On the homotopy of finite CW-complexes with polycyclic fundamental group

Let X be a finite CW-complex of dimension q. If its fundamental group $\pi_{1}(X)$ is polycyclic of Hirsch number h>q we show that at least one of the homotopy groups $\pi_{i}(X)$ is not finitely generated. If h=q or h=q-1 the same conclusion holds unless X is an Eilenberg-McLane space $K(\pi_{1}(X),1)$.


Introduction
Let X be a finite connected CW-complex of dimension q. Consider the homotopy groups π i (X) = [S i , X], for i ≥ 2. If all these (Abelian) groups are finitely generated we say that the homotopy of X is finitely generated. For simply connected complexes, a celebrated theorem of Serre [30] asserts that Theorem 1.1 If π 1 (X) = 1 the homotopy of X is finitely generated.
On the other hand simple examples as X = S 1 ∨ S 2 show that the homotopy of X is not always finitely generated.
One can then ask if there is a general negative statement asserting that the homotopy of X is not finitely generated under some hypothesis on computable invariants of X, such as its fundamental group. This is the aim of the present paper. The statements we will prove have the following form : If the fundamental group of X satisfies to the conditions (C) (which depend on the dimension q), then the homotopy of X is not finitely generated unless X is an Eilenberg-Mc.Lane space K(π 1 (X)).
If X is an Eilenberg-Mc Lane space, the cohomological dimension of π 1 (X) is less or equal than q = dim(X). So, we may add cd(π 1 (X)) > q to the hypothesis (C) in order to have the desired conclusion on the homotopy of X. For instance when π 1 (X) has non trivial torsion elements then cd(π 1 (X)) is infinite and the above conclusion is valid.

Statement of the results
Before stating the main theorem we remind the following : Definition. A group G is called polycyclic if it admits a series 1 = G 0 < G 1 < · · · < G k = G with cyclic factors G i+1 /G i . When all the factor groups are infinite cyclic we call G a poly-Z group. K.A. Hirsch proved in [14] that the number of infinite cyclic factors in such a series is an invariant of G. It is called Hirsch number of G and it is denoted by h(G).
Here are our main statements : Suppose that X is a finite connected CW complex of dimension q and that π 1 (X) is polycyclic. Then a) If h(π 1 (X)) > q, the homotopy of X is not finitely generated. b) If h(π 1 (X)) ∈ {q − 1, q} then the homotopy of X is not finitely generated unless X is a K(π 1 (X), 1). In particular, when π 1 (X) has torsion then the conclusion of a) holds.
Note that, by a result of J-P. Serre (see 4.2 below), when the homotopy of X is not finitely generated at least one of the groups π 2 (X), π 3 (X), . . . , π q (X) is not finitely generated. When X is a manifold we can improve this result : Theorem 1.3 Let M n be a closed manifold. Let r = max n 2 , 3 . Suppose that π 1 (M) is polycyclic. Then, if h(π 1 (M)) ≥ n − 1, the group π i (M) is not finitely generated for some i ≤ r unless h(π 1 (M)) = n and M is a K(π 1 , 1).
Moreover, if n ≥ 6 we can replace the assertion "M is a K(π 1 , 1)" by the stronger one "the universal cover of M is diffeomorphic to R n ".
When n = 4, we are able to prove the above statement for r = 2, namely : Theorem 1.4 Let M 4 be a closed connected manifold with polycyclic fundamental group. If h(π 1 (M)) ≥ 3 then π 2 (M) is not finitely generated unless h(π 1 (M)) = 4 and M is a K(π 1 , 1).
In the next theorem we weaken the hypothesis on the fundamental group but we suppose in addition that χ(X) = 0. Recall first the finiteness properties of a group, which were introduced by C.T.C. Wall in [35] : Definition. Let r ≥ 1 be an integer. A groupe G is of type F r if there is an Eilenberg-McLane space K(G, 1) whose r-skeleton has a finite number of cells. Equivalently a group is of type F r if it acts freely, properly, cellularly and cocompactly on an (r − 1)-connected cell complex.
A group is of type F ∞ if it is of type F r for any integer r > 0.

Remark.
A group G is of type F 1 if and only if it is finitely generated. G is of type F 2 if and only if it is finitely presented.
Theorem 1.5 Let X q be a finite connected CW-complex of dimension q with fundamental group of type F q+1 . Suppose that there is a non vanishing morphism u : π 1 (X) → Z such that Ker(u) is of type F q+1 . Then, if χ(X) = 0, the homotopy of X is not finitely generated. If X is a manifold with non-zero Euler characteristic we obtain the same holds for π 1 (X) and The hypothesis on u may be reformulated in terms of the Bieri-Renz invariants of π 1 (X) as follows : The Bieri-Renz invariants Σ i (G) of a group G are open subsets of the unit sphere of R rk(G) . We recall the definition and the properties of these invariants in Section 4.
In the next subsection we show that polycyclic fundamental groups satisfy the hypothesis of 1.5 and we give other examples of groups for which the condition ( * ) is fulfilled.

