I am no topologist, but I find pictures like this amusing: http://math.stackexchange.com/questions/479371/constructing-a-genus-2-surface-from-8-gon.

Also, this is a cool theorem (and I didn't know where else to put it): If $X$ is a complex manifold of dimension $n$ that is compact and connected, and $K(X)$ is the field of globally defined meromorphic functions on $X$, then the transcendental degree of $K(X)$ (viewed as a field extension of $\mathbb{C}$) is at most $n$.

http://math.stackexchange.com/questions/29935/f3-g3-1-for-two-meromorphic-functions/29936#29936