Hatcher-Algebraic-Topology · Problem 2.3.1

Homology

Mathematics — Graph Theory · homology-theory-axioms

Problem

If Tn(X,A)T_n(X, A) denotes the torsion subgroup of Hn(X,A;Z)H_n(X, A; \mathbb{Z}), show that the functors (X,A)↦Tn(X,A)(X, A) \mapsto T_n(X, A), with the obvious induced homomorphisms Tn(X,A)→Tn(Y,B)T_n(X, A) \to T_n(Y, B) and boundary maps Tn(X,A)→Tn−1(A)T_n(X, A) \to T_{n-1}(A), do not define a homology theory. Do the same for the 'mod torsion' functor MTn(X,A)=Hn(X,A;Z)/Tn(X,A)MT_n(X, A) = H_n(X, A; \mathbb{Z})/T_n(X, A).

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