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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.

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    I was describing homotopy-type, not homeomorphism type.2014-10-22

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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.