Algebraic Topology · Problem 2.2.26

Homology — Allen Hatcher

Mathematics — Topology · relative-homology

Problem

For a pair (X,A)(X, A), let X∪CAX \cup CA be XX with a cone on AA attached. (a) Show that XX is a retract of X∪CAX \cup CA iff AA is contractible in XX: There is a homotopy ft:A→Xf_t: A \to X with f0f_0 the inclusion A↪XA \hookrightarrow X and f1f_1 a constant map. (b) Show that if AA is contractible in XX then Hn(X,A)≈H~n(X)⊕H~n−1(A)H_n(X, A) \approx \widetilde{H}_n(X) \oplus \widetilde{H}_{n-1}(A), using the fact that (X∪CA)/X(X \cup CA)/X is the suspension SASA of AA.

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