We consider the genus two orientable surface $\Sigma_2$ and the map $r$ which is reflection along a plane cutting $\Sigma_2$ into two punctured tori. I am trying to read off the Lefschetz number of $r$. Well, what $r$ does on first homology is clear: We have four generators (two 'meridians' and two 'longitudes') and none of these is fixed by $r$. So the trace of $H_1(r)$ is zero. How do I quickly calculate $\operatorname{tr}(H_1(r))$ and $\operatorname{tr}(H_2(r))$?
My notes argue like this:
Clearly, $r$ has trace zero in degree $0$. For degree $2$ we use $H_2(\Sigma_2)\cong H_2(\Sigma_2,\Sigma_2\setminus\{x\})$ for some point $x$ in the intersection of the reflecting plane and $\Sigma_2$. Then the reflection $r$ changes the sign of the fundamental class in $H_2(\Sigma_2,\Sigma_2\setminus\{x\})\cong H_2(S^2,S^2\setminus\{x\})$ and we have $\operatorname{tr}(H_2(r))=-1.$
Why is $H_2(\Sigma_2)\cong H_2(\Sigma_2,\Sigma_2\setminus\{x\})$ true? The punctured genus two surface $\Sigma_2\setminus\{x\}$ is homotopy equivalent to the wedge of four circles. So the map $H_2(\Sigma_2)\rightarrow H_2(\Sigma_2,\Sigma_2\setminus\{x\})$ is injective but not surjective? Also how do you see that $r$ changes the sign of the fundamental class?
A quick way to see that $r$ has Lefschetz number zero is to note that the subspace of fixed points is a circle, which has Euler characteristic zero. As $r$ is (homotopic) to a simplicial homeomorphism this Euler characteristic is equal to the Lefschetz number.