Covering spaces

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how do you go about constructing covering spaces I know the definition of a covering and the usual ones for a circle and torus are easy to see but for example constructing a covering space of a sphere + a diameter how would you tackle something like this.
 

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  • #2
Office_Shredder
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What do you mean by a sphere plus a diameter?
 
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Well it depends on what kind of cover you are creating. If you want to construct the universal cover of, say, the cartesian product of two manifolds then it is not hard to see that its universal cover is the product of the universal cover of the first manifold with the universal cover of the second manifold. So, the universal cover of S^2 x S^1 is S^2 x R.

If you have some kind of non-trivial bundle, I don't actually know - haven't learned about it yet.
 
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If you want an actual sufficient condition for the existence of a covering space,
it is semilocal simple-connectedness. Yes, seems like a contrived concept, but it
works. Also, if you want to see the actual construction of a covering space of
a semilocally s,c space, look it up in Donald Kahn's book (Chaka's dad, and Genghis'
Great, Great, Great Grandfather :wink:).
 
  • #6
mathwonk
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covering spaces are constructed by taking paths in the base space. So the universal cover of a circle is given by the set of all paths on a circle starting at 1, mod homotopy. that gives sort of an infinite spiral over the circle.
 
  • #7
mathwonk
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i guess a sphere plus a diameter is the same as a sphere with a handle attached to the outside, so the universal cover should be long chain of such spheres.
 
  • #8
Hurkyl
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i guess a sphere plus a diameter is the same as a sphere with a handle attached to the outside, so the universal cover should be long chain of such spheres.
That doesn't sound right -- I think universal covers are supposed to be simply connected.

A sphere with a handle attached is a torus (?), so its universal cover should be the plane.

Edit: oh wait that doesn't work -- the universal cover is supposed to be a local homeomorphism, and the torus isn't locally homeomorphic to a sphere + diameter. :( I bet the universal cover is some quotient of the plane.
 
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What mathwonk described (a copy of R, with 2-spheres glued tangent to it at each integer point, for example) is simply connected, and I agree, that's what the universal cover of the OP's space would be.
 
  • #10
lavinia
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how do you go about constructing covering spaces I know the definition of a covering and the usual ones for a circle and torus are easy to see but for example constructing a covering space of a sphere + a diameter how would you tackle something like this.
there is a general existence proof using paths but explicit constructions are not easy.
 
  • #11
Hurkyl
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What mathwonk described (a copy of R, with 2-spheres glued tangent to it at each integer point, for example) is simply connected, and I agree, that's what the universal cover of the OP's space would be.
When I hear "handle" I think of cutting out two discs and attaching a cylinder.

But anyways, the space you describe isn't locally homeomorphic to the sphere + diameter: where the sphere meets the diameter, the space locally looks like a half-open interval with its endpoint attached to the center of an open disc.

But in the space you describe, locally to any lift of such a point, your space looks like an open interval with its midpoint attached to the center of an open disc.

And while I haven't fully wrapped my head around it, I think there is another serious problem with the fact that each of your spheres only touch the line once.



However, I think I now understand the right covering space -- alternate gluing intervals to spheres in a chain:
...-O-O-O-O-...​
 
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Yeah, you're right, I described the wrong thing. Yours is correct.
 
  • #13
lavinia
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what about the universal covering space of euclidean space minus a full lattice? is it already simply connected for euclidean space of dimension three or higher?
 
  • #14
mathwonk
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Hurkyl, by handle I meant a closed interval attached at both ends (i.e. just visualize the diameter on the outside), and by a long chain of spheres I meant exactly what you drew:

"However, I think I now understand the right covering space -- alternate gluing intervals to spheres in a chain:
...-O-O-O-O-..."

A real line with a string of spheres tangent to it would be the universal cover of a one point union of a sphere and a circle.

In general if X is simply connected, the universal cover of the one point union of X and a circle should be a real line with a string of X's each attached at one point, and the univ cover of X plus a (skinny) handle, would be your picture with X's instead of O's.

I agree a "handle" is usually something else, but I just tossed that word off informally thinking it was obvious what I meant, since a diameter is an interval. or maybe I'm losing my ability to communicate.

I.e. I meant a real world handle, like a wire handle on a bucket, not a mathematical handlebody (by the way, what's the univ cover of a bucket?). Sorry for the lack of clarity and precision. Most people have done this homework problem for a sphere and a circle joined at one point, so i was reducing it to that same picture by putting the diameter outside the sphere. I.e. you still do it by cutting the circle (the handle) apart in the middle, and then joining an infinite number of them together into a chain.
 
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  • #15
mathwonk
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lavinia, did you mean dimension 4 or higher? or am I too celebratory today?
 
  • #16
mathwonk
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oh, you mean the points of the lattice, not the edges joining the points. yes "clearly" the complement of the set of all points with integer coordinates, is simply connected in 3 space and higher. In 2 space the complement "obviously' retracts onto the union of the horizontal and vertical lines through the lattice points obtained by translating the previous set by the vector (1/2, 1/2), i.e. all vertical and horizontal lines of form x = a and y = b, where a,b are congruent to 1/2, mod 1. I'm having a little more trouble picturing the covering space. How about just a figure eight? what is the universal cover of that?
 

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