The Fundamental Group and Van Kampen's Theorem

• The fundamental group π₁(X, x₀) classifies loops in a space up to continuous deformation, providing an algebraic invariant that captures topological "holes"
• Van Kampen's Theorem computes π₁(X) by decomposing X into simpler open sets and combines their fundamental groups via free product with amalgamation
• For path-connected open sets U, V covering X: π₁(X) ≅ π₁(U) *_{π₁(U∩V)} π₁(V)
• Direct computation from definitions is hard; Van Kampen's Theorem enables practical calculations by exploiting space decompositions
• Introduced by Poincaré (1895) and formalized by van Kampen (1933), the theorem remains essential for distinguishing topological spaces algebraically

The Fundamental Group and Van Kampen's Theorem

Overview

The fundamental group is one of the most important algebraic invariants in algebraic topology, capturing information about the "holes" and "loops" in a topological space. Given a space $X$ and a basepoint $x_0\in X$, the fundamental group ${\pi }_1(X,x_0)$ consists of equivalence classes of loops based at $x_0$, where two loops are considered equivalent if one can be continuously deformed into the other. This construction provides a bridge between topology and group theory, allowing us to use algebraic techniques to distinguish topological spaces.

Computing fundamental groups directly from the definition can be challenging, even for relatively simple spaces. Van Kampen's Theorem provides a powerful tool for this computation by showing how to construct the fundamental group of a space from the fundamental groups of its subspaces. When a space $X$ can be decomposed into simpler open sets whose fundamental groups are known, Van Kampen's Theorem gives an explicit description of ${\pi }_1(X)$ in terms of a free product with amalgamation.

Historically, the fundamental group was introduced by Henri Poincaré in 1895 as part of his development of algebraic topology. Egbertus van Kampen published his theorem in 1933, though similar results were discovered independently by Herbert Seifert. The theorem has become indispensable in algebraic topology, providing one of the primary computational tools alongside covering space theory and homology theory.

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.

• $H(s,0)={\gamma }_0(s)$ 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: $[{\gamma }_1]\cdot [{\gamma }_2]=[{\gamma }_1\ast {\gamma }_2]$ where

A space is simply connected if ${\pi }_1(X,x_0)$ is trivial for some (equivalently, every) basepoint.

Core Theory

Van Kampen's Theorem

1. The homomorphisms $(j_U{)}_{\ast },(j_V{)}_{\ast }:{\pi }_1(U,x_0),{\pi }_1(V,x_0)\to {\pi }_1(X,x_0)$ induced by the inclusions satisfy $(j_U{)}_{\ast }\circ (i_U{)}_{\ast }=(j_V{)}_{\ast }\circ (i_V{)}_{\ast }$.

1. The fundamental group ${\pi }_1(X,x_0)$ is isomorphic to the amalgamated free product (also called pushout):

1. More precisely, ${\pi }_1(X,x_0)$ is the quotient group of the free product ${\pi }_1(U,x_0)\ast {\pi }_1(V,x_0)$ by the normal subgroup generated by elements of the form $(i_U{)}_{\ast }(\alpha )(i_V{)}_{\ast }(\alpha {)}^{-1}$ for all $\alpha \in {\pi }_1(U\cap V,x_0)$.

We must show that the diagram

π₁(U∩V, x₀) → π₁(U, x₀)
     ↓            ↓
π₁(V, x₀)   → π₁(X, x₀)

Let $\Phi :{\pi }_1(U,x_0)\ast {\pi }_1(V,x_0)\to {\pi }_1(X,x_0)$ be the homomorphism extending $(j_U{)}_{\ast }$ and $(j_V{)}_{\ast }$. 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 ${\delta }_i={\tau }_{i-1}^{-1}\ast {\gamma }_i\ast {\tau }_i$, which is a loop based at $x_0$ in either $U$ or $V$.

Then $\gamma \simeq {\delta }_1\ast {\delta }_2\ast \cdots \ast {\delta }_n$ (by homotopy using the paths ${\tau }_i$), so $[\gamma ]=[{\delta }_1]\cdots [{\delta }_n]$ where each $[{\delta }_i]$ is in the image of either $(j_U{)}_{\ast }$ or $(j_V{)}_{\ast }$. Thus $\Phi$ is surjective.

Let $N$ be the normal subgroup of ${\pi }_1(U,x_0)\ast {\pi }_1(V,x_0)$ generated by $\{(i_U{)}_{\ast }(\alpha )(i_V{)}_{\ast }(\alpha {)}^{-1}:\alpha \in {\pi }_1(U\cap V,x_0)\}$. We must show $\ker (\Phi )=N$.

Clearly $N\subseteq \ker (\Phi )$ since $(j_U{)}_{\ast }\circ (i_U{)}_{\ast }=(j_V{)}_{\ast }\circ (i_V{)}_{\ast }$.

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 $\gamma ={\gamma }_1\ast {\gamma }_2\ast \cdots \ast {\gamma }_m$ 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 $(i_U{)}_{\ast }(\alpha )(i_V{)}_{\ast }(\alpha {)}^{-1}$
• Using relations in ${\pi }_1(U,x_0)$ or ${\pi }_1(V,x_0)$

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

To complete the proof, we verify the universal property of the pushout. Suppose $G$ is a group with homomorphisms ${\varphi }_U:{\pi }_1(U,x_0)\to G$ and ${\varphi }_V:{\pi }_1(V,x_0)\to G$ satisfying ${\varphi }_U\circ (i_U{)}_{\ast }={\varphi }_V\circ (i_V{)}_{\ast }$.

By the universal property of the free product, there is a unique homomorphism $\psi :{\pi }_1(U,x_0)\ast {\pi }_1(V,x_0)\to G$ extending ${\varphi }_U$ and ${\varphi }_V$. The compatibility condition ensures $N\subseteq \ker (\psi )$, so $\psi$ factors through the quotient ${\pi }_1(X,x_0)\cong ({\pi }_1(U,x_0)\ast {\pi }_1(V,x_0))/N$, giving the unique homomorphism making the required diagram commute.

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Examples and Applications

1

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:

2

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.

• $U$ 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 ${\pi }_1(U\cap V)\cong \mathbb{Z}\ast \mathbb{Z}$

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, ${\pi }_1({\mathbb{R}}^2\setminus \{p_1,\dots ,p_n\})\cong F_n$, the free group on $n$ generators.

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:

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.

Historical Development

The fundamental group was introduced by Henri Poincaré in his seminal 1895 paper "Analysis Situs," which laid the foundations for algebraic topology. Poincaré recognized that loops and their homotopy classes provided algebraic invariants for topological spaces, though his treatment was not fully rigorous by modern standards.

The modern formulation using category theory and the language of pushouts emerged in the 1940s and 1950s with the work of Saunders Mac Lane and Samuel Eilenberg, who developed categorical algebra and recognized that Van Kampen's theorem expresses the fundamental group as a functor that preserves certain colimits. This perspective revealed deep connections between topology and algebra.

The theorem has been generalized in numerous ways, including versions for higher homotopy groups, fundamental groupoids, and in the context of algebraic geometry and schemes.

Connections to Other Areas

What You Need First

What This Opens Up

See Also