TITLE: Spaces with both "simple homology" and "simple homotopy" at the same time
QUESTION [14 upvotes]: Maybe every algebraic topology student, at some moment, will ask himself/herself the question: why are $\pi_*$ so difficult and mysterious, especially when compared with (co)homology? Think about the weird connections between $\pi_n(S^k)$ and number theory... it is insane!
But one day I realize this idea may be wrong and biased.
Homology, is probably not really easier than homotopy.
One tends to think $H_*$ is easy, only because most of us only care about finite-dimensional manifolds in our daily life. And then one have that nice vanishing result for higher-dimensional $H_*$.
Consider the infinite-dimensional Eilenberg−MacLane space $K(\mathbb{Z}, n)$ when $n>2$. Its $\pi_*$ is surely as easy as one could hope, but how about its $H_*$, especially the torsions?
It appears to me, the better (?) statement might be
"Spaces with simple $\pi_*$ tends to have complicated $H_*$.
Spaces with simple $H_*$ tends to have complicated $\pi_*$."
This gives one some strange feeling. It is almost like some kind of "Fourier transformation", some kind of duality.
And this makes some spaces particularly interesting: spaces with both simple $H_*$ and simple $\pi_*$ at the same time.
Let me start with some simple examples:
1) $S^1 \simeq K(\mathbb{Z}, 1)$.
2) $\Sigma^g$, that is, Riemann surfaces.
3) $K(\mathbb{Z}, 2) \simeq CP^\infty$.
4) $K(G, 1)$ when $G$ is a finite group.
And I am looking forward to your examples of such spaces, especially comments from AT experts. 
Thank you very much.

REPLY [5 votes]: For the nicest answers to this kind of question, you should fix a prime $p$ and look for $p$-local spaces $X$ such that 

$X$ is simply connected and admits an $H$-space structure
Each homotopy group $\pi_k(X)$ is free and finitely generated over $\mathbb{Z}_{(p)}$.
Each homology group $H_k(X)$ is also free and finitely generated over $\mathbb{Z}_{(p)}$.

These are called Wilson spaces, because they were first studied in Steve Wilson's paper "The $\Omega$-spectrum for Brown-Peterson homology II".  He showed that

For each $k\geq 2$ there is an indecomposable Wilson space $Y_k$ whose first homotopy group is in dimension $k$.
This $Y_k$ is unique up to (non-canonical) weak equivalence.  It can be constructed as a space in the $\Omega$ spectrum for $BP\langle n\rangle$ for suitably chosen $n$.
Any Wilson space can be written as a product $\prod_iY_{k_i}$, where the indices $k_i$ form a nondecreasing sequence that tends to infinity.