . . ,

q+r −1. By Whitehead’s theorem, the induced application between the universal covFq+p andQ0 is an isomorphism at the level of theith homology groups fori � q + r − 1.In particular, sinceq + r − 1� q + r − p, we have

Hq+r−p

(Fq+p

) ≈ Hq+r−p

(Q0

). (3)

But Fp+q is simply connected of dimensionq + r − p, so the left side of (3) isZ. On theother handq + r − p > q = dim(Q0), so the right side of (3) vanishes. This contradithe initial assumptionp � 1 and the first statement of the theorem is proved.

Suppose now thatβ1(Q) = q + p with p = 0,−1. As above we get a closed simpconnected manifoldFq+p of dimensionq + r − p such that the applicationHi(Fq+p) →Hi(Q0) is an isomorphism fori � q + r − 1. In particular, the homologyHi(Fq+p;Z)

vanishes fori = q + 1, . . . , q + r − 1.By the universal coefficients theorem we obtain that the cohomology

Hi(Fq+p;Z) = 0

for i = q + 2, . . . , q + r − 1, so Poincaré duality implies thatHi(Fq+p;Z) also vanishesfor i = −p + 1, . . . , r − p − 2. We infer (if r is chosen sufficiently large) that the integcohomology ofFq+p vanishes in the degreesi � q which means thatQ0 is contractible.ThereforeQ0 andQ are Eilenberg–Mac Lane spaces. In particular the cohomologicamension ofπ1(Q) cannot exceedq, which implies thatπ1(Q) is free Abelian and thereforQ has the homotopy type of the torusTβ1(Q).

Finally, suppose thatπ1(Q) = Z × G. Constructing the manifoldM as above, we haveif r is even:

χ(Q0) = χ(W) = 2χ(M)

by Poincaré duality. Ifr is large enough, we can apply Theorem 2.1 and obtain thatχ(M)

vanishes. This completes the proof.�3. Novikov homology and fibrations over the circle

If M is the total space of a fibrationf over the circle, thenf ∗dθ is a nonvanishingclosed one form onM . An analogue of Morse theory for circle valued functions westablished by Novikov in [11]. In the subsequent Morse–Novikov inequalities usuamology is replaced by the Novikov homology ofM associated to the cohomology clau := [f ∗dθ ]. So, iff :M → S1 is a fibration, the Novikov homologyH∗(M, [f ∗dθ ]) willvanish.

Conversely, Latour [8] and Pajitnov [12,13] proved that under some additional hyesis the vanishing of the Novikov homology implies the existence of a fibration ofM overthe circle. We will use this result in the proof of 2.1. It will be stated in Section 3.2.

Let us start by recalling the definition and some properties of Novikov homology:


on

n of

logy

um

gs

ho-

re

3.1. Novikov homology

Let u ∈ H 1(M;R). Denote byΛ the ringZ[π1(M)] and byΛ the ring of formal seriesZ[[π1(M)]]. Consider aC1-triangulation ofM which we lift it to the universal coverM .We get aΛ-free complexC•(M) spanned by (fixed lifts of) the cells of the triangulatiof M .

We define nowthe completed ringΛu:

Λu :={λ =

∑nigi ∈ Λ | gi ∈ π1(M), ni ∈ Z, u(gi) → +∞

}.

The convergence to+∞ means here that for allA > 0, u(gi) < A only for a finite numberof gi which appear with a nonzero coefficient in the sumλ.

Definition. Let C•(M,u) be theΛu-free complexΛu ⊗Λ C•(M). The Novikov homologyH∗(M,u) is the homology of the complexC•(M,u).

A purely algebraic consequence of the previous definition is the following versiothe universal coefficients theorem [7, p. 102, Theorem 5.5.1]):

Theorem 3.1. There is a spectral sequenceErpq which converges toH∗(M,u) and such

that

E2pq = TorΛp

(Λu,Hq

(M

)).

We will use this result to prove that in the hypothesis of 2.1, the Novikov homoassociated to some classu vanishes.

3.2. Morse–Novikov theory

Let α be a closed generic one form in the classu. Let ξ be the gradient ofα with respectto some generic metric onM . For every critical pointc of α we fix a pointc abovec in theuniversal coverM . We can define then a complexC•(α, ξ) spanned by the zeros ofα: theincidence number[d, c] for two zeros of consecutive indices is the (possibly infinite) s∑

nigi whereni is the algebraic number of flow lines which joinc andd and which arecovered by a path inM joining gi c andd . It turns out that this incidence number belonto Λu, soC•(α, ξ) is actually aΛu-free complex.

The fundamental property of the Novikov homology is that it is isomorphic to themology of the complexC•(α, ξ) above for any couple(α, ξ).