Comments on the results
An Abelian group is obviously polycyclic of Hirsch number equal to its rank. It is also of type F ∞ . For Abelian fundamental groups the results of the previous subsection were proved by the author in [9]. Here are other remarks about these statements.
Remarks. 1. The lower bound q − 1 for h(π 1 ) in the hypothesis of 1.2 is optimal. Indeed, the complex X = T n−2 × S 2 fulfills the conditions of the hypothesis of 1.2 except for h(π 1 (X)) = q − 2. It is a consequence of the theorem 1.1 of J.P.Serre, quoted above, that X does not satisfy to the conclusion of 1.2.
2. If G is a polycyclic group then [G, G] is obviously polycyclic and therefore for every morphism u : G → Z, Ker(u) has the same property. This implies that the hypothesis on the fundamental group of 1.5 is fulfilled when π 1 (X) is polycyclic. Indeed we have :

Proof
The proposition is an immediate corollary of the following : Lemma 1.7 Consider an exact sequence of groups If K and H are of type F ∞ then so is G.
To prove the above lemma one may use the following results : a) G is F ∞ iff H * (G, ·) commutes with direct products. This was proved by R. Bieri and B. Eckmann in [2]. b) The Hochshild-Serre spectral secquence [16] which satisfies and converges towards H * (G, R). (The complete statement 4.6 is given in Section 4). It is clear how a) and b) imply 1.7 proving thus 1.6. ⋄

3.
Other examples of groups satisfying the hypothesis of 1.5 were constructed by M. Bestvina, N. Brady in [1] and then in a more general statement by J.
Meier, H. Meinert and L. Van Wyk in [20]. Let us describe them briefly : Definition. A flag complex L is a finite simplicial complex with the property that any collection of q + 1 mutually adjacent vertices span a q-simplex in L. The right angled Artin group associated to L is the group G L spanned by the vertices v 1 , . . . , v s of L with relations [v i , v j ] = 1 whenever v i and v j are adjacent in L.
Note that the group G L admits a finite Eilenberg-McLane space [1] and therefore it is of type F ∞ . We have Theorem 1.8 (Bestvina, Brady, Meier, Meinert, van Wyk) Let L and G L as above and consider a morphism u : Considering appropriate flag complexes L (remark that the barycentric subdivision of any complex is a flag complex) we may find a lot of examples of groups which fulfil the hypothesis of 1.5.

Idea of the proof
The general idea of the proof of 1.3 is the following : we show first that a manifold M as in 1.3 whose dimension is greater or equal to 6 and whose homotopy groups π i (M) are of finite type for i ≤ r admits a fibration over the circle. This is the main difficulty of the proof. To overcome it, we will use Novikov homology theory and some of its applications on Morse functions f : M → S 1 .
Remark that if the manifold F n−1 is a fiber, the inclusion j : F ֒→ M induces isomorphisms in π i for i ≥ 2. The lift of j induces a homotopy equivalence between the universal covers of F and M. Now take a manifold M, as in 1.3 and consider the product M × S 3 in order to fulfill the condition on the dimension. Supposing that the homotopy groups π i (M) are of finite type for i ≤ r, apply the previous argument to this product and get a fiber F as above. Then check that F still satisfies the hypothesis of 1.3. If its dimension is still greater or equal than 6 we apply the same argument to F . By succesive iterations we find a manifold F 0 of low dimension whose universal cover has the same homotopy type as the one of M × S 3 . In particular the homology groups of F 0 and M × S 3 are isomorphic. By comparing them, we infer that the universal cover of M is acyclic, which means that M is Eilenberg-McLane. So, either the homotopy of M is not finitely generated, or M is Eilenberg-Mc.Lane, as in the statement of 1.3.
To prove 1.2 we embed X into an Euclidian space and thicken it to a manifold W , with boundary M. Then we apply the above argument to M.
In the hypothesis of 1.5, supposing that the homotopy of X is finitely generated, we are only able to prove that the Novikov homology H * (M, u) of the corresponding manifold M vanishes. But this implies that χ(M) = χ(X) = 0, yielding a contradiction.
The paper is organized as follows. In Section 2 we state the result 2.1 on the fibration over the circle which was roughly sketched above. Supposing 2.1 true, we show how it can be succesively applied in order to prove 1.3, 1.4 and 1.2. In Section 3 we recall the definition and some useful properties of the Novikov homology. We also recall some basic facts about Morse theory of circle-valued functions and point out the relation between the (vanishing of the) Novikov homology and the existence of a fibration over the circle. In Section 4 we prove 2.1 and 1.5. We will use the Bieri-Renz criterion which we recall in the Subsection 4.3, devoted to the Bieri-Renz invariants.

