 Seifert–Van Kampen theorem

Let $X=U\cup V$ be an open cover such that $U$, $V$ and $U\cap V$ are nonempty and path-connected, and choose a base point $x_0\in U\cap V$. The inclusions induce a pushout of groups:The maps from the overlap need not be injective. The amalgamated free product here means the universal group receiving the two inclusion-induced homomorphisms with their overlap relations.

A loop can be subdivided into pieces in $U$ or $V$; connecting their endpoints through the path-connected overlap expresses its class using the two local fundamental groups. Subdividing a homotopy shows that the overlap relations are exactly the identifications needed.

The overlap condition matters. Two open arcs can cover a circle while their intersection has two components. Both arcs have trivial fundamental group, but the circle has fundamental group $\mathbb{Z}$. The displayed single-basepoint formula cannot be applied to that disconnected overlap; a groupoid version keeps the extra component information.

The Seifert–Van Kampen theorem is a fundamental result in algebraic topology that provides a method for computing the fundamental group of a topological space that can be decomposed into simpler pieces. Specifically, it relates the fundamental group of a space to the fundamental groups of its subspaces when certain conditions are satisfied.