I want to prove that if a homeomorphism (a continuous bijection with continuous inverse) in a two dimensional manifold doesn't have a periodic point, then the Euler Characteristc of the manifold is zero.
In other words, if $M$ is a two dimensional compact manifold, and there exist a homeomorphism $h: M\longrightarrow M$ such that $h$ doesn't have a periodic point, then $\chi(M)=0$.
It implies for instance that a homeomorphism on a sphere always have periodic points.