Algebraic Topology · Problem 3F.7

Cohomology — Allen Hatcher

Mathematics — Topology · moore-space-short-exact-sequence-ext

Problem

Show that for a short exact sequence of abelian groups

0 \to A \to B \to C \to 0

, a Moore space

M(C, n)

can be realized as a quotient

M(B, n)/M(A, n)

. Applying the long exact sequence of cohomology for the pair

(M(B, n), M(A, n))

with any coefficient group

G

, deduce an exact sequence

0 \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