Results 1 to 2 of 2

Math Help - Hausdorff space, structure

  1. #1
    Junior Member
    Joined
    Nov 2008
    Posts
    25

    Hausdorff space, structure

    Let X be a Hausdorff space such that X= A \cup B where A and B are each homeomorphic to a torus, and A \cap B =\{x_0\}.

    What is the structure of \pi_1(X, x_0)?

    How do I describe the structure of this topological space's fundamental group? I know how to draw it, etc. But, I don't know much about this space's fundamental group.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by Erdos32212 View Post
    Let X be a Hausdorff space such that X= A \cup B where A and B are each homeomorphic to a torus, and A \cap B =\{x_0\}.

    What is the structure of \pi_1(X, x_0)?

    How do I describe the structure of this topological space's fundamental group? I know how to draw it, etc. But, I don't know much about this space's fundamental group.
    Lemma 1. Let X = U \cup V , where U and V are open sets of X. If U \cap V is simply connected, then \pi(X) is the free product of groups \pi(U) and \pi(V) with respect to the homomorphisms \phi_1 : \pi(U) \rightarrow \pi(X) and \phi_2 : \pi(V) \rightarrow \pi(X).

    If A and B were open subsets of X, we could apply Lemma 1 with U=A and V=B to determine the structure of \pi_1(X, x_0).

    Alternatively, we need to choose two open sets U and V such that A and B are deformation retracts of U and V, respectively.

    Choose the circles and lines S_1, l_1 \in A and S_2, l_2 \in B such that if you remove the S_1, l_1 from A and S_2,l_2 from B, A and B should become simply connected (Think of a reverse procedure of making a torus from a rectangle paper).

    Let U=X - S_2 -l_2 and V=X - S_1 -l_1, where x_0 does not belong to any S_1, S_2, l_1 and  l_2. Then, A and B are deformation retracts of U and V, respectively

    Now, we have X = U \cup V , and U \cap V = X -S_1 -S_2 -l_1 -l_2 , where U \cap V is contractible.

    Then, by lemma 1, we conclude that \pi_1(X, x_0) is the free product of groups \pi_1(U, x_0) and \pi_1(V, x_0), or the free product of groups \pi_1(A, x_0) and \pi_1(B, x_0).

    Since \pi_1(A) \cong Z \times Z and \pi_1(B) \cong Z \times Z, \pi_1(X, x_0) is isomorphic to the free product of groups Z \times Z and Z \times Z.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] A subspace of a Hausdorff space is Hausdorff
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: September 24th 2011, 01:40 PM
  2. [SOLVED] Hausdorff space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 19th 2011, 04:16 PM
  3. Hausdorff and Compact Space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 29th 2010, 03:13 AM
  4. Quotient map from Hausdorff to non-Hausdorff space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 28th 2009, 05:51 PM
  5. Hausdorff Space- Topology
    Posted in the Calculus Forum
    Replies: 0
    Last Post: April 5th 2008, 05:42 PM

Search Tags


/mathhelpforum @mathhelpforum