Results 1 to 3 of 3

Math Help - Deformation retracts?

  1. #1
    Member
    Joined
    Feb 2009
    Posts
    98

    Deformation retracts?

    Let x_0 \in S^1.

    Is S^1 X  \{ x_0 \} a retract of  S^1 X  S^1 ?

    Is it a deformation retract of S^1 X S^1 ?


    I would think one would use the fact that if S^1 X  \{ x_0 \} were a deformation retract of S^1 X S^1 , then their fundamental groups would be isomorphic.

    But the fundamental group of S^1 X S^1 is ZXZ, while I think the fundamental group of S^1 X \{ x_0 \} is ZX{0}. Would this be enough to justify it's not a deformation retract? If yes, I have a feeling that S^1 X \{ x_0 \} is a retract of S^1 X S^1 . But how do you prove this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member HappyJoe's Avatar
    Joined
    Sep 2010
    From
    Denmark
    Posts
    234
    You are quite right.

    A deformation retract is a special case of a homotopy equivalence. Hence, if a subspace A was a deformation retract of the space X, the two spaces would indeed have the same fundamental group (which they don't in your case).

    To show that S^1\times\{x_0\} is however a retract of S^1\times S^1, you just use the first map that comes to mind, i.e. use the map f\colon S^1\times S^1\rightarrow S^1\times\{x_0\} given by f(x,y) = (x,x_0), so you map all of the second copy of S^1 to the point x_0
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2009
    Posts
    98
    Thanks, this makes a lot of sense.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Deformation gradient tensor and FEA question
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: December 6th 2011, 06:24 AM
  2. Deformation gradient tensor from coordinate data?
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: June 16th 2011, 11:53 AM
  3. Finite Deformation Tensor in Cylindrical Coordinates
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 3rd 2009, 11:31 AM
  4. Retracts of the Mobius Band
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 27th 2009, 10:29 PM
  5. torus, homeomorphic, deformation retraction
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 19th 2009, 10:58 AM

Search Tags


/mathhelpforum @mathhelpforum