@Balarka Kreck just told me the story about the punctured torus' fundamental group. If you blow up the puncture, which even is a homeo, you get what looks like two crossing strips (which of course retract to two wedged circles). The loop around the puncture in the punctured torus corresponds to the loop traversing the entire boundary of this figure, which, in terms of the loops around the two circles in this figure, which are the generators of the fundamental group of the torus, is their commutator. On the fundamental square, puncturing can be thought of as removing all 4 corners and this l…