Vipul at 17:42, 20 December 2010

2010-12-20T17:42:42Z

← Older revision		Revision as of 17:42, 20 December 2010
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	* [[Union of two simply connected open subsets with path-connected intersection is simply connected]]		* [[Union of two simply connected open subsets with path-connected intersection is simply connected]]
			* [[Suspension of path-connected space is simply connected]] (this is a special case of the preceding application)
			* [[n-sphere is simply connected for n greater than 1]] (this is a special case of both the preceding applications)
			* [[Fundamental group of wedge of circles is free group of rank equal to the number of circles]]

Vipul: Created page with '==Statement== Suppose $X$ is a fact about::path-connected space. Suppose $U$ and $V$ are nonempty open subsets of $X$ whose union...'

2010-12-20T00:50:25Z

Created page with '==Statement== Suppose <math>X</math> is a fact about::path-connected space. Suppose <math>U</math> and <math>V</math> are nonempty open subsets of <math>X</math> whose union...'

New page

==Statement==

Suppose <math>X</math> is a [[fact about::path-connected space]]. Suppose <math>U</math> and <math>V</math> are nonempty open subsets of <math>X</math> whose union is <math>X</math> and whose intersection is also path-connected. Since <math>X</math> is path-connected, the intersection <math>U \cap V</math> must be a nonempty open subset of <math>X</math>, which we call <math>W</math>. Note that since all the four spaces <math>U,V,W,X</math> are path-connected, the fundamental groups of each of these spaces is independent of the choice of respective basepoint for each space.

We have natural homomorphisms for the [[fact about::fundamental group]]s induced by inclusion maps of the spaces(with respect to any chosen basepoint in <math>W</math>, which we choose not to write for brevity):

<math>\! \pi_1(W) \to \pi_1(U), \pi_1(W) \to \pi_1(V), \pi_1(U) \to \pi_1(X), \pi_1(V) \to \pi_1(X)</math>

The Seifert-van Kampen theorem has the following equivalent formulations:

* ''Category-theoretic version'': The commutative diagram formed by the above four inclusions is a pushout.
* ''Group-theoretic version'': <math>\pi_1(X)</math> is the [[amalgamated free product]] of <math>\pi_1(U)</math> and <math>\pi_1(V)</math> via identification of the images of <math>\pi_1(W)</math> in each. In other words:

<math>\pi_1(X) = \pi_1(U) *_{\pi_1(W)} \pi_1(V)</math>

Note that since the natural maps from <math>\pi_1(W)</math> to <math>\pi_1(U)</math> and <math>\pi_1(V)</math> need not be injective, this is not necessarily an amalgamated free product in the strict sense of a common ''subgroup'' being identified between <math>\pi_1(U)</math> and <math>\pi_1(V)</math>.
* ''Presentation version'': {{fillin}}

==Applications==

* [[Union of two simply connected open subsets with path-connected intersection is simply connected]]

Seifert-van Kampen theorem - Revision history

Vipul at 17:42, 20 December 2010

Vipul: Created page with '==Statement== Suppose X is a fact about::path-connected space. Suppose U and V are nonempty open subsets of X whose union...'

Vipul: Created page with '==Statement== Suppose $X$ is a fact about::path-connected space. Suppose $U$ and $V$ are nonempty open subsets of $X$ whose union...'