While embedding spaces are seductive at first blush, I've hope I've given you a taste of why they just don't work out like you would hope.

And finally, I've read that "GR is topologically incomplete". What that means exactly, I'm not sure, and we'd need the high priests to try to 'splain that (if they could). The way I understand it is that given space-times that are valid solutions of the EFE (such as our universe), there is some ambiquity in the "shape" in the sense that some topologist would use when he declared something topologically "complete".

IOW, all the information that GR provides is not enough to pin down the ultimate "shape of the universe".

Sorry for raining on the parade.

-Richard