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Theorem. Every finitely presentable group is the fundamental group of a closed 4-manifold. Sketch proof. Let $\langle a_1,\ldots,a_m\mid r_1,\ldots, r_n\rangle$ be a presentation. By van Kampen, th …

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 …

One can also see it using the theory of ends. If $\pi_1M$ were freely decomposable, then it would follow from the easy direction of Stallings' Ends Theorem that $\pi_1M$ had two or infinitely many en …

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 …

Apologies for the shameless self-promotion, but as you ask for necessary conditions, you seem to want a list of theorems of the form 'If G is a 3-manifold group then G has property P'. Aschenbrenner, …

This might help. Lemma If $A$ does not split freely and $C$ is a non-trivial subgroup of $A$ then the HNN extension $G=A*_C$ does not split freely. The proof uses Bass--Serre theory---see Serre's bo …

For a group $G$, $H_2(G,\mathbb{Z})$ is also called the Schur multiplier of $G$. Among other things, if $G$ is perfect (ie $H_1=0$) then it is a term in the universal central extension $\widehat{G}$f …

A presentation with the same number of generators and relations is called balanced. The triviality problem for balanced presentations (indeed, the word problem for balanced presentations) is a major …

This question is very vague, but here are some thoughts to add to Mark's answer. First, note that any finitely presented group arises as the fundamental group of a closed manifold of dimension 4 (see …

As Ricardo points out in the comments, there's an error in my sketched calculation below. I also didn't notice the requirement that $X$ should be finite, so the natural $BK$ fails on two counts! How …

Here's a result that gives some idea of how hard it is to characterise linear (let alone residually finite) groups of type $F$ (ie with a $K(G,1)$ that's a finite complex). Theorem: There is a sequen …

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 …

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_ …

The answer is 'yes' when your group $G$ is word-hyperbolic. This can be deduced from Sageev's theorem. I'll explain this here, but a good reference is Hruska--Wise's paper 'Finiteness properties of …

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 $n$- …