Algebraic Topology · Problem 2.2.40

Homology — Allen Hatcher

Mathematics — Topology · homology-with-coefficients

Problem

From the long exact sequence of homology groups associated to the short exact sequence of chain complexes 0→Ci(X)→nCi(X)→Ci(X;Zn)→00 \to C_i(X) \xrightarrow{n} C_i(X) \to C_i(X; \mathbb{Z}_n) \to 0 deduce immediately that there are short exact sequences 0→Hi(X)/nHi(X)→Hi(X;Zn)→n-Torsion(Hi−1(X))→00 \to H_i(X)/nH_i(X) \to H_i(X; \mathbb{Z}_n) \to n\text{-Torsion}(H_{i-1}(X)) \to 0 where n-Torsion(G)n\text{-Torsion}(G) is the kernel of the map G→nGG \xrightarrow{n} G, g↦ngg \mapsto ng. Use this to show that H~i(X;Zp)=0\widetilde{H}_i(X; \mathbb{Z}_p) = 0 for all ii and all primes pp iff H~i(X)\widetilde{H}_i(X) is a vector space over Q\mathbb{Q} for all ii.

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