Fundamental group has finite commutator subgroup

algebraic-topology differential-geometry differential-topology smooth-manifolds

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.

Related

The Laplacian of the squared length of a (0,2)-tensor
Show that the coinvariant ring is self-injective.
Calculation of de Rham complex for real projective space
Asymptotic behavior of $f(x) = \sum_{k=1}^{\infty} \sqrt{ 1 - \frac{k^2}{x^2} }$ as $x\to\infty$
$(x+1)^2y''+(x+1)y'+y=x^2+2\sin(\ln(x+1)), y(0)=\frac{1}{5},y'(0)=2$
Solving Non-Homogenous Recurrence Relation
Proving that sequentially compact spaces are compact.
Tree and total probability rule (two events)
If $c$ is a local extreme but not a strict local extreme of a continuous $f$, then does there exist $X$ such that $x \in X$ and $f(X)=f(x)$?
How Riemann found his hypothesis? [closed]
Linear independence of another set of infinite functions
What is the Variance of $(1-X)/X$ and the Expectation of $(1-X)/X^{2}$?