TITLE: Why is a Kummer surface simply-connected?
QUESTION [10 upvotes]: By a kummer surface, I mean a quotient of two-dimensional complex torus by multiplication by $-1$. It is a K3 surface and known to be simply connected.
But it is not clear to me why it is simply-connected from its construction, which is a quotient of a surface with infinite fundamental group.
Is there a direct explanation or any references? 
Thanks.

REPLY [4 votes]: I haven't checked details but the following might work. Coverings of the Kummer surface will lift to coverings of the abelian surface (whose fundamental group is clearly $\mathbb{Z}^4$), so the fundamental group of the Kummer surface is abelian. Now it's enough to show that the first homology group is trivial. The latter I expect you will find in textbooks, e.g. as a computation of the Hodge diamond of a K3 surface.