# Hatcher Exercise 2.2.30(a)

For the mapping torus $$T_f$$ of a map $$f:X\to X$$, we have a long exact sequence $$\cdots\to H_n(X)\xrightarrow{1-f_*}H_n(X)\to H_n(T_f)\to H_{n-1}(X)\to\cdots.$$ Use this to compute the homology of the mapping tori of the following maps: (a) A reflection $$S^2\to S^2$$, (b) ...

Anyway, I have $$0\to H_1(T_f)\to H_0(S^2)\to H_0(S^2)\to H_0(T_f)\to 0.$$ Since $$\deg(-1) = -1$$ at $$0$$ degree, $$1-(-1)_*$$ is multiplication by $$2$$. Hence, $$H_1(T_f) =0$$ and $$H_0(T_f) = \Bbb Z/2$$ which is weird since $$H_0$$ should be free. I think interpreting the middle map by multiplication by $$2$$ is wrong but don't know how to correct. Please help.

It is not true that $$\text{deg}(-1)= -1$$ at degree $$0$$. $$H_0(X)$$ is a free abelian group with generators being the path components of $$X$$. If $$f:X \to Y$$, then $$f_*$$ maps the path components of $$X$$ to those of $$Y$$. Thus for $$X=Y=S^2$$ we always get $$f_*= 1$$.