---
tag: S17DY98VF2
kind: statement
role: theorem
name: Seifert–van Kampen theorem
msc: [57M05, 55Q05]
uses: [S1XK3HXHFX, 0WB6DH499T, TM7QS7P7X9, 0CB0DX5BVV, 0JYTG5K9Y4]
about: [S1XK3HXHFX]
sources:
  - url: https://en.wikipedia.org/wiki/Seifert–Van_Kampen_theorem
same_as:
  wikidata: Q372037
  wikipedia: Seifert–Van Kampen theorem
---

## 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|0WB6DH499T]], 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|0JYTG5K9Y4]] of [[fundamental groups|S1XK3HXHFX]].
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|TM7QS7P7X9]] that restricts to $j_U$ and $j_V$.
Then $\Phi$ is surjective, and its kernel is the [[normal subgroup|0CB0DX5BVV]] 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.

## Explanations

### L1 human

The theorem computes the loops of a space glued from two overlapping pieces.
Each loop in the whole space is made of loops in the two pieces.
The only new relations come from loops that live in the overlap, since such a loop can be seen in either piece.

### L2 human

Together with the [[fundamental group of the circle|6SQXPE0JPN]], 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|0JGEQXQ4AC]] 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|ZBHGXJF24E]] and a set of base points removes the condition that $U \cap V$ is path-connected.