TITLE: Fundamental group of an orientable surface of infinite genus.
QUESTION [7 upvotes]: I am trying to calculate the fundamental group of an orientable surface $X$ of countably infinite genus.
The $1$-skeleton $Y$ of $X$ is infinite wedge of circles, so its fundamental group is free group on countably infinite generators, but I am not able to see how is the $2$-cell attached to $Y$. My guess is that it is attached by loop of product of commutators of generators but this product being infinite doesn't make sense in group.

REPLY [2 votes]: The fundamental group of every noncompact connected surface $S$ (oriented or not) is free and the surface itself is homotopy-equivalent to a graph.
See here.