Iterated fibrations
Our main result 1.2 is a consequence of the following : Theorem 2.1 Let M n a closed manifold of dimension n ≥ 6. Suppose that π 1 is of type F [ n 2 ] . Suppose also that π i (M) are of finite type for i ≤ n 2 . Suppose that there is a non zero cohomology class u ∈ H 1 (M; R) ≈ Hom(π 1 (M), R) such that Ker(u) is of type F [ n 2 ] . Suppose also that the Whitehead group W h(π 1 (M)) vanishes.
Then there is a fibration f : When M of arbitrary dimension n and Ker(u) is of type F r , where r = max n 2 , 2 the same conclusion holds if we replace M by M × S p for all p ≥ 6 − n.
Remarks. 1. The Whitehead group is defined as follows : 2. T. Farell and W. Hsiang proved in [11] that W h(π) vanishes when π is poly-Z. The Whitehead group also vanishes when π = G L is one of the examples of Bestvina and Brady. This result was proved by B. Hu [17]. It is conjectured that W h(π) = 0 for any torsion-free group π. 3. For π 1 = Z, Theorem 2.1 was proved by W. Browder and J. Levine in [6].
The proof of 2.1 will be given in Section 4. Let us now show how this theorem implies our main results 1.3, 1.4 and 1.2.

Proof of 2.1 =⇒ 1.3
Without restricting the generality of our statements, we may suppose that n ≥ 3. We begin with the statement of the following result, due to K. Hirsch ([15], Theorem 2). Theorem 2.2 Let G be a polycyclic group. There exists a normal subgroup N of finite index in G which is poly-Z and such that h(N) = h(G). Now, let π 0 1 ≤ π 1 a subgroup as in 2.2 and consider the associated finite cover M 0 → M. Let be a series with infinite cyclic factor groups : we have therefore h(π 1 ) = k, so by hypothesis k ≥ n−1. Denote by u 1 the projection Suppose that π i (M) is finitely generated for i ≤ r = max n 2 , 3 . Consider first the case n ≥ 6. Using 1.6 we find that u 1 fulfills the hypothesis of 2.1. We apply 2.1 and we get a fibration M 0 → S 1 . Let F 1 be a fibre of this fibration. So F 1 satisfies : We wish to apply 2.1 to F 1 . We use the cohomology class u 2 : The hypothesis on the higher homotopy groups of 2.1 is fulfilled by F 1 (because of the condition 2. above), therefore we may apply 2.1 to F 1 if its dimension is no less than 6.
We get thus a closed connected manifold F 2 ⊂ F 1 whose higher homotopy groups are those of M and whose fundamental group is G k−2 . If its dimension is greater or equal than 6 we may again apply 2.1 to the couple (F 2 , u 3 : By iterating this argument we get a secquence (1) F n−5 ֒→ F n−6 ֒→ · · · ֒→ F 1 ֒→ M 0 such that for j = 1, . . . , n − 1 : The manifold F n−5 is of dimension 5, so it does not verify the dimension hypothesis of 2.1. Its fundamental group is G k−n+5 . In order to continue to apply 2.1, we consider the product F = F n−5 × S 3 . As above, we have the cohomology class u k−n+4 : G k−n+5 → G k−n+5 /G k−n+4 which is non zero and has a (poly-Z) kernel of type F [ n 2 ] . By 2.1, we get a fibration of F over S 1 . Its fiber K 0 is a closed connected manifold of dimension 7 with finitely generated π i for i = 1, . . . , r = max n 2 , 3 (since S 3 has the same property by 1.1). Its fundamental group is G k−n+4 . Since [7/2] = 3, we may apply 2.1 to K 0 and then again to its submanifold K 1 given by 2.1, to obtain a sequence as above, where the maps are inclusions of fibers of fibrations over the circle which therefore induce isomorphisms at the level of π i for i ≥ 2 : It follows that the universal covers of K 2 and F are homotopically equivalent, in particular [5], p.346) ; We have therefore that H i ( K 2 ) vanishes for i > 4.
Using the Kunneth formula we infer from (2) that H i ( F n−5 ) vanishes for i > 0. Therefore F n−5 is contractible and, using the sequence (1) . The cohomological dimension of π 1 (M) is therefore equal to n. Now we use the following well-known result (for a proof, see [12], Lemma 8, p. 154) : Applying 2.3 we get that M cannot be a K(π 1 , 1) unless h(π 1 (M)) = n. If this relation is not valid, we get a contradiction and therefore the homotopy of M cannot be finitely generated, completing the proof of 1.3 for n ≥ 6. If it is and if M is a K(π 1 , 1), let us show that its universal cover is R n . Since according to 2.1 M fibers over S 1 , we have To finish the proof we just have to apply a celebrated theorem of J. Stallings [29] which asserts : If a manifold P n≥5 is a product of two open non-trivial contractible manifolds than P is diffeomorphic to R n . Suppose that π 2 (M) is finitely generated. As in the proof of 1.3, use 2.2 and consider a finite cover M 0 with poly-Z fundamental group π 0 1 of Hirsch number greater than 3. We apply first 2.1 to M 0 × S 2 and to the morphism We get a manifold F of dimension 5 which is a fiber of a fibration M 0 → S 1 . Then we apply again 2.1 to the product F × S 2 and the cohomology class which induces homotopic equivalences at the level of the universal covers. In particular Now, as above, since k ≥ 3, the fundamental group G k−2 of K is infinite, so the homology of K vanishes in degrees i > 5. This implies that the homology of M is zero, so M is a K(π 1 , 1). We conclude using 2.3. ⋄ Proof of 2.1 =⇒ 1.3 in the cases n = 3, 5 Suppose that π 2 (M) and π 3 (M) are finitely generated. We apply (n − 2 ) times Theorem 2.1 to the 8-dimensional manifold M × S 8−n . We get a secquence : as above. The manifold K n−2 is of dimension 8 − n + 2 and has an infinite fundamental group. So, H i ( K n−2 ) = 0 for i ≥ 8 − n + 2. Since this homology is isomorphic to H * ( M × S 8−n ), it follows that M is acyclic, and therefore that M is an Eilenberg-Mc.Lane. As above, we use 2.3 to finish the proof. ⋄

