The Fundamental Group and Van Kampen's Theorem

💡 The Core Insight

Van Kampen's Theorem shows that the fundamental group of a union of path-connected open sets is the free product of their fundamental groups, amalgamated over their intersection—transforming topological decompositions into algebraic constructions.

Definition

The Fundamental Group: Let $X$ be a topological space and $x_0 \in X$ a basepoint. A loop based at $x_0$ is a continuous map $\gamma: [0,1] \to X$ with $\gamma(0) = \gamma(1) = x_0$ . Two loops $\gamma_0, \gamma_1$ are homotopic (written $\gamma_0 \simeq \gamma_1$ ) if there exists a continuous map $H: [0,1] \times [0,1] \to X$ such that:

$H(s, 0) = \gamma_0(s)$ and $H(s, 1) = \gamma_1(s)$ for all $s \in [0,1]$
$H(0, t) = H(1, t) = x_0$ for all $t \in [0,1]$

The fundamental group $\pi_1(X, x_0)$ is the set of homotopy classes $[\gamma]$ of loops based at $x_0$ , with group operation given by concatenation: $[\gamma_1] \cdot [\gamma_2] = [\gamma_1 * \gamma_2]$ where

(\gamma_1 * \gamma_2)(s) = \begin{cases} \gamma_1(2s) & 0 \leq s \leq 1/2 \\ \gamma_2(2s-1) & 1/2 \leq s \leq 1 \end{cases}

A space is simply connected if $\pi_1(X, x_0)$ is trivial for some (equivalently, every) basepoint.

TheoremVan Kampen's Theorem

Theorem [1]: Let $X$ be a path-connected space and let $U, V \subseteq X$ be path-connected open sets such that $U \cup V = X$ and $U \cap V$ is path-connected and non-empty. Let $x_0 \in U \cap V$ be a basepoint. Let $i_U: U \cap V \to U$ , $i_V: U \cap V \to V$ , $j_U: U \to X$ , and $j_V: V \to X$ denote the inclusion maps. Then: 1. The homomorphisms $(j_U)_*, (j_V)_*: \pi_1(U, x_0), \pi_1(V, x_0) \to \pi_1(X, x_0)$ induced by the inclusions satisfy $(j_U)_* \circ (i_U)_* = (j_V)_* \circ (i_V)_*$ . 2. The fundamental group $\pi_1(X, x_0)$ is isomorphic to the amalgamated free product (also called pushout):

\pi_1(X, x_0) \cong \pi_1(U, x_0) *_{\pi_1(U \cap V, x_0)} \pi_1(V, x_0)

3. More precisely, $\pi_1(X, x_0)$ is the quotient group of the free product $\pi_1(U, x_0) * \pi_1(V, x_0)$ by the normal subgroup generated by elements of the form $(i_U)_*(\alpha) (i_V)_*(\alpha)^{-1}$ for all $\alpha \in \pi_1(U \cap V, x_0)$ .

Proof

Proof of Van Kampen's Theorem: We must show that the diagram ``` π₁(U∩V, x₀) → π₁(U, x₀) ↓ ↓ π₁(V, x₀) → π₁(X, x₀) ``` is a pushout in the category of groups.

Step 1: Surjectivity of the natural map. Let $\Phi: \pi_1(U, x_0) * \pi_1(V, x_0) \to \pi_1(X, x_0)$ be the homomorphism extending $(j_U)_*$ and $(j_V)_*$ . We show $\Phi$ is surjective. Let $[\gamma] \in \pi_1(X, x_0)$ . Since $\gamma([0,1])$ is compact and covered by the open sets $U$ and $V$ , by the Lebesgue number lemma there exists a partition $0 = s_0 < s_1 < \cdots < s_n = 1$ such that for each $i$ , $\gamma([s_{i-1}, s_i])$ lies entirely in either $U$ or $V$ . Let $\gamma_i$ denote the restriction of $\gamma$ to $[s_{i-1}, s_i]$ , reparametrized to $[0,1]$ . Each $\gamma_i$ is a path in either $U$ or $V$ from $\gamma(s_{i-1})$ to $\gamma(s_i)$ . Since $U$ and $V$ are path-connected, we can choose paths $\tau_i$ in $U$ or $V$ (whichever contains $\gamma_i$ ) from $x_0$ to $\gamma(s_i)$ . Define $\delta_i = \tau_{i-1}^{-1} * \gamma_i * \tau_i$ , which is a loop based at $x_0$ in either $U$ or $V$ . Then $\gamma \simeq \delta_1 * \delta_2 * \cdots * \delta_n$ (by homotopy using the paths $\tau_i$ ), so $[\gamma] = [\delta_1] \cdots [\delta_n]$ where each $[\delta_i]$ is in the image of either $(j_U)_*$ or $(j_V)_*$ . Thus $\Phi$ is surjective.

