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Another implememntation of van Kampen thoerem.

  1. Sep 27, 2008 #1


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    There this question that someone gave me an answer to, in the ask topology forum, but I feel it's an immediate conclusion of this theorem mentioned in the title, it goes like this:

    Let X=X1UX2 and X1,X2 are open and simply connected in X, show that if the intersection of both sets X1 and X2 is composed of two path components, then [tex]\pi_1(X)[/tex] is isomorphic to Z the additive group of integers.

    Now I think, that what I need to find is that [tex]\pi_1(X1)[/tex] and [tex]\pi_1(X2)[/tex] are isomorphic to [tex]\pi_1(S^1)[/tex] which is isomorphic to Z, and then just use the above theorem, the question which isomorphism will do the job?

    any hints?

    thanks in advance.
  2. jcsd
  3. Oct 2, 2008 #2
    If [tex]X_1,X_2[/tex] are simply-connected, then by definition [tex]\pi_1(X_1) = \pi_1(X_2) = 0[/tex] which certainly will cause difficulties if you're trying to prove they are isomorphic to Z.
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