Van Kampen's Theorem - Core Definitions

Van Kampen's theorem (also called the Seifert-van Kampen theorem) provides a method to compute the fundamental group of a space by decomposing it into simpler pieces.

Definition

Let $X = U \cup V$ where $U, V$ are open subsets of $X$ . We say this is an open cover if $U$ and $V$ are both open in $X$ . If additionally $U$ , $V$ , and $U \cap V$ are path-connected with $x_0 \in U \cap V$ , the configuration is suitable for Van Kampen's theorem.

The key insight is that loops in $X$ can be decomposed into loops that stay within $U$ or $V$ , glued together at intersection points. This geometric decomposition translates into an algebraic operation on fundamental groups.

Definition

Given groups $G$ and $H$ with homomorphisms $\phi : K \to G$ and $\psi : K \to H$ from a common group $K$ , the free product with amalgamation (or amalgamated product) is denoted $G *_K H$ and defined as: $G *_K H = (G * H) / N$ where $N$ is the normal subgroup generated by all elements of the form $\phi(k)^{-1}\psi(k)$ for $k \in K$ .

This construction identifies elements of $G$ and $H$ that come from the same element in $K$ . When $K = \{e\}$ , we recover the free product $G * H$ .

Example

If $K = \mathbb{Z}$ and $\phi : \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}$ is $\phi(n) = (n, 0)$ while $\psi(n) = (0, n)$ , then $(\mathbb{Z} \times \mathbb{Z}) *_\mathbb{Z} (\mathbb{Z} \times \mathbb{Z}) \cong \mathbb{Z} \times \mathbb{Z}$ This reflects gluing two cylinders along a circle to form a torus.

Definition

The pushout (or fibered coproduct) of groups in category theory provides another perspective on the amalgamated product. Given $K \xrightarrow{\phi} G$ and $K \xrightarrow{\psi} H$ , the pushout is the universal group $P$ with maps $G \to P$ and $H \to P$ making the diagram commute.

In the category of groups, the pushout is precisely the amalgamated product $G *_K H$ . This categorical viewpoint clarifies why Van Kampen's theorem is a "gluing" theorem for fundamental groups.

Example

For the wedge sum $X = A \vee B$ (gluing two spaces at a point), take $U = A$ with a small neighborhood of the basepoint and $V = B$ with a similar neighborhood. Then $U \cap V$ is contractible with trivial $\pi_1$ , giving: $\pi_1(A \vee B) = \pi_1(A) * \pi_1(B)$

Remark

The amalgamated product generalizes both the free product (when $K$ is trivial) and the direct product (when the maps $\phi, \psi$ are injective and their images commute with everything). Van Kampen's theorem translates geometric gluing into this algebraic gluing.

Understanding amalgamated products requires comfort with group presentations and quotients. The normal subgroup $N$ imposes relations that force agreement on the overlap.