Hatcher Exercise 2.2.30(a)

Question

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.

Answer 1

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 $f_*= 1$.