About a conjecture of Daniel H. Gottlieb

Thomas Schick Mathematisches Institut

Bunsenstr. 3-5 37073 Göttingen, Germany.

Andreas Thom Andreas Thom Mathematisches Institut Bunsenstr. 3-5 37073 Göttingen, Germany.

April 26, 2007

Abstract

We give a counterexample to a conjecture of D.H. Gottlieb and prove a strengthened version of it.

The conjecture says that a map from a finite CW-complex X to an aspherical CW-complex Y with non-zero Euler characteristic can have non-trivial degree (suitably defined) only if the centralizer of the image of the fundamental group of X is trivial.

As a corollary we show that in the above situation all components of non-zero degree maps in the space of maps from X to Y are contractible.

1 A version of Gottlieb’s conjecture

Let X and Y be finite CW-complexes. In [4, 3], Gottlieb defines a notion of degree of a continuous map f: X → Y as follows. Let f_{∗}: H_{∗}(X, ℤ) → H_{∗}(Y, ℤ) be the induced map in reduced integral homology. The degree deg(f) of f is the least integer n ∈ ℕ, such that there exists a group homomorphism τ: H_{∗}(Y, ℤ) → H_{∗}(X, ℤ) which satisfies f_{∗} ∘ τ = n · id. He conjectures the following (compare [3]):

Conjecture 1 (Gottlieb). Let (Y, y) be a finite aspherical CW-complex which is not acyclic. Let f: (X, x) → (Y, y) be a continuous map with deg(f) ≠ 0. If χ(Y) ≠ 0, then the centralizer of f_{∗}(π_{1}(X, x)) in π_{1}(Y, y) is trivial.

In this note we give a counterexample to this form of the conjecture (see Example 11) and prove a version with a stronger hypothesis, see Theorem 3. Let us rephrase one important consequence of non-vanishing degree in the case of mappings between closed oriented manifolds, so that it is applicable in a more general setting.

Definition 2. Let f: (X, x) → (Y, y) be a continuous map. We say that f is a superposition, if for any ℚπ_{1}(Y, y)-module L, the induced map

f_{∗}: H_{∗}^{π_{1}(X,x)}(X̃; f^{∗}L) → H_{∗}^{π_{1}(Y,y)}(Ỹ; L)

is surjective.

We will see in Theorem 7, that a map of non-vanishing degree between closed oriented manifolds, or more generally between oriented Poincaré duality complexes, is a superposition. Moreover, an equivariant version of the Becker-Gottlieb transfer gives plenty of examples of maps between CW-complexes which are not Poincaré complexes. Clearly, if one adds in Gottlieb’s definition of degree that arbitrary twisted coefficients have to be used, a map of non-zero degree in this stronger sense is a superposition, too.

Our main result is the following.

Theorem 3. Let (Y, y) be a finite aspherical CW-complex. Let f: (X, x) → (Y, y) be a continuous superposition. If χ(Y) ≠ 0, then the centralizer of f_{∗}(π_{1}(X, x)) in π_{1}(Y, y) is trivial.

Assuming Theorem 3, we can show some corollaries which generalize results from [3]. Let f be a continuous map from X to Y. We denote by map(X, Y, f) the space of continuous maps from X to Y which are homotopic to f.

Corollary 4. Let Y be a finite aspherical CW-complex. Let f: X → Y be a continuous superposition. If χ(Y) ≠ 0 and Y is aspherical, then the mapping space map(X, Y, f) is contractible.

Proof. If Y is aspherical, then map(X, Y, f) is also aspherical because of the following reasoning: we have to extend a given map from S^{n} to map(X, Y, f) to D^{n+1}. By the exponential law, this means to extend a map from X × S^{n} to X × D^{n+1}. The latter space is obtained from the former by attaching cells of dimension n + 1 or higher. If n ≥ 2, because π_{k}(Y) = 0 for k ≥ 2, we can extend the map cell by cell as required.

Gottlieb showed in [2] that π_{1}(map(X, Y, f), f) is naturally isomorphic to the centralizer of f_{∗}(π_{1}(X, x)) in π_{1}(Y, y). Hence the claim follows from Theorem 3.

