Results 1 to 6 of 6

Math Help - Cylinder and Torus minus a point

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    18

    Cylinder and Torus minus a point

    Hey everyone,
    I have to show that the cylinder S^1 \times \mathbb{R} minus a point is homotopy equivalent to the torus T^2 minus a point. What is the quickest way to do it?
    I would really appreciate any help you can give me. Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,790
    Thanks
    1531
    Quote Originally Posted by DJDorianGray View Post
    Hey everyone,
    I have to show that the cylinder S^1 \times \mathbb{R} minus a point is homotopy equivalent to the torus T^2 minus a point. What is the quickest way to do it?
    I would really appreciate any help you can give me. Thanks.
    Well, look at the closed paths not homotopic to a point. On a cylinder minus a point, we can have paths that go around the cylinder, paths that do not go around the cylinder but do go around the missing point, and paths that go around either the cylinder or the point. Do you see that that gives three distinct homotopy classes? Do the same thing for the torus minus a point.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by HallsofIvy View Post
    Well, look at the closed paths not homotopic to a point. On a cylinder minus a point, we can have paths that go around the cylinder, paths that do not go around the cylinder but do go around the missing point, and paths that go around either the cylinder or the point. Do you see that that gives three distinct homotopy classes? Do the same thing for the torus minus a point.

    I guess that three lines before the end it should be "...paths that go NEITHER around the..."

    Tonio
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Nov 2009
    Posts
    54
    Quote Originally Posted by tonio View Post
    I guess that three lines before the end it should be "...paths that go NEITHER around the..."

    Tonio

    Paths that go neither around the cylinder nor the removed point are homotopic to a point, right? So you really only need concern yourself with the other two types of paths.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by cribby View Post
    Paths that go neither around the cylinder nor the removed point are homotopic to a point, right? So you really only need concern yourself with the other two types of paths.

    Well, a null-homotopic path is ALSO a path, but yes: the interesting ones are the other two kinds.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by DJDorianGray View Post
    Hey everyone,
    I have to show that the cylinder S^1 \times \mathbb{R} minus a point is homotopy equivalent to the torus T^2 minus a point. What is the quickest way to do it?
    I would really appreciate any help you can give me. Thanks.
    My suggestion is to use polygons with identifications (link).



    If we pairwise identify both As and Bs, then the resulting space is a torus. Now, WLOG, let's make a hole in the center. Then it deformation retracts to its boundary. When we pairwise identify both As and Bs, the resulting space is a wedge sum of two circles, a.k.a, figure 8 space.

    For a space S^1 \times R, we can use the above figure with a slight modification. We don't pairwise identify either As or Bs. Assume we pairwise identify Bs only. Actually, B is the real line, so we cannot draw using a polygon. This is just for illustration purpose. The resulting space having a hole at the center after deformation retract is "O-O", where "-" is a real line. A real line can be deformation retracted to a point, "O-O" is homotopy equivalent to "OO", which is a figure 8 space.

    Thus we conclude that both spaces are homotopy equivalent.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Difference between plus minus and minus plus
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: December 23rd 2011, 04:14 AM
  2. torus
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 2nd 2009, 06:32 AM
  3. plus/minus in TeX
    Posted in the LaTeX Help Forum
    Replies: 2
    Last Post: May 7th 2009, 03:44 PM
  4. phi(x/2) and w(x) minus pi(x)=???
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: April 7th 2009, 03:30 AM
  5. Volume of sphere minus cylinder
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 3rd 2009, 12:10 PM

Search Tags


/mathhelpforum @mathhelpforum