Algebraic Topology · Problem 2.3.1

Homology — Allen Hatcher

Mathematics — Topology · axioms-for-homology

Problem

T_n(X, A)

denotes the torsion subgroup of

H_n(X, A; \mathbb{Z})

, show that the functors

(X, A) \mapsto T_n(X, A)

, with the obvious induced homomorphisms

T_n(X, A) \to T_n(Y, B)

and boundary maps

T_n(X, A) \to T_{n-1}(A)

, do not define a homology theory. Do the same for the 'mod torsion' functor

MT_n(X, A) = H_n(X, A; \mathbb{Z})/T_n(X, A)