Corollary 5. Let (Y, y) be a finite aspherical CW-complex with χ(Y) ≠ 0. Every subgroup of finite index in π_{1}(Y, y) has trivial centralizer.

Proof. Let G be a finite index subgroup of π_{1}(Y, y). The induced map f: BG → Y is a superposition. Hence, Theorem 3 implies the claim.

Proof of Theorem 3. Because Y = Bπ_{1}(Y) is finite dimensional, π_{1}(Y) is torsion free. Therefore every non-trivial subgroup is infinite. Let us assume that the centralizer of f_{∗}(π_{1}(X, x)) in π_{1}(Y, y) is infinite.

If π is a discrete group, let Lπ be its group von Neumann algebra. If χ(Y) is not zero, then the equivariant L^{2}-homology

H_{∗}^{π_{1}(Y,y)}(Ỹ; Lπ_{1}(Y, y))

cannot be zero-dimensional in all degrees. Indeed,

0 ≠ χ(Y) = ∑_{i=0}^{∞}(−1)^{k}β_{k}^{(2)}(Y),

by Atiyah’s L^{2}-index theorem, see [7, Theorem 6.80]. Here β_{k}^{(2)}(Y) denotes the k-th L^{2}-Betti number

β_{k}^{(2)}(Y) = dim_{Lπ_{1}(Y,y)} H_{k}^{π_{1}(Y,y)}(Ỹ; Lπ_{1}(Y, y)).

By assumption, the map f: (X, x) → (Y, y) is a superposition, so it induces a surjection

H_{k}^{π_{1}(X,x)}(X̃; f^{∗}Lπ_{1}(Y, y)) → H_{k}^{π_{1}(Y,y)}(Ỹ; Lπ_{1}(Y, y)),

for every k ∈ ℕ. However, since Y is aspherical, for every subgroup G of π_{1}(Y, y) which contains f_{∗}(π_{1}(X, x)), this map can be factorized through

H_{k}^{G}(EG; res_{G}^{π_{1}(Y,y)}Lπ_{1}(Y, y)) = H_{k}^{G}(EG; LG) ⊗_{LG} Lπ_{1}(Y, y).

Here, we used that Lπ_{1}(Y, y) is flat as LG-module, compare [7, Theorem 6.29]. By the same theorem, the Lπ_{1}(Y, y)-dimension of the right hand side is equal to β_{k}^{(2)}(G). To derive a contradiction, it suffices to construct a sub-group as above which has only vanishing L^{2}-Betti numbers.

If the centralizer of f_{∗}(π_{1}(X, x)) intersects non-trivially with π_{1}(X, x), then the intersection is infinite, since π_{1}(Y, y) is torsion-free. In this case f_{∗}(π_{1}(X, x)) has an infinite center and all its L^{2}-Betti numbers are zero by Theorem 7.2 of [7]. If the intersection is trivial, we may pick a non-torsion element which centralizes f_{∗}(π_{1}(X, x)). Together with f_{∗}(π_{1}(X, x)), it generates a copy of f_{∗}(π_{1}(X, x)) × ℤ, which has trivial L^{2}-Betti numbers by Künneth’s Theorem, see Theorem 6.54 (4) in [7]. Hence, we arrive at a contradiction.

Since π_{1}(Y, y) is torsion-free, we conclude that the centralizer of f_{∗}(π_{1}(X, x)) is trivial. This finishes the proof.

Remark 6. Note that, because the classifying space of im(f_{∗}) ⊂ π_{1}(Y) is in general not a finite CW-complex, we have to use the generalization of L^{2}-Betti numbers to arbitrary CW-complexes of Lück as developed in [5, 6].

The next theorem gives examples of maps which are superpositions.

Theorem 7. The following classes of maps are superpositions.

1. retractions,

2. continuous maps between oriented closed manifolds, or more generally Poincaré duality spaces, which have non-vanishing degree,

3. continuous maps f: (X, x) → (Y, y), whose homotopy fiber has the homotopy type of a finite CW-complex and non-vanishing Euler characteristic.

