Hatcher 2.1.15

Hatcher, Algebraic Topology, Chapter 2, Section 1

15. For an exact sequence A\to B\to C\to D\to E show that C=0 if and only if the map A\to B is surjective and D\to E is injective. Hence, for a pair of spaces (X,A), the inclusion A\hookrightarrow X induces isomorphisms on all homology groups if and only if H_n(X,A)=0 for all n.
Proof.

\square

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