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I will explain why the general answer to this question is negative in dimensions $\geq 4$. The next remark expresses the fundamental obstruction. We say a simplicial tree $T$ equipped with an action o …

This basic question is unfortunately not well explained anywhere in the literature that I know of, although the answer is well known to lots of people. When $\pi_1(S)$ embeds into $\pi_1(M)$ and $\pi_ …

At the risk of stating the obvious, perhaps it’s also worth addressing the final question, of “why one needs to take Čech cohomology on the left”, with an example. The typical object of study in the f …

As mentioned in comments, your picture is not entirely accurate. But perhaps this is what you're looking for? (Note that, if you had chosen a different gluing pattern for your once-punctured genus-tw …

It seems to me that one does get $H_3(X/G)$ finitely generated under some natural assumptions, namely: the action of $G$ on $H_2(X)$ makes it into a free $G$-module; the group homology $H_3(G)$ is f …

Collins and Miller proved that it is algorithmically undecidable whether or not $\pi_2(X)$ is trivial, for $X$ a finite 2-complex. As Achim Krause points out, in general $\pi_2(X)$ is only a module ov …

As suggested in the comments, what you are asking for is essentially the presentation complex of a finitely presented, infinite, simple group. Thus it suffices to exhibit a presentation for such a gro …

Gabai proved that homotopy hyperbolic 3-manifolds are virtually hyperbolic, in the paper of that name: Gabai, David, Homotopy hyperbolic 3-manifolds are virtually hyperbolic. J. Amer. Math. Soc. 7 (1 …

I presume by "acyclic" you are referring to homology with $\mathbb{Z}$ coefficients. There are many such examples. For instance, you can take two elements $u,v$ in the free group $F_2$ of rank 2 tha …

The answer is 'yes' by Britton's lemma (see wikipedia and, more generally, Serre's book Trees and Scott and Wall's article 'Topological methods in group theory'). Since $M$ and $U$ are smooth and c …

See pp. 449--457 of Peter Scott's article The geometries of 3-manifolds for a complete description of all 3-manifolds with finite fundamental group. The article is available on his website. There don …

They arise naturally all the time. The Galois correspondence tells us that a covering space is normal if and only if the corresponding subgroup of the fundamental group is normal. So whenever you hav …

There are situations in which surfaces are the "unique" examples with big mapping class groups. One such is closed manifolds of negative sectional curvature. Theorem (Paulin): If $M$ is a closed $n$- …

The answer to both questions is 'no'. This was proved by Greenberg for Fuchsian groups. One outline of the proof is as follows. Any finitely generated subgroup $H$ of a surface group $\Gamma$ is qua …

This is certainly true, though one needs to be sufficiently careful about one's definition of Seifert fibred---it's important to allow fibres with a neighbourhood that looks like a fibred solid Klein …