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

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