Topology question

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Homework Statement


What is the fundamental group of A where A is the 2-sphere with two disjoint disks removed. It has the same homotopy type as a familiar space.


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The Attempt at a Solution


When I first looked at this problem, and saw how it was drawn out (in Munkres book,) it looked like a squashed sphere with two holes in it, so my first thought that it was homotopic to the double tours T#T. However, since the problem states that it is not the solid 2-sphere, I'm having second thoughts about it. To me it seems like its a sphere missing two holes in one hemisphere. It doesn't say anything about performing some surgery on the space and adding a cylinder or Mobius band to it, so it seems to me that it should be homotopic to the 2-sphere and therefore it's fundamental group is trivial. Am I on the right track here?
 

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  • #2
Hurkyl
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I think the easiest way to do this problem is to actually visualize a rubber sphere, removing two disks, and seeing what shape remains -- in my mind, at least, it's clear what shape that is. (It may help to visualize removing antipodal disks)


If you can't visualize it, then you could try computing it. A sphere with two disks removed is the same as a (sphere with one disk removed) with one disk removed. So first, can you say what a sphere with one disk removed looks like?


it seems to me that it should be homotopic to the 2-sphere and therefore it's fundamental group is trivial.
How do you construct that homotopy? What happened to the holes?
 
  • #3
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If you can't visualize it, then you could try computing it. A sphere with two disks removed is the same as a (sphere with one disk removed) with one disk removed. So first, can you say what a sphere with one disk removed looks like?
Wouldn't a sphere with one disk removed look like a disk? (For instance, chopping off the lower hemisphere or at least cutting a hole and stretching it out to a disk?
 
  • #4
quasar987
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The sphere with one disk removed is topologically like an open disk. Now, following Hurkyl's idea, what is an open disk with one disk removed like?

(topologically you can't go much further.. find what the holed disk is homotopically equivalent to)
 

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