Seifert–van Kampen theorem

Builds on Fundamental group Path-connected space Free product Normal subgroup Group homomorphism

Statement

Let $X$ be a topological space with $X = U \cup V$ , where $U$ and $V$ are open in $X$ . Assume that $U$ , $V$ and $U \cap V$ are path-connected, and let $x_0 \in U \cap V$ . Write $\pi_1(-)$ for $\pi_1(-, x_0)$ . Let $i_U \colon \pi_1(U \cap V) \to \pi_1(U)$ , $i_V \colon \pi_1(U \cap V) \to \pi_1(V)$ , $j_U \colon \pi_1(U) \to \pi_1(X)$ and $j_V \colon \pi_1(V) \to \pi_1(X)$ be maps induced by inclusions. They are homomorphisms of fundamental groups. Let $\Phi \colon \pi_1(U, x_0) * \pi_1(V, x_0) \to \pi_1(X, x_0)$ be the homomorphism from the free product that restricts to $j_U$ and $j_V$ . Then $\Phi$ is surjective, and its kernel is the normal subgroup generated by the elements $i_U(\omega)\, i_V(\omega)^{-1}$ for $\omega \in \pi_1(U \cap V, x_0)$ . Hence $\pi_1(X, x_0) \cong \pi_1(U, x_0) *_{\pi_1(U \cap V, x_0)} \pi_1(V, x_0),$ the free product with amalgamation.

Remarks

Surjectivity comes from cutting a loop in $X$ into pieces that each lie in $U$ or in $V$ , by a Lebesgue number argument. The cut points are then joined to $x_0$ by paths inside $U \cap V$ . The description of the kernel comes from cutting a path homotopy in $X$ into small squares in the same way. Example: for the figure eight $S^1 \vee S^1$ , let $U$ be the wedge minus a point of the first circle, and $V$ the wedge minus a point of the second circle. Each of them deformation retracts onto a circle, and $U \cap V$ is contractible. So $\pi_1(S^1 \vee S^1)$ is the free group on two generators. Example: for $n \ge 2$ , cover $S^n$ by the complements of the two poles; both sets are contractible and their intersection is path-connected, so $S^n$ is simply connected. The hypothesis that $U \cap V$ is path-connected is needed: for $S^1$ covered by two open arcs, the intersection has two components, both arcs are contractible, and yet $\pi_1(S^1) \cong \Z$ is not trivial.

Going deeper

For a student

Together with the fundamental group of the circle, the theorem computes the fundamental group of every connected CW complex from its $2$ -skeleton. Attaching a $2$ -cell adds one relation. The result is a presentation with one generator for each $1$ -cell outside a maximal tree and one relation for each $2$ -cell. It gives the fundamental groups of closed surfaces and of knot complements. A version with the fundamental groupoid and a set of base points removes the condition that $U \cap V$ is path-connected.

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Tag	`S17DY98VF2`
URI	`m2://S17DY98VF2`
Kind	statement, theorem
Writing	W0 stub
MSC	57M05, 55Q05
Status	This copy of the site has no build data. Run `m2 status S17DY98VF2`.
About	Fundamental group
Uses	Fundamental group, Path-connected space, Free product, Normal subgroup, Group homomorphism
Same as	Wikidata Q372037 Wikipedia Seifert–Van Kampen theorem
Changed	2 November 2026