Algebraic Topology · Problem 2.1.27

Homology — Allen Hatcher

Mathematics — Topology · homology-induced-maps

Problem

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 - \{0\})

, show that

f

is not a homotopy equivalence of pairs --- there is no

g: (D^n, D^n - \{0\}) \to (D^n, S^{n-1})

such that

fg

and

gf

are homotopic to the identity through maps of pairs. [Observe that 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.]