Proof of 2.1 =⇒ 1.2
Assume that the homotopy of X is finitely generated. a) Suppose that h(π 1 (X)) = k for some integer k > q. Using 2.2, consider a finite cover X 0 of X such that π 1 (X 0 ) is poly-Z and h(π 1 (X 0 )) = k.
Embed X 0 in an Euclidian space R 2q+r+1 for some r ≥ 0 which will be fixed later in the proof. Let W be a tubular neighbourhood of X 0 and denote by M 2q+r the smooth manifold ∂W . Since M is a deformation retract of W \ X 0 , using a general position argument we get isomorphisms between π i (W ) and π i (M) for i ≤ q + r − 1. In particular, for r > 1 the inclusion M ֒→ W induces an isomorphism at the level of fundamental groups and, since X 0 is a retract of W , the higher homotopy groups π i (M) are finitely generated for i ≤ q +r −1. If r ≥ 1 then q +r −1 ≥ 2q+r 2 and the hypothesis on the higher homotopy groups of 2.1 is satisfied by the manifold M. Let be a series with infinite cyclic factor groups. For r large enough (r ≥ k − 2q + 5), we may apply k times 2.1, as in the proof of 1.3, and get a sequence of closed connected manifolds : The constant r ≥ 1 must be chosen large enough to have : a) dim(F j ) ≥ 5 ∀j : needed for the dimension hypothesis in 2.1 (this means 2q + r − k ≥ 5). b) r + q − 1 ≥ 2q+r 2 , (which is true for r ≥ 1) to insure the hypothesis on the higher homotopy groups in 2.1, as we explained above. c) r > k − q.
It follows that the first manifold F k in the sequence (1) is closed and simply connected, of dimension 2q + r − k > q. Using the property 3 we find that the composition (4) F k ֒→ M ֒→ W → X 0 induces isomorphisms at the level of the higher homotopy groups π i for i = 2, . . . , q + r − 1. By Whitehead's theorem, the induced application between the universal covers of F k and X 0 is an isomorphism at the level of the i th homology group for i ≤ q + r − 1. In particular, since q + r − 1 ≥ 2q + r − k (since by hypothesis k > q), we have But F k is simply connected of dimension 2q + r − k, so the left side of (5) is Z. On the other hand 2q + r − k > q = dim(X 0 ), so the right side of (5) vanishes. This contradicts the initial assumption on the finite generation of the homotopy of X and the statement a) of the theorem is proved. b) Suppose now that k = h(π 1 (X)) ∈ {q−1, q}. As above we get a closed simply connected manifold F k of dimension 2q+r−k (i.e. q+r+1 or q+r) such that the application In particular, the homology H i (F k ; Z) vanishes for i = q + 1, . . . , q + r − 1.
By the univesal coefficients theorem we obtain that the cohomology H i (F k ; Z) is zero for i = q + 2, . . . , q + r − 1, so Poincaré duality implies that H i (F k ; Z) also vanishes for for i = q −k + 1, . . . , q + r −k −2. Now q −k + 1 ∈ {1, 2} and F k is simply connected : We infer after choosing r sufficiently large that the integer homology of F k vanishes in the degrees i ≤ q which means that X 0 is contractible. Therefore X 0 and X are Eilenberg-Mac Lane spaces. In particular the cohomological dimension of π 1 (X) cannot exceed q, which implies that π 1 (X) is torsion free.
This completes the proof. ⋄

