mod torsion functor does define/does not define a homology theory.

Question

(Hatcher Exercise 2.3.1) If $T_n(X,A)$ denote the torsion subgroup of $H_n(X,A;\Bbb Z)$, 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 $MT_n(X,A) = H_n(X,A;\Bbb Z)/T_n(X,A)$.

This question is already posted before. From this answer, to show 'mod torsion' functor does not define a homology theory, I need to find some pair $(X,A)$ that fails to induce a l.e.s. but I couldn't find. Umm... does it define a homology theory?

Answer 1

We use the same counterexample from the linked answer. Let $X$ be the Mobius band and $A$ its boundary. Looking at part of the long exact sequence $$\cdots \to H_1(A) \to H_1(X) \to H_1(X,A) \to \cdots,$$ we have $$\cdots \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \to \mathbb{Z}/2 \to \cdots.$$

Modding out torsion, we get the sequence $$\cdots \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \to 0 \to \cdots,$$ which is not exact. Therefore "mod torsion homology" is not a homology theory.