I've come across another exact sequence, where (I guess) I need to deduce the result using some properties of torsion.
I am calculating the homology of the Klein bottle using attaching maps. I start by defining $\Phi:I \times I \to K$ as the natural map and denote $\partial(I \times I)$ as the boundary, then let $f=\Phi|\partial(I \times I)$. We can then regard $f$ as a function $S^1 \to S^1 \times S^1$.
I think I can show that the induced map $f_*: H_1(S^1) \to H_1(S^1 \vee S^1)$ has degree 2 (i.e. is multiplication by 2)
It boils down to the following exact sequence and have come across the following exact sequence (I am trying to calculate $H_1(K)$)
$0 \to \mathbb{Z} \stackrel{f_*}{\to} \mathbb{Z} \oplus \mathbb{Z} \stackrel{i_*}{\to} H_1(K) \to \mathbb{Z}$
I know that $H_1(K)$ must have rank 1 (from the Euler characteristic of the Klein bottle)
I note that previously when I had a sequence $H_1(S^1 \vee S^1) \to H_1(T) \to H_0(S^1)$ the book concluded that $H_1(T)$ was torsion free (here $T$ is the torus), but as Jim pointed out to me, $H_1(K)$ is not torsion free this time.