Van Kampen's Theorem - Examples and Constructions

The power of Van Kampen's theorem lies in its applications to concrete computations. We examine key examples that illustrate the technique.

Example

Fundamental group of the torus: Write $T^2$ as a square with opposite sides identified. Decompose $T^2 = U \cup V$ where:

$U$ is the torus minus a point (deformation retracts to a figure-eight)
$V$ is a small disk around the removed point (contractible)
$U \cap V$ deformation retracts to $S^1$

Then $\pi_1(U) \cong \mathbb{Z} * \mathbb{Z}$ , $\pi_1(V) = \{e\}$ , and $\pi_1(U \cap V) = \mathbb{Z}$ . The inclusion $U \cap V \to U$ corresponds to the commutator $[a,b]$ in the free group. Van Kampen gives: $\pi_1(T^2) \cong \langle a, b \mid [a,b] = 1 \rangle = \mathbb{Z} \times \mathbb{Z}$

This computation reveals the abelian structure of $\pi_1(T^2)$ , reflecting the fact that loops on the torus commute.

Example

Orientable surfaces of genus $g$ : The surface $\Sigma_g$ can be decomposed as $\Sigma_g = \Sigma_{g-1} \cup_{\text{disk}} (\text{torus minus disk})$ . Iteratively applying Van Kampen: $\pi_1(\Sigma_g) \cong \langle a_1, b_1, \ldots, a_g, b_g \mid [a_1,b_1] \cdots [a_g,b_g] = 1 \rangle$ This is a $2g$ -generated group with one relation, showing the rich structure of surface fundamental groups.

Example

Complement of knots: For a knot $K \subseteq S^3$ , decompose $S^3 \setminus K = U \cup V$ where $U$ is a tubular neighborhood of $K$ (with $\pi_1 \cong \mathbb{Z}$ ) and $V$ is the exterior. Van Kampen relates the knot group $\pi_1(S^3 \setminus K)$ to the structure of $U \cap V$ , a torus. This gives: $\pi_1(S^3 \setminus K) \cong \langle \text{generators} \mid \text{Wirtinger relations} \rangle$

The Wirtinger presentation can be read directly from a knot diagram, making Van Kampen a practical computational tool in knot theory.

Example

Real projective spaces: Write $\mathbb{RP}^n = U \cup V$ where $U$ is a contractible neighborhood of a point and $V$ is a neighborhood of the complement, deformation retracting to $\mathbb{RP}^{n-1}$ . Then $U \cap V \simeq S^{n-1}$ , and for $n \geq 3$ : $\pi_1(\mathbb{RP}^n) \cong \pi_1(\mathbb{RP}^{n-1}) *_{\{e\}} \{e\} = \pi_1(\mathbb{RP}^{n-1})$ By induction from $\pi_1(\mathbb{RP}^2) = \mathbb{Z}/2\mathbb{Z}$ , we get $\pi_1(\mathbb{RP}^n) \cong \mathbb{Z}/2\mathbb{Z}$ for all $n \geq 2$ .

Remark

Van Kampen's theorem is most powerful for CW complexes and spaces built by attaching cells. The attaching maps translate directly into relations in the fundamental group presentation, making the algebraic structure explicit.

Example

Bouquet of circles: The wedge sum of $n$ circles has $\pi_1 = \mathbb{F}_n$ , the free group on $n$ generators. This follows from iteratively applying Van Kampen with trivial overlaps. The figure-eight is $\mathbb{F}_2 = \langle a, b \mid \rangle$ .

These examples demonstrate that Van Kampen's theorem reduces fundamental group computations to combinatorial group theory, making previously intractable problems solvable.