Algebraic Topology · Problem 2.2.43

Homology — Allen Hatcher

Mathematics — Topology · homology-with-coefficients

Problem

(a) Show that a chain complex of free abelian groups CnC_n splits as a direct sum of subcomplexes 0→Ln+1→Kn→00 \to L_{n+1} \to K_n \to 0 with at most two nonzero terms. [Show the short exact sequence 0→ker⁡∂→Cn→Im⁡∂→00 \to \ker\partial \to C_n \to \operatorname{Im}\partial \to 0 splits and take Kn=ker⁡∂K_n = \ker\partial.] (b) In case the groups CnC_n are finitely generated, show there is a further splitting into summands 0→Z→00 \to \mathbb{Z} \to 0 and 0→Z→mZ→00 \to \mathbb{Z} \xrightarrow{m} \mathbb{Z} \to 0. [Reduce the matrix of the boundary map Ln+1→KnL_{n+1} \to K_n to echelon form by elementary row and column operations.] (c) Deduce that if XX is a CW complex with finitely many cells in each dimension, then Hn(X;G)H_n(X; G) is the direct sum of the following groups: • a copy of GG for each Z\mathbb{Z} summand of Hn(X)H_n(X) • a copy of G/mGG/mG for each Zm\mathbb{Z}_m summand of Hn(X)H_n(X) • a copy of the kernel of G→mGG \xrightarrow{m} G for each Zm\mathbb{Z}_m summand of Hn−1(X)H_{n-1}(X)

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