7
$\begingroup$

A famous conjecture in topology asserts:

The Euler characteristic of a closed aspherical $2n$-manifold $M$ satisfies $(-1)^n\chi(M) \geq 0$.

This was conjectured by Hopf for manifolds with non-positive sectional curvature and (much later) by Thurston for all aspherical manifolds.

By the classification of surfaces, it holds in dimension $2$.

I am curious about the status of this conjecture in dimension 4.

It seems as if the special case for manifolds with negative sectional curvature was settled by Milnor, as explained in this paper by Chern. Undoubtedly, the more general conjecture by Thurston is much younger. I wonder if the general version has some kind of reformulation in less manifoldy and more algebraic terms. More precisely, I am curious about the following.

  • For which fundamental groups of aspherical $4$-manifolds is the $4$-dimensional Thurston conjecture settled?

One approach to the Thurston conjecture that I am aware of is the employment of $\ell^2$ invariants: the Singer conjecture implies the Hopf conjecture in general. However, the Singer conjecture is open.

Improve this question
$\endgroup$

2 Answers 2

Reset to default
8
$\begingroup$

There has been a lot of work on cases of this conjecture connected to Coxeter groups. M. Davis and R. Charney made a conjecture that comes from these cases in 1995 in The Euler characteristic of a non-positively curved piecewise Euclidean manifold. Also have a look at the article Vanishing theorems and conjectures for the $\ell^2$-homology of right-angled Coxeter groups by M Davis and B Okun. They prove the case when $M^4$ has a non-positively curved piecewise Euclidean cubical structure. This is a far larger class than it sounds: every flag triangulation of the 3-sphere gives rise to such a manifold via the Davis construction.
Davis-Okun also give a conjecture for the $\ell^2$-homology of groups that act properly cocompactly on contractible manifolds that extends the Singer conjecture, which is equivalent to the case when the group acts freely properly cocompactly.

Improve this answer
$\endgroup$
5
$\begingroup$

Since aspherical manifolds with isomorphic fundamental groups are homotopy equivelant (Theorem 2.1 here http://www.map.mpim-bonn.mpg.de/Aspherical_manifolds), for each group $G$ it is necessary to only check it for one manifold with $\pi_1=G$.

So is true for any known examples $G= \mathbb{Z}^4$ or the fundamental group of the Davis hyperbolic 4-manifold, products of surface groups etc.

For hyperbolic 4-manifold groups it follows from Gauss-Bonnet.

Improve this answer
$\endgroup$
6
  • $\begingroup$ Could you elaborate on your last paragraph? It’s rather cryptic. $\endgroup$
    – HJRW
    Commented Feb 9, 2022 at 22:13
  • $\begingroup$ The Guass-Bonnet theorem for hyperbolic $4$-manifolds states $\chi(M) = \frac{4 \pi^2}{3} Vol(M)$, Since the volume is positve, it implies the Euler characteristic is positive. $\endgroup$
    – Nick L
    Commented Feb 9, 2022 at 22:19
  • $\begingroup$ Thanks. A little Googling suggests that the statement for 2n>2-dimensional hyperbolic manifolds was proved by Hopf in 1925. $\endgroup$
    – HJRW
    Commented Feb 10, 2022 at 8:46
  • $\begingroup$ Is it known that any aspherical 4-manifold whose fundamental group is word-hyperbolic is homotopy-equivalent to one that supports a hyperbolic metric? (In domensions bigger than 4, the analogous statement fails.) $\endgroup$
    – Jens Reinhold
    Commented Feb 10, 2022 at 17:40
  • 4
    $\begingroup$ @JensReinhold This is of course false. $\endgroup$
    – Moishe Kohan
    Commented Feb 13, 2022 at 16:47

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .