Describes how the fundamental group of a topological space formed by gluing two open, path-connected pieces together can be computed from the fundamental groups of the two pieces and of their overlap. Named for Egbert van Kampen, it is a basic computational tool of algebraic topology.
Facts
StatementThe fundamental group of a topological space that is the union of two open, path-connected subspaces whose intersection is path-connected and nonempty can be computed from the fundamental groups of the two subspaces and of their intersection. 1 Proof YearHerbert Seifert proved a related special case independently in 1931; van Kampen's 1933 paper is the namesake general result. Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Sources
1. Van Kampen's Theorem (Wikipedia)
Wikimedia FoundationReferences sectionQuote, References section
E. R. van Kampen. On the connection between the fundamental groups of some related spaces. American Journal of Mathematics, vol. 55 (1933), pp. 261-267.
Seifert-Van Kampen theorem (Wikipedia)
In Group: Algebraic Topology, Lead paragraphQuote, In Group: Algebraic Topology, Lead paragraph
In mathematics, the Seifert-Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space X {\displaystyle X} in terms of the fundamental groups of two open, path-connected subspaces that cover X {\displaystyle X} .
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