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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.

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    @Mariano - right again! (I am not having a great day). I think what I meant was compose $f$ with either projection $S^1 \vee S^1 \to S^1$ to give a map $\phi: S^1 \to S^1$ (which I am not so sure has degree$2$anymore - I think that might be the projective plane). I was going to use a degree argument to try and calculate the induced map $f_*:H_1(S^1) \to H_1(S^1 \vee S^!)$ - which is really the crux of the problem2011-03-25

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The composition of the map $f:S^1\to S^1\vee S^1$ with one of the projections $S^1\vee S^1\to S^1$ is a map $S^1\to S^1$ of degree two: indeed, this composition turns around the codomain $S^1$ twice. Now it is easy to see that if $p_1$, $p_2:S^1\vee S^1\to S^1$ are the projections, then the map $\left(\begin{array}{c}p_{1*}\\p_{2*}\end{array}\right):H_1(S^1\vee S^1)\to H_1(S^1)\oplus H_1(S^1)$ is an isomorphism. It follows that the composition $\left(\begin{array}{c}p_{1*}\\p_{2*}\end{array}\right)\circ f_*$ is, in matrix terms, $\left(\begin{array}{c}2\\2\end{array}\right).$

It follows that your exact sequence is $0 \to \mathbb{Z} \stackrel{\left(\begin{array}{c}2\\2\end{array}\right)}{\to} \mathbb{Z} \oplus \mathbb{Z} \stackrel{i_*}{\to} H_1(K) \to \mathbb{Z}$

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    @Mariano - I will break this off into another question (hopefully that is OK) - this is new, and somewhat intriguing to me2011-03-25