TITLE: Why do finite homotopy groups imply finite homology groups?
QUESTION [36 upvotes]: Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. $\mathbb{R}P^2$ is a counterexample.
Do finitely generated homotopy groups imply finitely generated homology groups? I can proof this only for connected spaces with trivial fundamental group. The converse is false. $S^1\vee S^2$ is a counterexample.

REPLY [9 votes]: Using transfer map the question "why finite homotopy groups implies finite homology groups" can be reduced to the simply-connected case, where Serre's mod finite groups theory applies.
Recall that for a finitely-sheeted covering map $p:\tilde X\to X$ the homology transfer 
$t: H_*(X)\to H_*(\tilde X)$ is the homomorphism induced by associating to an oriented cell in $X$ the sum of its (finitely many) lifts in $\tilde X$. So the composition 
$p_* t$ is the self-map
of $H_*(X)$ that simply multiplies by the order of the cover; this map is an isomorphism when we use rational coefficients. Thus the tranfer is injective on rational homology.
In particular, if $X$ is a space with finite homotopy groups and some rational homology group nonzero, then the same is true for its universal (finite) cover.