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](https://link.springer.com/content/pdf/10.1007/BF02960745.pdf). 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](https://people.math.osu.edu/davis.12/papers/singer-hopf.pdf) the Hopf conjecture in general. However, the Singer conjecture is open.



Status of the Hopf-Thurston sign conjecture in dimension 4
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.


