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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!

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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.

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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".

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