By comparing the complexesC•(α, ξ) andC•(−α,−ξ) we get the following dualityproperty (see Proposition 2.8 in [4] and 2.30 in [8]):

Theorem 3.2. Let Mn be a closed connected manifold,u ∈ H 1(M;R) and let l be aninteger. IfHi(M,−u) = 0 for i � l, thenHi(M,u) = 0 for i � n − l.

If the formα has no zeroes thenC•(α, ξ) vanishes and therefore we haveH∗(M, [α]) =0. Conversely, one can ask if the vanishing ofH∗(M,u) implies the existence of a nowhe


-1’]):

nce

ebraiced 1-ite

from] (see

s a

t

the

vanishing closed 1-form belonging to the classu ∈ H 1(M). Forn � 6 this problem was independently solved by Latour [8] and Pajitnov [12,13]. The statement is [8, Theorem

Theorem 3.3. For dim(M) � 6 the following set of conditions is equivalent to the existeof a nowhere vanishing closed1-form inu ∈ H 1(M,Z):

(1) Vanishing Novikov homologyH∗(M,u).(2) Vanishing Whitehead torsionτ(M,u) ∈ Wh(M,u).(3) Finitely presentedKer(u) ⊂ π1(M).

Remarks.

(1) The definition of the generalized Whitehead groupWh(M,u) and of the Whiteheadtorsion is given in [8].

(2) In the statement of [12] the first two conditions are replaced by:1’ C•(M,u) is simply equivalent to zero.

Actually, one can show (see [9]) that 1’ is equivalent to “1 and 2”.(3) In earlier works on the subject as those of Farell [5] and Siebenmann [18] the alg

conditions which are equivalent to the existence of a nowhere vanishing closform in a rational cohomology classu were stated in the hypothesis that the infincyclic coverX associated tou is finitely dominated. (X → M is defined to be thepull-back of the universal coveringR → S1 defined by a functionf :M → S1 suchthat [f ∗dθ ] = u). The relation between the finite domination ofX and vanishing ofthe Novikov homology (as well as between the Whitehead “fibering obstruction”[5] and [18] and the condition 2 above) was first established by Ranicki in [15,16also [14]).

The hypothesis onπi(M) in our Theorem 2.1 can be seen in this framework asufficient condition for thealgebraic finite domination of theZ[t, t−1]-free complexC•�[ n

2 ](X).

In order to prove 2.1 we will use a particular case of 3.3, namely:

Theorem 3.4. Let Mn be a closed connected manifold of dimensionn � 6. Suppose thaπ1(M) = Z × G and Wh(π1(M)) = 0.

Let u be the cohomology class which corresponds to the projectionG × Z → Z in thenatural isomorphismH 1(M;Z) ∼ Hom(π1(M),Z). Suppose thatH∗(M,u) vanishes.

Then there exists a fibrationf :M → S1 such that[f ∗dθ ] = u.

In the weaker statement 3.4 the third condition of 3.3 is obviously fulfilled andsecond follows from the work of Ranicki [14, Chapters 5, 14 and 15].


ishesng

nd part

bove.t

4. Proof of 2.1

It suffices to prove the following propositions:

Proposition 4.1. Let Mn be a closed connected manifold. Suppose thatπi(M) are finitelygenerated fori = 2, . . . , [n

2] andπ1(M) = Z × G. ThenH∗(M,u) = 0, whereu = prZ.

Remark. There is a theorem of Ranicki [16] (see also [14, Chapter 8]) which establan equivalence between the algebraic finite domination of the infinite cyclic coveriX

and the vanishing of the Novikov-type homologyH∗(C•(X) ⊗Z[t,t−1] Z(t)[t−1]. As ourhypothesis onπi(M) has an influence on the algebraic finite domination ofX this result isstrongly related to 4.1.

Proposition 4.2. If H∗(M,u) = 0 for some classu thenχ(M) = 0.

It is easy to see that 4.1 and 3.4 imply the first part of the statement 2.1. The secois an immediate consequence of 4.1 and 4.2.

Proof of Proposition 4.1. By 3.2 it suffices to prove thatHi(M,±u) = 0 for i � [n2]. The

proof is based on a Cayley–Hamilton argument as in the result of [14] mentioned aThe idea is to find a endomorphism�Φ of Hi(M,u) which is both invertible and nilpotenfor i � [n

2]. The morphism�Φ will be induced by a chain morphism

Φ = Id ⊗ ψ :Λu ⊗Λ C•(M) → Λu ⊗Λ C•(M).

