How Do We Know the Universe is Flat? Discovering the Topology of the Universe

Wish you had explained why you imposed a right hand turn rule, since the easiest way to get back to where you started on a sphere is to simply keep walking in a "straight" line.

Also the universe may be very close to flat on the largest scales, but locally it is a bit lumpy where there are mass concentrations, as evidenced by gravitational lensing.

Actually topology does not have much to do with curvature, but rather with the way parts of a surface or space are connected. A water glass has the same topology as a sphere, but a coffee cup, with a hole in it, does not. I think the topo- word Fraser was looking for was probably topography.

When I hear the term, "flat", I think of a 2 dimensional plane. The way you describe it here confuses me.