Hatcher-Algebraic-Topology · Problem 3F.7

Cohomology

Mathematics — Graph Theory · limits-and-ext

Problem

Show that for a short exact sequence of abelian groups 0→A→B→C→00 \to A \to B \to C \to 0, a Moore space M(C,n)M(C, n) can be realized as a quotient M(B,n)/M(A,n)M(B, n)/M(A, n). Applying the long exact sequence of cohomology for the pair (M(B,n),M(A,n))(M(B,n), M(A,n)) with any coefficient group GG, deduce an exact sequence 0→Hom(C,G)→Hom(B,G)→Hom(A,G)→Ext(C,G)→Ext(B,G)→Ext(A,G)→00 \to \mathrm{Hom}(C, G) \to \mathrm{Hom}(B, G) \to \mathrm{Hom}(A, G) \to \mathrm{Ext}(C, G) \to \mathrm{Ext}(B, G) \to \mathrm{Ext}(A, G) \to 0

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