Novikov homology and fibrations over the circle
In the preceeding section we showed that our main results 1.  We define now the completed ring Λ u :

Novikov homology
The convergence to + ∞ means here that for all A > 0, u(g i ) < A only for a finite number of g i which appear with a non-zero coefficient in the sum λ.
Remark Let λ = 1 + n i g i where u(g i ) > 0 for all i. Then λ is invertible in Λ u . Indeed, if we denote by λ 0 = n i g i then it is easy to check that k≥0 (−λ 0 ) k is an element of Λ u and it is obvious that it is the inverse of λ.
A purely algebraic consequence of the previous definition is the following version of the universal coefficients theorem ( [13], p.102, Th 5.5.1) : Theorem 3.1 There is a spectral sequence E r pq which converges to H * (M, u) and such that We will use this result in Section 4 to prove that in the hypothesis of 1.3, the Novikov homology associated to some class vanishes.

Morse-Novikov theory
We recall in this subsection the relation between Novikov homology and closed one forms. In dimension n ≥ 6, when the Novikov homology vanishes, some hypothesis on π 1 stated below imply the existence of a nowhere vansihing closed one form on M. It is well-known (see [34]) that the existence of such a form is equivalent to the existence of a fibration of M over S 1 .
Let α be a closed generic one form in the class u. Let ξ be the gradient of α with respect to some generic metric on M. For every critical point c of α we fix a pointc above c in the universal cover M . We can define then a complex C • (α, ξ) spanned by the zeros of α : the incidence number [d, c] for two zeros of consecutive indices is the (possibly infinite) sum n i g i where n i is the algebraic number of flow lines which join c and d and which are covered by a path in M joining g ic andd. It turns out that this incidence number belongs to Λ u , so C • (α, ξ) is actually a Λ u -free complex.
The fundamental property of the Novikov homology is that it is isomorphic to the homology of the complex C • (α, ξ) above for any couple (α, ξ).
By comparing the complexes C • (α, ξ) and C • (−α, −ξ) we get the following duality property (see Prop. implies the existence of a nowhere vanishing closed 1-form belonging to the class u ∈ H 1 (M). For n ≥ 6 this problem was independently solved by F. Latour [18] and A. Pajitnov [23], [24]. The statement is ( [18], Th.1') :  It follows that under the hypothesis of 2.1 the condition 2 of 3.3 is always satisfied.
In order to prove 1.5 we use Proof The complex C • (M, u) is acyclic. Like in [19] one can then show that this complex is simply equivalent to a complex of the form This means that the second complex is isomorphic to the first after adding or cancelling afinite number of trivial summands In particular χ(M) = χ(C • (M, u)) = 0.

Novikov homology and finiteness properties of groups
In this section we will achieve the proof of our results 1. Recall that, by 3.1, there is a spectral secquence E r pq which converges to H * (M, u) and whose term E 2 pq is equal to T or Λ p (H q ( M ), Λ u ). Recall also that, by the duality result 3.2 we have the implication : It suffices therefore to prove the following statement : ] . Suppose also that π i (M) are of finite type for i ≤ n 2 . Suppose that there is a non zero cohomology class u ∈ H 1 (M; R) ≈ Hom(π 1 (M), R) such that Ker(u) is of type F [ n 2 ] . Then, for all integers 0 ≤ p, q ≤ n 2 we have Note that this result is purely algebraic. In order to proof it we use some facts about

