# Hatcher 2.1.27

Hatcher, Algebraic Topology, Chapter 2, Section 1

27. Let $f:(X,A)\to(Y,B)$ be a map such that both $f:X\to Y$ and the restriction $f:A\to B$ are homotopy equivalences.

(a) Show that $f_*:H_n(X,A)\to H_n(Y,B)$ is an isomorphism for all $n$.

(b) For the case of the inclusion $f:(D^n,S^{n-1})\hookrightarrow(D^n,D^n-\left\{0\right\})$, show that $f$ is not a homotopy equivalence of pairs – there is no $g:(D^n,D^n-\left\{0\right\})\to(D^n,S^{n-1})$ such that $fg$ and $gf$ are homotopic to the identity through maps of pairs. [Observethat a homotopy equivalence of pairs $(X,A)\to(Y,B)$ is also a homotopy equivalence for the pairs obtained by replacing $A$ and $B$ by their closures.]
Proof.

$\square$