In an article I am currently reading, the author tells us that for compact (finite dimensional topological) manifolds X and finite groups $\Gamma$, the set $\mathrm{Hom}(\pi_1X,\Gamma)/\Gamma$ where $\Gamma$ acts on the set $\mathrm{Hom}(\pi_1X,\Gamma)$ via $(\varphi\cdot\gamma)([c]):=\gamma^{-1}\varphi([c])\gamma$, is finite.
This quotient space is naturally in bijection with the isomorphism classes of principal bundles over $X$ with structure group $\Gamma$.
How do you show that $\mathrm{Hom}(\pi_1X,\Gamma)/\Gamma$ is finite?
I can see one pathway which would be to show that the fundamental group of $X$ is finitely generated, which would entail that the set $\mathrm{Hom}(\pi_1X,\Gamma)$ itself is finite. I thought I remembered a proof of this fact from John M. Lee's Introduction to Topological Manifolds, and I tried in vain to reproduce it. He actually shows that the fundamental group is countable.
I have two "proofs" or rather "plausible arguments" that work in the differentiable setting. The first uses Morse theory : $X$ is homotopy equivalent/homeomorphic (which one is it?) to a finite CW-complex, and those have finitely generated fundamental groups.
The other one uses a riemannian metric on $X$ and a (finite) cover $\mathcal{O}_1,\dots,\mathcal{O}_N$ by geodesically convex open sets. I recall reading in Bott and Tu's Differential Forms in Algebraic Topology (where it is cited without proof but with a reference) that those exist. It seems plausible that such open sets $\mathcal{O}$ should be contractible : you can take some fixed point $x\in\mathcal{O}$ and define $C:\mathcal{O}\times[0,1]\to\mathcal{O},~(y,t)\mapsto c_{x,y}(1-t)$ where $c_{x,y}:[0,1]\to\mathcal{O}$ is the (unique) shortest geodesic that joins $x\rightarrow y$. By definition of being geodesically convex, there is indeed a unique shortest geodesic between any two points in $\mathcal{O}$ and it stays inside $\mathcal{O}$ at all times, and so $C$ is well defined (and smooth).
The next step copies a proof of Van Kampens's theorem : fix a basepoint $x_0\in X$. For any $i,j$, $\mathcal{O}_i\cap\mathcal{O}_j$ is either empty or non empty and geodesically convex. Pick points $x_{ij}\in\mathcal{O}_i\cap\mathcal{O}_j$ and paths $c_{ij}$ that link $x_o\rightarrow x_{ij}$. Also, for any i,j,j' with $\mathcal{O}_i\cap\mathcal{O}_j\neq\emptyset$ and \mathcal{O}_i\cap\mathcal{O}_{j'}\neq\emptyset define a path c_{j,i,j'}:x_0\underbrace{\longrightarrow}_{c_{ij}} x_{i,j}\underbrace{\longrightarrow}_{*} x_{ij'}\underbrace{\longrightarrow}_{\overline{c_{ij'}}} x_0 where $*$ is any path inside $\mathcal{O}_i$ that links x_{ij}\rightarrow x_{ij'} and \overline{c_{ij'}} is c_{ij'} in reverse. The c_{j,i,j'} should span $\pi_1X$, thus showing that it is finitely generated (contractability of $\mathcal{O}_i\cap\mathcal{O}_j$ is key).
My worry with both proofs is that I don't really know Morse theory, and I don't know why geodesically convex open sets exist around every point in a riemannian manifold. Of course, this could be remedied by studying the subjects, but I wonder :
Are there more elementary proofs of the fact that compact (possibly smooth) manifolds have finitely generated fundamental group?
Also,
is it true that compact topological manifolds have finitely generated fundamental groups?