Hurewicz-type morphisms
Recall that the classical Hurewicz's theorem asserts that for q ≥ 2 the cannonical morphism I q : π q (M) → H q (M) is an isomorphism provided that M is (q − 1)-connected.
In [30] J-P. Serre generalized this theorem (see also [33], p. 504) : For some "admissible" classes of groups C, he showed that, if X is simply connected such that π i (X) ∈ C for i = 1, . . . , q − 1, where q ≥ 2, then I q is an isomorphisme mod C : This means that Ker(I q ) and Coker(I q ) are in C.
The class of finitely generated Abelian groups is such an admissible class. In particular we have : Theorem 4.2 Let X be a simply connected space. Then π i (X) is finitely generated for i ≤ q iff H i (X) is finitely generated for i ≤ q.
In particular any closed, simply connected CW-complex has finitely generated homotopy groups (which is 1.1).
By applying this theorem to M we may replace the hypothesis on π i (M) by the analogue hypothesis on H i ( M). From now on we will suppose that for all i ≤ n 2 , H i ( M ) = Z r i ⊕ T i , where T i is a torsion finitely generated Z-module. The proof of 4.1 relies on the following statement : Proposition 4.3 Let π be a group of type F p for some positive integer p and u : π → Z a morphism whose kernel is of type F p . Suppose that Z r is a π-module. Denote by Λ the ring Z[π] and by Λ u the completed ring, as above. Let π 0 ≤ π be a normal subgroup of finite index and Λ 0 the corresponding group ring Z[π 0 ]. Then for i ≤ p we have Remark For π 0 = π and r = 1 (and therefore for arbitrary r and trivial action of π on Z r ) the statement above was proved by J-C. Sikorav in this thesis [32].
We postpone the proof of 4.3 to Subsection 4.4. We now show :

Proof of 4.3 =⇒ 4.1
Fix p, q ≤ n 2 . For g ∈ π 1 (M), denote by φ g the automorphism of H q ( M ) = Z rq ⊕ T q , given by the action of π 1 (M). We have : where a g : Z rq → Z rq , b g : Z rq → T q and c g : T q → T q . Note that a g and c g are automorphisms (of inverses a g −1 , resp. c g −1 ). Let Since T q is finite, there is only a finite number of automorphisms c : T q → T q . Therefore π 0 is a normal subgroup of π 1 (M) of finite index (It is the kernel of the morphism π 1 (M) → Aut(T q ) defined by g → c g ). We use the following : Let G a group of type F p and G 0 ≤ G a normal subgroup of finite index. Then G 0 is of type F p .
The proof of 4.4 is obvious : If Q is a K(G, 1) with finite p-skeleton, then the finite cover of Q corresponding to G 0 ≤ G will be a K(G 0 , 1) with finite p-skeleton. Now let u : π 1 (M) → R, as in the statement of 4.1 and let u 0 = u| π 0 . Obviously, Ker(u 0 ) has finite index in Ker(u) so, using 4.4, both π 0 and Ker(u 0 ) are of type F p .
Consider now the short exact sequence : One immediately checks that this is an exact sequence of π 0 -modules (where the action of π 0 on Z rq is x → a g (x)). Now consider Λ 0 = Z[π 0 ], and view the completed ring Λ u as a Λ 0 -module. The tensor product of Λ u and the exact sequence above yields a long exact sequence : As a consequence of 4.3, the right term in the sequence above vanishes. In order to prove that the left term is zero, we consider an exact sequence of the form 0 → Z s → Z m → T q → 0, which is viewed as a sequence of π 0 -modules with trivial actions (recall that, by construction, π 0 acts trivially on T q ). The corresponding long exact sequence given by the tensor product with Λ u writes : Applying again 4.3 we infer T or Λ 0 p (T q , Λ u ) = 0, therefore the middle term in (1) vanishes.
We have thus established that for each p, q ≤ n which is the assertion required in 4.1 ... with Λ 0 instead of Λ. To complete the proof we need the following results : Proposition 4.5 Let G be a groupe and let L be a Z[G] right module and N be a Z[G] left module. Assume that N is Z-torsion-free. Then where G acts diagonally on L ⊗ Z N : g(x ⊗ y) = xg −1 ⊗ gy.
Theorem 4.6 For any group extension and any G-module R there is a spectral sequence of the form This theorem is due to G. Hochschild and J-P. Serre [16] (see also [7], p. 171).
We apply first 4.5 and infer from (3) that for each p, q ≤ n 2 we have : Note that the hypothesis of 4.5 is fulfilled by N = Λ u .
We fix q and denote by R the π 1 (M)-module H q ( M ) ⊗ Z Λ u (for the diagonal action). Then we apply 4.6 to the extension 1 → π 0 → π 1 (M) → π 1 (M)/π 0 → 1 and to the module R ; We find using (4) that E 2 ij = 0 for all j ≤ n 2 and for all i ∈ N. According to 4.6, this implies that Finally, we apply once again 4.5 and we get that for all i, q ≤ n 2 we have : T or Λ i (H q ( M ), Λ u ) = 0. The analogous relation for Λ −u instead of Λ u can be established in the same way, so 4.1 follows. ⋄ So we only have to proof 4.3 to complete the proof of 1.2, 1.3 and 1.4. The proof involves some facts about Bieri-Renz invariants. We recall the definition and some properties of these invariants in the subsection below :

