I was looking at the following problem:
For which connected compact surfaces $X$ without boundary does there exist a continuous map $f: X\rightarrow X$ with no fixed point?
There is also a hint to the problem which says the following:
Verify by inspection that if $\mathbb{Z}$ is a direct summand of $H_1(X)$, then $S^1$ is a retract of $X$.
I was having no idea on this problem, and even don't know how to derive the hint. The only thought that I have is that both $T^2$ and $S^2$ would allow such self-map, e.g. a rotation map on $T^2$, and antipodal map on $S^2$. I have no idea what happens to $\mathbb{R}P^2$ or the connected sums.
Any help would be appreciated!