Another implememntation of van Kampen thoerem.

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The discussion centers on the application of the van Kampen theorem in topology, specifically regarding the fundamental group π₁ of a space X composed of two open, simply connected subsets X₁ and X₂. It concludes that if the intersection of X₁ and X₂ consists of two path components, then π₁(X) is isomorphic to the additive group of integers Z. The user seeks clarification on which isomorphism can be applied, given that both π₁(X₁) and π₁(X₂) equal zero due to their simply connected nature.

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  • Understanding of the van Kampen theorem in algebraic topology
  • Familiarity with fundamental groups, specifically π₁
  • Knowledge of simply connected spaces
  • Basic concepts of path components in topological spaces
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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 \pi_1(X) is isomorphic to Z the additive group of integers.

Now I think, that what I need to find is that \pi_1(X1) and \pi_1(X2) are isomorphic to \pi_1(S^1) 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.
 
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If X_1,X_2 are simply-connected, then by definition \pi_1(X_1) = \pi_1(X_2) = 0 which certainly will cause difficulties if you're trying to prove they are isomorphic to Z.
 

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