Bieri-Renz invariants. Proof of 4.3
Let G be a group of type F m . We call two non zero homomorphisms u, v : G → R equivalent if u = λv for some positive λ ∈ R. We denote by S(G) the quotient Hom(G, R)/ ≈ (which is a (rk(G) − 1)-dimensional sphere). The Bieri-Renz invariants Σ i (G) and Σ i (G, Z), defined for i = 1, . . . , m are open subsets of S(G). They were introduced by R. Bieri, W Neuman and R. Strebel in [3] for i = 1 and by R. Bieri and B. Renz in [4] for i ≥ 2.
These invariants are defined as follows. Let X be a K(G, 1) which is a complex with finite m-squeleton and let u : G → R be a non zero homomorphism. Then there exists an equivariant height function f : X → R, i.e. a function which satisfies f (gx) = f (x) + u(g). (If X is a manifold then f is a primitive of the pullback of some 1-form in the class u). It can be shown that the difference of two such functions is bounded. Consider the maximal subcomplex X f of X whose image by f is contained in [0, +∞[. Definition. Let u ∈ S(G) and i ∈ {1, . . . , m}. Then u belongs to Σ i (G) (resp. to Σ i (G, Z)) if for some couple (X, f ) as above the subcomplex X f is (i − 1)-connected (resp. (i-1)-acyclic).
Bieri and Renz show that the definition does not depend on (X, f ). The following properties of the Bieri-Renz invariants are obvious from the definition : The most striking application of these invariants is stated in the following : Note that {u, −u} ⊂ S(G, Ker(u)) ; When Im(u) is cyclic it is easy to show that these two sets coincide. Note also that S(G, [G, G]) = S(G). So, one immediately infers the following corollary : Corollary 4.8 i) Let u ∈ S(G) and i an integer as above.
In the sequel we give an algebraic description of the invariants Σ i (G, Z) following [4], Section 4. Fix a non zero homomorphism u : G → R. Let F be a finitely generated free Z[G]-module, and {e i } i=1,...,k a basis of F . We define an application v : F → R as follows : v is defined arbitraily on the elements e i ; Then for any g ∈ G we put v(ge i ) = v(e i ) + u(g). Finally, if λ ∈ F writes λ = i,j n ij g j e i in the basis and v(0) = +∞. Following Bieri and Renz we call v a valuation extending u. Now suppose that G is of type F m and let -free, finitely generated resolution (which exists since there exists a K(G, 1) with finite m-skeleton). Define as above v i : P i → R which are valuations extending u. We may suppose in addition that for any i = 1, . . . , m and for any x ∈ P i we have Indeed, one easily sees that it suffices to check the relation above on the basis {e i } of P i . We can construct v inductively, by choosing v(e i ) sufficiently negative in order to satisfy the inegality (1).
Bieri and Renz proved the following theorem ( [4], theorem 4.1) : Theorem 4.9 Let P • → Z be a Z[G]-free, finitely generated resolution of length m and let v : P • → R be a valuation extending u satisying (1). Then u ∈ Σ m (G, Z) if and only if there exists a chain endomorphism Φ : P • → P • which lifts the identity of Z and which satisfies the property We will reformulate this theorem as follows. It is obvious that one can construct the valuation v : P • → R such that it satisfies an additional feature : for all j = 1, . . . , m v is constant on the set {e j i } of the basis elements of P j . Denote this constant by ν j . Then, the valuation v j : P j → R is given by v j (λ) = ν j + inf {u(g k ) | n j ik = 0}, where λ = i,k n j ik g k e j i , g k ∈ G, n j ik ∈ Z. Thus, for an endomorphism Φ given by 4.
(where g j ik ∈ G and n j ik ∈ Z), the inegality (2) implies that u(g j ik ) > 0 for all elements of G appearing with non zero coefficient n j ik .
We call an element λ = i n i g i of Z[G] u-positive if u(g i ) > 0 for any g i which has a non zero coefficient n i in the writing of λ. We call a matrix A ∈ M k (Z[G]) u-positive if all its entries are u-positive. Taking into account the preceeding remarks, theorem 4.9 can be stated as follows : Applying 4.5 we infer that T or Λ 0 i (Z r , Λ u ) is isomorphic to H i (π 0 , Z r ⊗ Z Λ u ), where the action of π 0 on Z r ⊗ Z Λ u is given by x ⊗ λ → xg −1 ⊗ gλ. It suffices therefore to prove that the latter vanishes for i ≤ p.
Let P p → P p−1 → · · · → P 1 → P 0 → Z be a Z[π 0 ]-free resolution which we may suppose finitely generated since π 0 is F p . By definition we have We will prove that the right term of (3) vanishes for all i ≤ p. Fix a basis {e j i } i for each module P j . Since Ker(u| π 0 ) is of type F p , it follows by 4.8.i that u ∈ Σ p (π 0 , Z), so, applying 4.10 we obtain an endomorphism Φ : P • → P • such that for all j = 1, . . . , m the matrix of Φ j associated to the basis {e j i } is u-positive.
The proof of 4.3 will be complete if we prove the following : The homomorphism Id − Ψ is invertible and it induces zero in homology.