Step 2: Determining the kernel of $\Phi$ . Let $N$ be the normal subgroup of $\pi_1(U, x_0) * \pi_1(V, x_0)$ generated by $\{(i_U)_*(\alpha) (i_V)_*(\alpha)^{-1} : \alpha \in \pi_1(U \cap V, x_0)\}$ . We must show $\ker(\Phi) = N$ . Clearly $N \subseteq \ker(\Phi)$ since $(j_U)_* \circ (i_U)_* = (j_V)_* \circ (i_V)_*$ . For the reverse inclusion, suppose $w = g_1 g_2 \cdots g_m \in \ker(\Phi)$ where each $g_i \in \pi_1(U, x_0) \cup \pi_1(V, x_0)$ (written in reduced form). Then the loop $\gamma = \gamma_1 * \gamma_2 * \cdots * \gamma_m$ is null-homotopic in $X$ , where $\gamma_i$ represents $g_i$ in either $U$ or $V$ . Let $H: [0,1] \times [0,1] \to X$ be a null-homotopy of $\gamma$ . Since $[0,1] \times [0,1]$ is compact, by the Lebesgue number lemma applied to the cover $\{H^{-1}(U), H^{-1}(V)\}$ , we can subdivide the square into smaller rectangles, each mapped entirely into $U$ or $V$ . Using a careful inductive argument on these subdivisions (which we omit for brevity, but follows standard techniques in homotopy theory), we can show that $w$ can be transformed into the identity by:

Inserting or removing elements of the form $(i_U)_*(\alpha)(i_V)_*(\alpha)^{-1}$
Using relations in $\pi_1(U, x_0)$ or $\pi_1(V, x_0)$

Therefore $w \in N$ , giving $\ker(\Phi) = N$ .

Step 3: Universal property. To complete the proof, we verify the universal property of the pushout. Suppose $G$ is a group with homomorphisms $\phi_U: \pi_1(U, x_0) \to G$ and $\phi_V: \pi_1(V, x_0) \to G$ satisfying $\phi_U \circ (i_U)_* = \phi_V \circ (i_V)_*$ . By the universal property of the free product, there is a unique homomorphism $\psi: \pi_1(U, x_0) * \pi_1(V, x_0) \to G$ extending $\phi_U$ and $\phi_V$ . The compatibility condition ensures $N \subseteq \ker(\psi)$ , so $\psi$ factors through the quotient $\pi_1(X, x_0) \cong (\pi_1(U, x_0) * \pi_1(V, x_0))/N$ , giving the unique homomorphism making the required diagram commute. □

∎

⚠️ Common mistake

Common Error: Van Kampen's Theorem requires both $U$ and $V$ to be open and their intersection $U \cap V$ to be path-connected. If $U \cap V$ is not path-connected but has multiple components, a more general version of the theorem applies involving free products over multiple amalgamations. Without path-connectedness of the intersection, the simple form of the theorem fails completely.

mathematician

Egbertus van Kampen (1933) — Published the seifert-van Kampen theorem providing the fundamental tool for computing fundamental groups via decomposition.

visualization

Example1

Computing $\pi_1(S^1)$ , the circle. Decompose $S^1 = U \cup V$ where:

$U = S^1 \setminus \{-1\}$ (the circle with the left-most point removed)
$V = S^1 \setminus \{1\}$ (the circle with the right-most point removed)

Both $U$ and $V$ are homeomorphic to $(0,1)$ , hence contractible, so $\pi_1(U, x_0) = \pi_1(V, x_0) = \{e\}$ (the trivial group). The intersection $U \cap V$ consists of two connected components (top and bottom arcs), each contractible. However, we can instead use the decomposition where $U$ and $V$ are slightly larger open sets, so $U \cap V$ is path-connected (a single arc). With this adjustment, $U \cap V$ is contractible, so $\pi_1(U \cap V, x_0) = \{e\}$ . By Van Kampen's Theorem:

\pi_1(S^1, x_0) \cong \{e\} *_{\{e\}} \{e\} = \{e\} * \{e\} = \{e\}

This approach fails! The issue is that our decomposition doesn't capture the essential loop. Instead, we need a different method (like covering space theory) to show $\pi_1(S^1) \cong \mathbb{Z}$ .

