Van Kampen's Theorem

Cut a space into overlapping pieces and read off a presentation.

The idea

Theorem (Van Kampen's theorem).

Let $X = U \cup V$ with $U$ and $V$ open, path-connected and containing a common basepoint $x_{0}$, and with $U \cap V$ path-connected as well. Let $i_{U} \colon \pi_{1}(U \cap V) \to \pi_{1}(U)$ and $i_{V} \colon \pi_{1}(U \cap V) \to \pi_{1}(V)$ be induced by the inclusions of the overlap into the two pieces. Then $\pi_{1}(X) \cong \bigl(\pi_{1}(U) * \pi_{1}(V)\bigr)\big/N,$ where $N$ is the smallest normal subgroup containing $i_{U}(\omega)\, i_{V}(\omega)^{-1}$ for every $\omega \in \pi_{1}(U \cap V)$.

The theorem earns its place because $\pi_{1}(X)$ is defined by quantifying over every loop in $X$ and every homotopy between loops, a computation nobody can carry out by hand except on the simplest spaces. Van Kampen replaces it with algebra. Cut $X$ into two pieces whose groups are already known; the free product joins those groups with no interaction between them; and $N$ imposes the only interaction there is, that a loop lying in the overlap must name the same element of $\pi_{1}(X)$ whether it is read inside $U$ or inside $V$.

The output is a presentation, by generators and relations, and a presentation is enough to tell spaces apart. The sphere's group is trivial and the torus's is not; the torus and the wedge of two circles each need two generators, and the two spaces are separated by whether those generators commute.

Ways to work on it

• Walkthrough. Compute a fundamental group from a two-set cover, and see why the overlap must be connected.
• Proof. See why a 2-cell kills its attaching loop — puncture the cell and apply the theorem.
• Practice. Compute _1 of a wedge, of a circle with a cell attached, and from an attaching word.
• Hardest. Compute fundamental groups of spaces glued along a circle, including closed surfaces.

Not sure where to start? Take the ten-question placement test.