let $ M $ be a connected manifold. Suppose that $ \pi_1(M) $ has finite commutator subgroup $ [\pi_1(M),\pi_1(M)] $ . Then $ H_1(M) $ is related to $ \pi_1(M) $ by Hurewicz
$$
H_1(M) \cong \pi_1(M)/[\pi_1(M),\pi_1(M)]
$$
what else can we say about $ M $ given just the fact that $ \pi_1(M) $ has finite commutator subgroup?
Here is some motivation for this question: Every Grassmannian, even every flag manifold, even every generalized flag manifold, even every Riemannian homogeneous manifold has $ \pi_1(M) $ with finite commutator subgroup. These are all very nice spaces, I'm curious if any of the nice properties of these spaces can be deduced just from this fact about the fundamental group.
For example, If $ \pi_1(M) $ has finite commutator subgroup then can we conclude that the manifold $ M $ is homotopy equivalent to a compact manifold?
Best Answer
Certainly not. Let $X= S^2\times S^1 \# S^2\times S^1$ the connected sum of two copie of $S^2\times S^1$. Its universal cover $M$ is simply connected, so satisfy your hypothesis, but has infinitely generated $H_2$, so has not the same homotopy type than a compact CW complex.