Let $f : (X, A) → (Y, B)$ be a map such that both $f : X → Y$ and $f : A → B$ are
homotopy equivalences. Show that $f_∗ : H_n(X, A) → H_n(Y, B)$ is an isomorphism
for all $n$.
I know that there is a function $g:Y\to X, g:B\to A$ such that $fg\simeq Id_Y, gf\simeq Id_X$ and that $fg\simeq Id_B$ and $gf\simeq Id_A$. Is it true $g:(Y,B)\to (X,A)$ is the homotopic inverse of $f:(X,A)\to (Y,B)$? Thanks!
Best Answer
It is not necessarily true that $f$ is a homotopy equivalence of pairs--the homotopy $fg\simeq Id_Y$ may not map $B$ to itself at all times, for instance. (We know that there does exist a homotopy $fg\simeq Id_B$ restricted to $B$, but that homotopy need not be the restriction of our homotopy on all of $Y$.)
Instead, you can use the long exact sequence of homology for the pairs. There is a commutative diagram $$\require{AMScd} \begin{CD} H_n(A) @>{}>> H_n(X) @>>> H_n(X,A) @>>>H_{n-1}(A) @>{}>> H_{n-1}(X)\\ @VVV @VVV @VVV @VVV @VVV\\ H_n(B) @>{}>> H_n(Y) @>>> H_n(Y,B) @>>>H_{n-1}(B) @>{}>> H_{n-1}(Y) \end{CD}$$ where the rows are exact and the vertical maps are all induced by $f$. By hypothesis, all the vertical maps except the middle one are isomorphisms, so the middle one is an isomorphism too by the five lemma.