Proof. The statement about retractions follows from functoriality; for the second statement one uses the transfer given by Poincaré duality, and for the third the Gottlieb-Becker transfer (with twisted coefficients).

Remark 8. The proofs of the results presented so far show that the assumptions can be weakened as follows:

• χ(Y) ≠ 0 can be replaced by the assumption that Y has at least one non-vanishing L^{2}-Betti number

• The map f being a superposition can be replaced by the assumption that f induces a surjective homomorphism in L^{2}-cohomology.

That surjectivity in L^{2}-cohomology is true for inclusions of finite index subgroups and therefore Corollary 5 holds under the weaker assumptions follows e.g. from [8].

Remark 9. One should observe that Gottlieb’s theorem, stating that the center of an acyclic finite CW-complex with non-trivial Euler characteristic is trivial, has been generalized considerably. Its strongest version now reads that such a group does not contain an infinite amenable normal subgroup.

Our main application states that the centralizer of an image group is trivial; again we expect a generalization similar to the one about normal amenable subgroups. However, the correct notion of “amenable centralizer” still has to be developed.

2 A counterexample to a strong form of the conjecture

We finish this note by giving the desired counterexample to Gottlieb’s Conjecture. The tools in the construction are the techniques from the work of Baumslag, Dyer and Heller, see [1].

Theorem 10 (Baumslag-Dyer-Heller). There exists a finite aspherical and acyclic CW-complex (D, ∗) whose fundamental group π_{1}(D, ∗) contains a copy of ℤ.

Example 11. There exists a finite aspherical CW-complex Y with χ(Y) = 2 and a continuous map f: 𝕋^{2} → Y which is of degree one (taking Gottlieb’s definition) and injective on the fundamental group. In particular, the centralizer of f_{∗}(π_{1}(𝕋^{2}, ∗)) = ℤ^{2} is infinite.

Construction of the Example. Let D be as in Theorem 10. Starting with 𝕋^{2}, we glue in two copies of D, along a circle in D representing the generator of ℤ ⊂ π_{1}(D, ∗), and along each of the generators of the fundamental group of 𝕋^{2}, to obtain the new space Y. Let f: 𝕋^{2} → Y be the natural map, induced by the glueing process.

This is exactly the type of construction of Baumslag-Dyer-Heller; since we glue along inclusions on the level of fundamental groups, the resulting space Y is aspheric and the map f_{∗}: ℤ^{2} → π_{1}(Y, y) is injective. On the other hand, a look at the Mayer-Vietoris sequence shows that the map from 𝕋^{2} to Y is an isomorphism in second integral homology, whereas H_{1}(Y, ℤ) = 0.

References

[1] G. Baumslag, E. Dyer, and A. Heller. The topology of discrete groups. J. Pure Appl. Algebra, 16(1):1–47, 1980. 2

[2] Daniel H. Gottlieb. Covering transformations and universal fibrations. Illinois J. Math., 13:432–437, 1969. 1

[3] Daniel Henry Gottlieb. Self coincidence numbers and the fundamental group. arXiv:math.AT/0702236. 1, 1 [4] Daniel Henry Gottlieb. The trace of an action and the degree of a map.

Trans. Amer. Math. Soc., 293(1):381–410, 1986. 1

[5] Wolfgang Lück. Dimension theory of arbitrary modules over finite von Neumann algebras and L^{2}-Betti numbers. I. Foundations. J. Reine Angew. Math., 495:135–162, 1998. 6

[6] Wolfgang Lück. Dimension theory of arbitrary modules over finite von Neumann algebras and L^{2}-Betti numbers. II. Applications to Grothendieck groups, L^{2}-Euler characteristics and Burnside groups. J. Reine Angew. Math., 496:213–236, 1998. 6

[7] Wolfgang Lück. L^{2}-invariants: theory and applications to geometry and Ktheory, volume 44 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2002. 1

[8] Thomas Schick. L^{2}-determinant class and approximation of L^{2}-Betti numbers. Trans. Amer. Math. Soc., 353(8):3247–3265 (electronic), 2001. 8