Results 1 to 2 of 2

Math Help - Help understanding van Kampen's theorem

  1. #1
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,935
    Thanks
    784

    Help understanding van Kampen's theorem

    Greetings,

    I am trying to figure out van Kampen's theorem. I understand it in some instances, but not in others. Here is an example:

    Let X be the space obtained by two tori S^1\times S^1 by identifying the circle S^1 \times \{x_0\} of each torus. (The space looks like one torus stacked on top of a second torus). In order to compute the fundamental group, I want to apply van Kampen's theorem. So, let A equal the whole space minus a circle S^1 \times \{x_1\} where x_1\neq x_0. Only remove the circle from one torus. Then, let B be the whole space minus the same circle from the other torus. Obviously, their union is the whole space, and both A and B deformation retract to a torus. Their intersection deformation retracts to a circle. So, the fundamental group is isomorphic to the quotient of the free product of the fundamental groups of two tori and the fundamental group of a circle. Hence, it is isomorphic to \mathbb{Z}^3.

    Now, if I calculate it a different way, I get confused about the quotient of the free product of A and B. Let x_2,x_3 be points on different tori, neither one is a point on the circles S^1\times \{x_0\} nor \{x_0\}\times S^1. Since X is path connected, let p be a path from x_2 to x_3. Let Y be the space obtained by attaching I\times I to X with the identification that I\times \{0\} follows the path p, and the rest is unidentified. Obviously, Y deformation retracts to X, so the two spaces are homotopic. Let A=Y\setminus\{x_2,x_3\} and let B=Y\setminus X \cup N_\epsilon(x_2) \cup N_\epsilon(x_3) (where \epsilon is very small). While it is not obvious, I can show that A deformation retracts to the wedge sum of three circles. It is obvious that B is simply connected. However, the intersection of A and B is homotopic to the wedge sum of two circles. This is where I run into problems with van Kampen's theorem, since the quotient of \mathbb{Z}^3 by \mathbb{Z}^2 is \mathbb{Z}, which I know is not the correct fundamental group of X. What am I missing?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,935
    Thanks
    784

    Re: Help understanding van Kampen's theorem

    I understand what I was doing wrong. Thank you anyway if you tried to offer advice.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help understanding what this theorem means
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: April 9th 2011, 01:48 AM
  2. Need help understanding proof of theorem
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: March 8th 2010, 09:41 AM
  3. Problem understanding the theorem
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: January 5th 2010, 08:30 PM
  4. Help understanding this theorem/proof
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: December 8th 2009, 04:39 PM
  5. Need help understanding the Chinese Remainder Theorem
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: November 20th 2009, 12:55 PM

Search Tags


/mathhelpforum @mathhelpforum