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In mathematics, the Seifert–van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space , in terms of the fundamental groups of two open, path-connected subspaces and that cover . It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones. The underlying idea is that paths in can be partitioned into journeys: through the intersection of and , through but outside , and through outside . In order to move segments of paths around, by homotopy to form loops returning to a base point in , we should assume , and are path-connected and that is not empty. We also assume that and are open subspaces with union . == Equivalent formulations == In the language of combinatorial group theory, is the free product with amalgamation of and , with respect to the (not necessarily injective) homomorphisms and . Given group presentations: : : and : the amalgamation can be presented as : In category theory, is the pushout, in the category of groups, of the diagram: : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Seifert–van Kampen theorem」の詳細全文を読む スポンサード リンク
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