Denote byg the elementt × 1 ∈ π1(M), where t is the generator ofZ the morphismψ :C•(M) → C•(M) will be the left multiplication byP(g) whereP is a polynomial withinteger coefficients. Here are the details:

As g ∈ Z(π1(M)) the left multiplication byg induces a isomorphism ofΛ-complexes

g :C•(M) → C•(M),

which yields aΛ-isomorphism in homology

g :

[ n2 ]⊕

i=0

Hi

(M

) →[ n

2 ]⊕i=0

Hi

(M

).

We will use the following lemma:

Lemma 4.3. There exists some polynomialP(X) ∈ Z[X], such that:

(A) The morphismP(g) is zero.(B) The morphism Id⊗ P(g) :Λu ⊗Λ C•(M) → Λu ⊗Λ C•(M) is invertible.

Proof. (A) Denote byA the sum⊕[ n

2 ]i=0 Hi(M). Because of 1.1 we have thatA is finitely

generated as aZ-module. ThereforeA is isomorphic toZr ⊕ T , whereT is a direct sumof modulesZ/kZ (k ∈ N∗).


no-

e-

.

The restrictiong : Zr → Zr is an isomorphism and therefore its characteristic polymial is of the form

R(X) = ±1+ XR0(X).

This property will be useful for the proof of (B). The restrictiong :T → T is again anisomorphism and, since Aut(T ) is finite, there is an integerp such that(gp)|T = Id.

The polynomialP(X) = (1− Xp)R(X) has the desired property.(B) SinceP(X) = ±1+ XP0(X), the elementP(g) ∈ Λ is a unit ofΛu: its inverse is

λ0 = ±(

1+∑k�1

(±1)k(gP0(g)

)k)

.

It is easy to check that the morphism defined byλ ⊗ m → λ0λ ⊗ m is an inverse forId ⊗ P(g). The fact thatg is an element of the centerZ(π1(M)) is important here. �

We finish now the proof of 4.1. Denote byΦ the chain homomorphism

Id ⊗ P(g) :Λu ⊗Λ C•(M) → Λu ⊗Λ C•(M).

It follows from 4.3(B) that the morphism�Φ = Id ⊗ P(g) induced byΦ is an automor-phism ofH∗(M,u). On the other hand, the following lemma shows that�Φ is nilpotent indegrees lower than[n

2], thereforeHi(M,u) must vanish fori � [n2].

Lemma 4.4. There exist an integers such that

�Φs :Hi(M,u) → Hi(M,u)

vanishes fori � [n2].

Proof. Consider a polynomialP as in 4.3. The chain morphismP(g) :C•(M) → C•(M)

naturally induces a morphism

TorΛp(Id,P (g)q

): TorΛp

(Λu,Hq

(M

)) → TorΛp(Λu,Hq

(M

))which, by 4.3(A), is zero forq � [n

2]. By naturality in the spectral sequenceErpq of 3.1 we

obtain that the induced endomorphism�Φ∞pq :E∞

pq → E∞pq vanishes forq � [n

2]. Now recallthat by 3.1 the first-quadrant spectral sequenceEpq converges toH∗(M,u). We have thusa filtration

0= A−1 ↪→ A0 ↪→ A1 ↪→ ·· · ↪→ Ak ↪→ ·· ·of the Novikov homology groupsH∗(M,u) such that for every couple of integers(i,p)

there is an exact sequence:

0→ Ap−1(Hi(M,u)

) → Ap

(Hi(M,u)

) → E∞p,i−p → 0. (4)

Actually, the above filtration is stationary fork � n, as one can infer from the exact squence (4).

We claim that, by naturality, the morphism�Φ :H∗(M,u) → H∗(M,u) preserves thefiltration and yields, together withΦ∞, a morphism from the exact complex (4) to itself


,ftedlogy

d by

nd

t

a

ism

nt

.

To see this one has to consider aΛ-free resolutionR• of Λu (as in the proof of 3.1see [7]). We define the bigraduated complexK• = R• ⊗Λ C•(M). Using the graduation oR• we get the filtrationBk = ⊕

s�k Rs ⊗ C•(M) for K•. The spectral sequence associato this filtration is exactlyEpq , which we considered above. It converges to the homoof K•. Since the morphismId ⊗ P(g) :K• → K• obviously preserves the filtration(Bk),it induces an endomorphism ofH∗(K•) which preserves the induced filtration(Ak) inhomology. By naturality, this endomorphism is compatible with the morphism induce�Φ∞ which is also induced byId ⊗ P(g).

On the other hand, considering the filtration ofK• given by the graduation of the secofactor we get another spectral sequenceEpq converging to the same limit such thatE2

pq iszero except forp = 0 and

E20q = Hq(M,u).