Proof
Let us prove first that Id − Ψ vanishes in homology. Since Φ : P • → P • lifts the identity of Z, and P • is free it is well known and easy to prove that there exists a homotopy s : P • → P •+1 between Φ and Id. It follows that s⊗Id is a homotopy between Ψ and Id, so Id−Ψ induces the zero morphism in homology. Now let us prove the first assertion of the lemma. Let {f 1 , f 2 , . . . f r } be the canonical basis of Z r . We can see Z r ⊗ Z Λ u as a right Λ u -module endowed with the canonical structure : (x ⊗ λ)µ = x ⊗ λµ. It is a free module of rank r and {f 1 ⊗ 1, f 2 ⊗ 1, . . . , f r ⊗ 1} is a basis for this module.
Recall that we denoted by Λ 0 the ring Z[π 0 ]. For any j = 1, . . . m, the product P j ⊗ Λ 0 (Z r ⊗ Z Λ u ) inherits of the structure of right Λ u -module described above. On the other hand, since P j is free we have (4) P j ⊗ Λ 0 (Z r ⊗ Z Λ u ) ≈ (Z r ⊗ Z Λ u ) rk(P j ) ≈ (Λ u ) r·rk(P j ) .
If {e j i } i=1,...rk(P j ) is the given basis of P j then ..,rk(P j ), s=1,...r will be a basis for P j ⊗ Λ 0 (Z r ⊗ Z Λ u ). It is easy to check that the isomorphisms (4) preserve the right Λ u -module structure. Moreover, the differential ∂⊗ Λ 0 Id of the complex P • ⊗ Λ 0 (Z r ⊗ Z Λ u ) respects this structure, therefore it is a complex free right Λ u -modules.
We claim that the matrices of Ψ in the basis (5) are u-positive. Indeed, for fixed j, let (λ ik ) i,k=1,...rk(P j ) be the matrix of Φ j in the basis {e j i } ; This matrix is known as u-positive. We dropped down the index j from the coefficients of the matrix to simplify the notations. Denote byλ the image of an element λ ∈ Λ 0 under the endomorphism Λ 0 → Λ 0 induced by the involution of π 0 : g → g −1 . The right action ofλ ik on Z r , evaluated on the basis {f s } writes : n l iks f l , for some integers n l iks . We infer that so the matrix of Ψ j is u-positive. Denote this matrix by A j and define another matrix B j by : B j = Id + +∞ k=1 A k j . As A j is u-positive it is easy to check that the matrix B j belongs to M r·rk(P j ) (Λ u ). It therefore defines using the basis (5) an endomorphism of P j ⊗ Λ 0 (Z r ⊗ Z Λ u ). It is actually an automorphism since obviously (Id − A j )B j = Id. Finally, as Ψ is a morphism of complexes, the morphism Id + Ψ + Ψ 2 + · · · induced by B j will also commute with the differential. We have finally got an automorphism of P • ⊗ Λ 0 (Z r ⊗ Z Λ u ) whose inverse is Id − Ψ. This completes the proof of 4.11 and hence the proof of 4.3 which implies our main theorems 1.2, 1.3 and 1.4. ⋄

Proof of 1.5
Assume that the homotopy of X is finitely generated. By embedding X in R 2q+3 construct a manifold M = ∂W of dimension 2q + 2 as in the proof of 1.2. We have χ(X) = χ(W ) = 2χ(M). By general position π i (M) ≈ π i (X) for i ≤ q + 1. These groups are therefore finitely generated. The hypothesis of 4.1 is fulfilled and, by applying this result, we get H * (M, u) = 0. But according to 3.5 this implies χ(M) = 0, contradicting thus the hypothesis on χ(X). If X is a manifold, the result follows directly from 4.1, 3.1 and 3.5.
The proof is finished. ⋄