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?


1 Answer 1


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 $i=r+s+1$.

  • $\begingroup$ Nice answer: +1. However I think that $\pi_2(D^2,S^1)=\mathbb Z$, not zero as you claim. $\endgroup$ Jun 13, 2016 at 8:58
  • $\begingroup$ I'm lazy to compute, ok I edited, thanks :) $\endgroup$ Jun 13, 2016 at 9:06
  • $\begingroup$ @GeorgesElencwajg I said $k\geq 3$ :P $\endgroup$ Jun 13, 2016 at 9:11
  • $\begingroup$ Perfect now! ${}{}$ $\endgroup$ Jun 13, 2016 at 9:13

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