Could someone show an example of two spaces $X$ and $Y$ which are not of the same homotopy type, but nevertheless $\pi\_q(X)=\pi\_q(Y)$ for every $q$? Is there an example in the CW complex or smooth category?



Are there two non-homotopy equivalent spaces with equal homotopy groups?
There are such spaces, for example $X = S^2 \times \mathbb{R}P^3$, $Y = S^3 \times \mathbb{R}P^2.$
(These are both smooth and CW-complexes.)


Whitehead's Theorem says that for CW-complexes, if a map $f : X \to Y$ induces an isomorphism on all homotopy groups then it is a homotopy equivalence. But, as the example above shows,
you need the map. Such a map is called a weak homotopy equivalence.


(Whitehead's Theorem is not true for spaces wilder than CW-complexes. The Warsaw circle
has all of its homotopy groups trivial but the unique map to a point is not a homotopy equivalence.)


