If we take a square and identify opposite sides, we get a torus. If we change the direction of one pair of sides we get the Klein bottle. If we change the direction of both sides, we get first a Möbius strip, then when I tried to glue the opposite sides I got a sphere with a singularity (ie the band is twisted into a point), is this process allowed, ie is it a topological sphere, is it possible to glue this together without a singularity if I had some extra spatial dimensions?
And given an $2n$-gon of sides $a_1$ to $a_{2n}$, which identifications of sides (pairwise and with $4$ possible directions), result in a smooth compact manifold, is there some rule to determine which manifold, and are all identifications allowed? How many diemsnions are needed for a smooth gluing process?