Hatcher-Algebraic-Topology · Problem 2.1.15

Homology

Mathematics — Graph Theory · exact-sequences-algebra

Problem

For an exact sequence A→B→C→D→EA \to B \to C \to D \to E show that C=0C = 0 iff the map A→BA \to B is surjective and D→ED \to E is injective. Hence for a pair of spaces (X,A)(X, A), the inclusion A↪XA \hookrightarrow X induces isomorphisms on all homology groups iff Hn(X,A)=0H_n(X, A) = 0 for all nn.

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