Another implememntation of van Kampen thoerem.

  • #1
MathematicalPhysicist
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Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
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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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