It is known that if $(X,A)$ is a good pair, for example a $CW$ pair, then $H_k(X,A)\simeq H_k(X/A)$ for every $k$. Is it true for homotopy groups of $CW$ pairs? If not, what is the counter-example?
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No it is false . $\pi_k(D^2,S^1)=0$ for $k\geq 3$ ( follows from long exact sequence of Homotopy pairs). But $\pi_k(D^2/S^1)=\pi_k(S^2)$ which is non-zero for infinitely many $k$.
But there is a partial result which says , if a CW pair $(X,A)$ is $r-$connected and $A$ is $s-$connected, with $r,s\geq 0$, then the map $\pi_i(X,A)\to \pi_i(X/A)$ induced by quotient map $X\to X/A$ is an isomorphism for $i\leq r+s$ and a surjection for $i=r+s+1$.
Anubhav Mukherjee
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Nice answer: +1. However I think that $\pi_2(D^2,S^1)=\mathbb Z$, not zero as you claim. – Georges Elencwajg Jun 13 '16 at 08:58
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I'm lazy to compute, ok I edited, thanks :) – Anubhav Mukherjee Jun 13 '16 at 09:06
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@GeorgesElencwajg I said $k\geq 3$ :P – Anubhav Mukherjee Jun 13 '16 at 09:11
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Perfect now! ${}{}$ – Georges Elencwajg Jun 13 '16 at 09:13