I wonder if there exists a compact closed smooth aspherical manifold $M$ of dimension at least $4$, so that for any covering space $\tilde{M}$ over $M,$ we always have $H\_2(\tilde{M},\mathbb{Z})=0$ and $H\_2(\tilde{M},\mathbb{Z}/v\mathbb{Z})=0$ for all integers $v\ge 2$.


Of course, when $H\_2$ is replaced with $H\_1,$ then the answer is negative since we can always choose a covering with homotopy type of $S^1$. If we drop the closedness assumption, the answer is true by considering a regular neighborhood of a bouquet of circles in $\mathbb{R}^4.$


The motivation comes from the calculus of variations. I'm sorry if this problem has trivial answers since my field is somewhat far away from algebraic topology. Many thanks!



Compact closed aspherical manifolds with vanishing second homology in all the covering spaces
I think that the answer to this question is unknown in general. If one had a closed aspherical manifold with this property, then it could not contain a [Baumslag-Solitar subgroup](https://en.wikipedia.org/wiki/Baumslag%E2%80%93Solitar_group?wprov=sfti1) since such a group has a subgroup with non-trivial $H\_2$ (possibly with finite field coefficients, eg $BS(1,2)$ contains a $BS(1,4)$ subgroup with $H\_2(BS(1,4);\mathbb{F}\_3)\neq 0$). Examples of groups that do not contain Baumslag-Solitar subgroups are [hyperbolic groups](https://en.wikipedia.org/wiki/Hyperbolic_group?wprov=sfti1). However, Gromov has conjectured that hyperbolic groups contain a surface subgroup, and hence would have a subgroup with non-trivial $H\_2$ if that conjecture is true.


Moreover, [until recently](https://arxiv.org/abs/2105.14795) a group of finite type (such as the fundamental group of an aspherical manifold) which was non hyperbolic and contained no Baumslag-Solitar subgroup was not known.
Thus I would argue that there is probably no example of this sort in the literature, since it would have resolved one of two possible well-known open questions.


On the other hand, maybe it is known that any aspherical n-manifold admits a cover with non-trivial $H\_2$? I think a theorem of this sort in the literature for $n>3$ would be well-known if it existed (and at least @MoisheKohan and myself are not aware of such a theorem, if that carries any weight).


