What is the Genus of Riemann Surface?

[yicong2011]

I learned something about genus in Topology. The concept Genus in Topology is intuitive and lucid. Now I am confronted with the Genus in Riemann Surface. I do not know what is Genus on Riemann Surface. Is it relevant to "singularity"?

Anyone can help me make it a bit clear?
Thanks.

 

[homeomorphic]

Same thing as genus in topology. A Riemann surface is just a topological surface with extra structure on it. It therefore has a genus.

 

[mathwonk]

here is a set of class notes from my course on RS.