3 ms·

I'm really confused. Sperical wraparound can be viewed as a disk with it's perimeter identified to a point. Maybe I'm just missing something obvious in the di

I'm really confused. Sperical wraparound can be viewed as a disk with it's perimeter identified to a point. Maybe I'm just missing something obvious in the discussion here? Also, how is Teichmüller Theory relevant? Is there some deep insight I'm just missing?

tgb 9y ago
Yes, that works. But it loses the geometry of the plane when you do that. Or in terms of topology it's not a covering space of the sphere. When you represent the torus as the plane cut up into squares, any small piece of the torus looks just like a small piece of the plane. But with your example, the North Pole (say) comes from this extra point you add in. So it's a topological quotient of the disk plus a point, but that's not a covering space. So it's much less useful, particularly if you want to compute distances. Distances (and angles and other geometric properties) behave well under coverings but not under arbitrary topological quotients like you have. In particular, there's no flat sphere so you can't possibly have it covered by the flat plane. Teichmuller theory isn't actually important here, just a connection to a piece of mathematics that's been in the news.
- xelxebar 9y ago
  Thanks. The discussion just says "torus" leaving everyone to infer what category we are working in. It seems like people are inferring different things, and that's what confused me. I'm not super familiar with homotopy theory but do know that universal covers pull back a lot of nice structure from the base space. Is there a theorem that makes this precise? Anyway, since the original article here discusses distances, I'd have thought that we'd be wanting to hold ourselves to isometries or something, which universal covers usually aren't, e.g. R^2/Z^2 isn't isometric to a doughnut.
  - tgb 9y ago
    Of course, you're right that what mathematical structures are the equivalent of the everyday terms we're talking about here depends on exactly what we're trying to accomplish. (Though note that the definition of trapezoid versus a square inherently makes use of lengths or angles hence implies we need to be working on geometric and not purely topological structures to be able to make any statement about trapezoids that are meaningful. I.e. a trapezoid can be the domain of a covering of a torus but only when we are interested only in the topology and not the geometry and hence only when "trapezoid" is indistinguishable from "square.") Yeah coverings can't be isometries (unless they're trivial), but they can always be local isometries which is what we'd most likely want here. R^2/Z^2 is (isometric to) a flat torus, which is, I'd argue, the nicest (though not the tastiest) torus. (Though the question of which flat torus is nicest is another matter and many would say it's not the torus that comes from the square!) I don't know of any general statement for you about what pulls back by coverings but in general anything you can describe as "local" pulls back. That's pretty vague though, sorry!

tgb 9y ago

Yes, that works. But it loses the geometry of the plane when you do that. Or in terms of topology it's not a covering space of the sphere. When you represent the torus as the plane cut up into squares, any small piece of the torus looks just like a small piece of the plane. But with your example, the North Pole (say) comes from this extra point you add in. So it's a topological quotient of the disk plus a point, but that's not a covering space. So it's much less useful, particularly if you want to compute distances. Distances (and angles and other geometric properties) behave well under coverings but not under arbitrary topological quotients like you have. In particular, there's no flat sphere so you can't possibly have it covered by the flat plane. Teichmuller theory isn't actually important here, just a connection to a piece of mathematics that's been in the news.

xelxebar 9y ago
Thanks. The discussion just says "torus" leaving everyone to infer what category we are working in. It seems like people are inferring different things, and that's what confused me. I'm not super familiar with homotopy theory but do know that universal covers pull back a lot of nice structure from the base space. Is there a theorem that makes this precise? Anyway, since the original article here discusses distances, I'd have thought that we'd be wanting to hold ourselves to isometries or something, which universal covers usually aren't, e.g. R^2/Z^2 isn't isometric to a doughnut.
- tgb 9y ago
  Of course, you're right that what mathematical structures are the equivalent of the everyday terms we're talking about here depends on exactly what we're trying to accomplish. (Though note that the definition of trapezoid versus a square inherently makes use of lengths or angles hence implies we need to be working on geometric and not purely topological structures to be able to make any statement about trapezoids that are meaningful. I.e. a trapezoid can be the domain of a covering of a torus but only when we are interested only in the topology and not the geometry and hence only when "trapezoid" is indistinguishable from "square.") Yeah coverings can't be isometries (unless they're trivial), but they can always be local isometries which is what we'd most likely want here. R^2/Z^2 is (isometric to) a flat torus, which is, I'd argue, the nicest (though not the tastiest) torus. (Though the question of which flat torus is nicest is another matter and many would say it's not the torus that comes from the square!) I don't know of any general statement for you about what pulls back by coverings but in general anything you can describe as "local" pulls back. That's pretty vague though, sorry!