Results 1 to 4 of 4

Math Help - Fundmental Group

  1. #1
    Junior Member
    Joined
    Apr 2008
    Posts
    48

    Fundmental Group

    How would I go about calculating the fundamental group of R^3 minus the x-y plane? I'm guessing the Van Kampen Theorem will be used somewhere, not really sure, any ideas? Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by skamoni View Post
    How would I go about calculating the fundamental group of R^3 minus the x-y plane? I'm guessing the Van Kampen Theorem will be used somewhere, not really sure, any ideas? Thanks.
    What is the deformation retract of R^3 minus the x-y plane? It is just two points whose fundamental group is a trivial group.
    I don't think Van Kampen theorem is needed here.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2008
    Posts
    48
    Thanks.

    How about the union of a sphere (in R3) and a disc (in the x-y plane), am I right in thinking that since both are simply connected, the union will be, and so the fundamental group trivial?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by skamoni View Post
    Thanks.

    How about the union of a sphere (in R3) and a disc (in the x-y plane), am I right in thinking that since both are simply connected, the union will be, and so the fundamental group trivial?
    Whether your union is a union having an intersection or one-pointed union (wedge sum) or disjoint union, the fundamental group of your space is trivial, implying that every loop in your space is homotopic to the constant map.
    For example, if your union is one-pointed union (wedge sum) and you apply the Van Kampen theorem, the fundamental group of your space is still trivial since the free product of two trivial groups is trivial.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Order of Group. Direct Product of Cyclic Group
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: November 19th 2011, 01:06 PM
  2. another book about fundmental theorem of algebra problem...
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: July 15th 2011, 03:03 PM
  3. Replies: 1
    Last Post: November 4th 2009, 09:52 AM
  4. Quick questions on Group Theory - Cosets / Normal Group
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 16th 2009, 08:39 AM
  5. Group Theory Question, Dihedral Group
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 4th 2008, 10:36 AM

Search Tags


/mathhelpforum @mathhelpforum