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Office_Shredder

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What do you mean by a sphere plus a diameter?

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If you have some kind of non-trivial bundle, I don't actually know - haven't learned about it yet.

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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 ).

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Just to give a quick relevant link:

http://mathworld.wolfram.com/SemilocallySimplyConnected.html

HTH

http://mathworld.wolfram.com/SemilocallySimplyConnected.html

HTH

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mathwonk

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mathwonk

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Hurkyl

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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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lavinia

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there is a general existence proof using paths but explicit constructions are not easy.

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Hurkyl

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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.

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lavinia

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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.

"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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mathwonk

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lavinia, did you mean dimension 4 or higher? or am I too celebratory today?

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