# Let $f : (X, A) → (Y, B)$ be a map such that both $f : X → Y$ and $f : A → B$ are homotopy equivalences.

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!

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.

The question about homology is answered by Eric.

The question of obtaining a homotopy equivalence of pairs assuming $$A \to X, B \to Y$$ are cofibrations is answered by 7.4.2 of Topology and Groupoids,(T&G), and the Addendum to that result shows how the homotopy inverse to $$f$$, and the homotopies, may be constructed from the given data. That answers your question on $$g$$. Here is a slightly transliteratwed version of that Addendum_:

Let $$g^{0} : B \to A$$ be any homotopy inverse of $$f^{0}:A \to B$$ and let $$H^{0}_{t} : f^{0}g^{0} \simeq 1$$, $$K^{0}_{t} : g^{0}f^{0} \simeq 1$$ be homotopies. Then $$g^{0}$$ extends to a homotopy inverse $$g':Y \to X$$ of $$f$$ such that the homotopy $$fg'\simeq 1$$ extends $$H^{0}_{t}$$ while the homotopy $$g'f\simeq 1$$ extends the sum $$K^{0} + g^{0}H^{0}f^{0} - g^{0}f^{0}K^{0}$$ of the homotopies $$g^{0}f^{0} = g^{0}f^{0}1_{A} \simeq g^{0}f^{0}g^{0}f^{0} \simeq g^{0}1_{B}f^{0} \simeq 1_{A}$$ determined by $$H^{0}_{t}$$ and $$K^{0}_{t}$$.

I do not know of examples showing that the description of the homotopies cannot be simplified. However the explicit form has useful implications, for example if $$H,K$$ are constant homotopies, and also to a gluing theorem (7.4.3 of T&G). It is not clear if $$g'$$ is homotopic to $$g$$.

These results were found in the 1960s by generalising the well known result that a homotopy equivalence $$f: X \to Y$$ induces an isomorphism of homotopy groups $$\pi_n(X,x) \to \pi_n(Y, f(x))$$. That proof involves an operation of fundamental groups on homotopy groups, so the more general result uses a more general operation, as in T&G, compare Section 5.2 of tom Dieck's "Algebraic Topology".