Let $(X,A)$ be a pair of topological spaces, where $X$ is path-connected and $A$ is a path-connected subespace of $X$ with a base point. So, we have a long exact sequence of homotopy groups $$... \to \pi_n A \to \pi_n X \to \pi_n (X,A) \to \pi_{n-1} A \to \pi_{n-1} X \to ...$$ Let $i:A \to X$ be the inclusion, with homotopy fiber $F(i) = F$, sequence $F \to A \to X$ and long exact sequence of fibration $$... \to \pi_n A \to \pi_n X \to \pi_{n-1} F \to \pi_{n-1} A \to \pi_{n-1} X \to ...$$ Is it true that $\pi_n (X,A) \cong \pi_{n-1} F$? How can I prove that?

Thanks in advance


If I am not confusing anything, you can prove it using 5-lemma.

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  • $\begingroup$ I suspected that, Sasha. Do you know the map $\pi_n (X,A) \to \pi_{n-1} F$ which turns out to be the isomorphism? Thanks $\endgroup$ – Gustavo Nov 11 '14 at 20:47

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