I know that the fundamental groups of homeomorphic spaces are isomorphic. Whithout assuming any furthr strucutre, is there any relation between higher homotopy groups of such spaces? If not, is there some set of conditions on the relevant spaces so that such a relation can be established?
I am particularly interested in the second homotopy groups of diffeomorphic spaces. Is the differential structure able to provide a relation between higher homotopy groups?
To be more specific, let $G$ be a Lie group, $M$ a differentiable manifold and $G\times{M}\to{M}$ a (left) transitive action. If $G_p$ is the isotropy group of some $p\in{M}$, then it is well known that the map: $$G/{G_p}\to{M}\quad,\quad\left[g\right]\mapsto{g}\cdot{p}$$ is a $G$-equivariant diffeomorphism. In this case, is there any relation between ${\pi}_{2}\left(G/{G_p},[e]\right)$ and ${\pi}_{2}\left(M,p\right)$ ? If not, is there any set of conditions (e.g. on the connectedness of the relevant spaces) so that an isomorphism between the above homotopy groups can be established?
We can write these things down concretely.
Given two manifolds $M$ and $N$, which are diffeomorphic through $f:M\rightarrow N$, then $f$ induces a map called $f_\ast$ on all homotopy groups by postcomposition (a homotopy class of maps $g:S^n\rightarrow M$ is taken to $f\circ g$). So we have a map $f_\ast:\pi_n(M)\rightarrow \pi_n(N)$ for all $n\geq 0$. This induced map has an inverse, namely the induced map of the inverse of $f$. To unravel this, if we have a representative $g:S^n\rightarrow M$, apply $f_\ast$ and then $(f^{-1})_\ast$, we obtain, $$S^n\overset{g}{\longrightarrow}M\overset{f}{\longrightarrow}N\overset{f^{-1}}{\longrightarrow}M.$$ Since $f^{-1}\circ f=id_M$, then $(f^{-1})_\ast\circ f_\ast=id_{\pi_n(M)}$. The same argument can be made for the other composition.
This works not just for diffeomorphism manifolds, but homeomorphic, and even homotopic spaces.
The converse is not always true however, there are spaces with isomorphic homotopy groups which are not homotopic spaces. Page 348 in Hatcher has an example of this.
The mathematical formalism that Qiaochu Yuan refers of functors and category theory is helpful to understand homotopy groups, algebraic topology, and lots of other areas of mathematics.