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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?

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    $\begingroup$ Diffeomorphic spaces are homeomorphic. Homeomorphic spaces are homotopy equivalent. All homotopy groups will be the isomorphic. $\endgroup$ – Neal Jan 21 '17 at 19:31
  • $\begingroup$ Oh, I thought that this holds only for the fundamental groups. I couldn't find any references on higher homotopy groups. Do you have any in mind? $\endgroup$ – user3257624 Jan 21 '17 at 19:34
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    $\begingroup$ @user3257624 I'm sure it's probably covered in Hatcher $\endgroup$ – Alex Mathers Jan 21 '17 at 19:43
  • $\begingroup$ Thank's, I found it as an exercise in chapter 4 (exercise 2), so I am assured of its validity. Do you have any other reference where it's actually proven? $\endgroup$ – user3257624 Jan 21 '17 at 20:01
  • $\begingroup$ There is no need to hunt down a reference. Everything follows straightforwardly from the fact that taking homotopy groups is a functor, and like all functors, it sends isomorphisms to isomorphisms. In this case, it sends isomorphisms of topological spaces (homeomorphisms) to isomorphisms of groups. $\endgroup$ – Qiaochu Yuan Jan 21 '17 at 20:06
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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.

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