Seifert–Van Kampen theorem

The Seifert–Van Kampen theorem computes the fundamental group of a union from the fundamental groups of overlapping path-connected pieces and their intersection, subject to suitable openness or neighborhood conditions.

Fundamental group: An algebraic group that records based loops in a space up to homotopy. The theorem's output is the fundamental group of the union.
Loop space: The space of continuous maps from a circle to a topological space, based at a specified point when required. Loops in the covered pieces are the raw data from which the theorem builds.
Fundamental group of the circle: The fundamental group of the circle, isomorphic to the integers under addition. Cutting a circle into overlapping arcs illustrates how an overlap can create a loop generator.
Mayer–Vietoris sequence: A long exact sequence relating the homology groups of a union, its pieces, and their intersection. It performs a parallel union calculation in homology rather than directly computing fundamental groups.
Walther von Dyck: A German mathematician who developed early ideas about topological groups and fundamental groups. His work anticipated forms of the theorem before its standard modern attribution.
Free product: The coproduct of groups, formed by combining their elements without imposing relations beyond those within each group. Groups from separate pieces combine freely before intersection relations are imposed.
Path-connected space: A topological space in which every pair of points can be joined by a continuous path. Path-connectedness lets loops and basepoints compare across the pieces.
Fundamental group of the torus: The fundamental group of the torus, isomorphic to the free abelian group on two generators. A square decomposition computes the two commuting loop directions through edge identifications.
Van Kampen diagram: A planar diagram representing a word that is trivial in a group presentation. It studies consequences of group relations after a presentation is known, rather than deriving the union group.
Herbert Seifert: A German mathematician known for work in topology, especially 3-manifolds and fibered spaces. His contributions helped establish the theorem's place in algebraic topology.

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