Note that the morphismId ⊗ P(g) induces�Φ at the level ofE20q ≈ E∞

0q . This means thathe filtration preserving endomorphism ofH∗(K•) which is induced byId ⊗ P(g) is ex-actly �Φ under the identificationH∗(K•) ≈ H∗(M,u) given byEpq . Therefore we havecommutative diagram:

0 Ap−1(Hi(M,u))

�Φ

Ap(Hi(M,u))

�Φ

E∞p,i−p

�Φ∞p,i−p

0

0 Ap−1(Hi(M,u)) Ap(Hi(M,u)) E∞p,i−p 0

Fix some i � [n2]. By 4.3(A) the vertical arrow�Φ∞ vanishes for everyp. Pro-

ceeding by induction onp, we easily infer from the above diagram that the morph�Φ :Ap(Hi(M,u)) → Ap(Hi(M,u)) is nilpotent.

Finally, for p = n, An(Hi(M,u)) = Hi(M,u), so the morphism�Φ is nilpotent inNovikov homology fori � [n

2]. �It follows that �Φ = Id ⊗ P(g) : Hi(M,u) → Hi(M,u) is nilpotent fori � [n

2] and, onthe other hand, by 4.4(B) the same morphism is onto. This means thatHi(M,u) vanishesfor i � [n

2].An analogous proof shows thatHi(M,−u) = 0 for i � [n

2]. Indeed, the same argumeyields a polynomialQ as in 4.3 above, which satisfies the conditions A and B withg−1

instead ofg. As above, we get an endomorphism ofHi(M,−u) which is both nilpotentand onto.

This completes the proof of 4.1 (using the duality 3.2).�Proof of Proposition 4.2. We call aelementary trivial complexa complex

0→ Ci+1Id−→ Ci → 0

with Ci ≈ Ci+1 ≈ Λu and atrivial complexa finite sum of elementary trivial complexesIf H∗(M,u) vanishes, one can show, as in [8], that the complexC•(M,u) = Λu ⊗Λ

C•(M) is isomorphic to a complex of the form

0→ Λru → Λr

u → 0,


c ofc-

ce

..Math.

Norm.

–740.rential

e Rham,

7.

Math.

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tical

94.1970)

modulo stabilization with trivial complexes. It follows that the Euler characteristiC•(M,u) vanishes. But the construction ofC•(M,u) shows that this characteristic is atually χ(M). �Remark. In the proof of 4.1 the only property ofπ1(M) which is needed is the existenof an elementg ∈ Z(π1(M)) and of a morphismu :π1(M) → R with u(g) �= 0. Thereforeone can sharpen the statement 4.1 in:

Theorem 4.5. Let Mn be a closed connected manifold. Suppose thatπi(M) are finitelygenerated fori = 2, . . . , [n

2] and that there exists an elementg ∈ Z(π1(M)) and a mor-phismu :π1(M) → R with u(g) �= 0. ThenH∗(M,u) = 0 andχ(M) = 0.

References

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(2) 96 (1972) 413–443.[4] M. Damian, Formes fermées non singulières et propriétés de finitude des groupes, Ann. Sci. École

Sup. 33 (3) (2000) 301–320.[5] F.T. Farell, The obstruction to fibering a manifold over the circle, Bull. Amer. Math. Soc. 73 (1967) 737[6] F.T. Farell, L.E. Jones, Stable pseudoisotopy spaces of non-positively curved manifolds, J. Diffe

Geom. 38 (1991) 769–834.[7] R. Godement, Topologie Algébrique et Théorie des Faisceaux, Hermann, Paris, 1958.[8] F. Latour, Existence de 1-Formes Fermées Non-Singulières dans une Classe de Cohomologie d

Publ. Math. IHES, vol. 80, 1994.[9] S. Maumary, Type Simple d’Homotopie, Lecture Notes in Mathematics, vol. 48, Springer, Berlin, 196

[10] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358–426.[11] S.P. Novikov, Multivalued functions and functionals. An analogue of the Morse theory, Soviet.

Dokl. 24 (2) (1981) 222–226.[12] A. Pajitnov, Surgery on the Novikov complex,K-Theory 10 (1996) 323–412.[13] A. Pajitnov, Surgery on the Novikov complex, Rapport de Recherche, Nantes, 1993,

193.52.98.6/~pajitnov.[14] A. Ranicki, High-Dimensional Knot Theory, Springer, Berlin, 1998.[15] A. Ranicki, LowerK- andL-Theory, London Mathematical Society Lecture Notes, London Mathema

Society, Cambridge, 1992.[16] A. Ranicki, Finite domination and Novikov rings, Topology 34 (1995) 619–632.[17] J.P. Serre, Groupes d’homotopie et classes de groupes Abéliens, Ann. of Math. (2) 58 (1953) 258–2[18] L. Siebenmann, A total Whitehead obstruction to fibering over the circle, Comment. Math. Helv. 45 (

1–48.[19] E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.

On the higher homotopy groups of a finite CW-complex

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