Correct application: Let $X = S^1 \vee S^1$ (two circles joined at a point). Decompose as $U =$ (slightly thickened first circle) and $V =$ (slightly thickened second circle). Then $U \cap V$ is contractible, $\pi_1(U) \cong \mathbb{Z}$ , $\pi_1(V) \cong \mathbb{Z}$ , and:

\pi_1(S^1 \vee S^1) \cong \mathbb{Z} *_{\{e\}} \mathbb{Z} \cong \mathbb{Z} * \mathbb{Z} = F_2

the free group on two generators.

Example2

Computing $\pi_1(\mathbb{R}^2 \setminus \{p_1, p_2\})$ , the plane with two points removed. Let $p_1, p_2$ be two distinct points in $\mathbb{R}^2$ . Define:

$U = \mathbb{R}^2 \setminus \{p_1\}$ (plane with first point removed)
$V = \mathbb{R}^2 \setminus \{p_2\}$ (plane with second point removed)

Then $U \cup V = \mathbb{R}^2 \setminus \{p_1, p_2\}$ and $U \cap V = \mathbb{R}^2 \setminus \{p_1, p_2\}$ union an open disk around one of the points—wait, this is incorrect.

Correct setup: Choose $U$ and $V$ as open sets that are deformation retracts of circles around each point:

$U$ deformation retracts to a circle around $p_1$ , so $\pi_1(U) \cong \mathbb{Z}$ , generated by $[\alpha]$
$V$ deformation retracts to a circle around $p_2$ , so $\pi_1(V) \cong \mathbb{Z}$ , generated by $[\beta]$
$U \cap V$ deformation retracts to a wedge of two circles, so $\pi_1(U \cap V) \cong \mathbb{Z} * \mathbb{Z}$

Actually, the standard approach uses homotopy equivalence: $\mathbb{R}^2 \setminus \{p_1, p_2\}$ deformation retracts to a wedge $S^1 \vee S^1$ , giving $\pi_1(\mathbb{R}^2 \setminus \{p_1, p_2\}) \cong F_2$ as computed in Example 1. More generally, $\pi_1(\mathbb{R}^2 \setminus \{p_1, \ldots, p_n\}) \cong F_n$ , the free group on $n$ generators.

Counterexample

When the intersection is not path-connected. Let $X$ be the union of two circles tangent at a point (the "figure-eight" or $S^1 \vee S^1$ ). Suppose we try to apply Van Kampen naively with $U$ and $V$ being small open neighborhoods of each circle, where $U \cap V$ consists of two disjoint arcs near the tangent point. Since $U \cap V$ has two path components, the standard Van Kampen theorem does not apply directly. Each component is contractible, but we cannot treat $\pi_1(U \cap V)$ as a single group. The generalized Van Kampen theorem for disconnected intersections shows that:

\pi_1(X) \cong \pi_1(U) * \pi_1(V)

since the amalgamation is trivial over the multiple components. However, using the wrong basepoint (one not in every component of $U \cap V$ ) can lead to incorrect conclusions.

Lesson: The path-connectedness hypothesis ensures a well-defined amalgamation. Without it, care must be taken with basepoints and the generalized version of the theorem.

🧠 Intuition

Geometric intuition: Imagine building a space $X$ by gluing together two simpler pieces $U$ and $V$ along their intersection $U \cap V$ . A loop in $X$ can be "pushed" into pieces that lie mostly in $U$ or mostly in $V$ , crossing the overlap region only when necessary. Van Kampen's Theorem says that any loop in $X$ can be decomposed into "words" alternating between loops in $U$ and loops in $V$ . When a loop passes through the intersection $U \cap V$ , we get a relation: a loop in $U \cap V$ can be viewed either as a loop in $U$ or as a loop in $V$ , and these two viewpoints must be identified. The amalgamated free product precisely captures this: we take all formal words built from $\pi_1(U)$ and $\pi_1(V)$ (the free product), then impose exactly those relations that identify the same loop viewed from the $U$ side and the $V$ side (the amalgamation). Think of it as a "dictionary" where loops in the "shared region" have synonyms in both $U$ -language and $V$ -language.

connection

Connection to Graph Theory and Combinatorial Group Theory: Via Van Kampen's theorem, computing the fundamental group of a CW complex reduces to understanding the Cayley graph structure of free products. For a 2-dimensional CW complex with one 0-cell, $k$ 1-cells (edges), and $m$ 2-cells (faces), the fundamental group has a group presentation with $k$ generators and $m$ relations. This connects directly to combinatorial group theory, where understanding word problems and normal forms in amalgamated free products is central. The theorem thus transforms topological questions into purely algebraic problems about groups defined by generators and relations, bridging geometric topology and computational group theory.

The Fundamental Group and Van Kampen's Theorem

Overview

Core Theory

Examples and Applications

Historical Development

Connections to Other Areas

What You Need First

What This Opens Up

See Also