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Let $f$ be a map from $X= S^1 \times S^1$ to itself that is the identity on one factor and a reflection on the other. Then how does the induced map $f_* :H_2(X)\to H_2(X)$ look like?

Since $f$ is just a reflection, I guess $f_* = -\mathbb{1}$, but then I have no idea how I can prove it. I encountered this problem when I was solving exercises in Hatcher, where I had to compute the homology groups of the mapping torus $T_f$ of $f$. In Hatcher, we have the long exact sequence $$ 0\to H_3(T_f) \to H_2(X) \xrightarrow{\mathbb{1} -f_*} H_2(X) \to H_2(T_f) \to H_{1}(X) \to \ldots $$ so I have to know the map $\mathbb{1}-f_*$.

For spheres, we have a very simple simplicial structure on it, so it is easy to calculuate the degree of the reflection map, but how do I calculate it for general complexes like $S^1 \times S^1$? I tried to give a symmetric-looking simplicial structure on $S^1 \times S^1$, but it requires too many simplices.

I also have no idea for the induced map on $H_1(X)$, but I guess I can get some idea if I know the case of second homology group.

JWL
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  • One way to understand the map would be to write down what this second homology group explicitly. In this case, it's easy because most terms in the Kunneth formula vanish. It might be easier to use a CW complex instead of a simplicial structure (it's in Hatcher) – Michael Burr Mar 07 '15 at 13:28
  • @MichaelBurr Well, I don't know much about Kunneth (googling about it right now), but I know that the second homology group of a torus is $\mathbb{Z}$. – JWL Mar 07 '15 at 13:31
  • Usually, it's easier to see these maps (at least for me) is to look at them on the level of the chain complexes and then bring the homology into the picture. – Michael Burr Mar 07 '15 at 13:38
  • @MichaelBurr Well I tried to give a CW complex structure on the torus by cutting the torus into half so that I can visualize the action of the reflection map on the two 2-cells $A,B$. The reflection map swaps $A$ and $B$. However, I cannot figure out if $H_2(S^1 \times S^1)$ is generated by $A+B$ or $A-B$. The issue doesn't matter when computing the homology group of a torus, and I am very confused about sign issues in boundary maps of computation of cellular homology. – JWL Mar 07 '15 at 14:01
  • I think I have the answer for the $f_*$ on the second homology, by identifying $H_2(S^1\times S^1)$ by $H_2(U, U-x)$ for some neighborhood of $U$ and computing the local degree of $x$. Since it is a reflection, it has degree $-1$. – JWL Mar 07 '15 at 14:55
  • I think the concept of local degree helps me for the rest of the problem. Thanks for your help Michael. – JWL Mar 07 '15 at 14:58

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