# Fundamental group

The **fundamental group** of a [topological space](https://www.edgepedia.org/topological-space), written \( \pi_1(X, x_0) \), is the group formed by the loops based at a chosen point \( x_0 \), taken up to continuous deformation (homotopy), with the operation of concatenating one loop after another. It is also called the first homotopy group or, historically, the [Poincaré group](https://www.edgepedia.org/poincare-group), and it records the ways a loop can wind around the holes of a space.<sup>[1](https://ncatlab.org/nlab/files/NicolaescuGeometryOfManifolds.pdf)</sup><sup> • </sup><sup>[2](https://msl.cs.uiuc.edu/planning/node143.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \pi_1(X, x_0) \) is the set of homotopy classes of loops based at \( x_0 \), with the group operation given by concatenation of loops.<sup>[1](https://ncatlab.org/nlab/files/NicolaescuGeometryOfManifolds.pdf)</sup> |
| Origin | Introduced by Henri Poincaré in the 1895 paper *Analysis situs* (Journal de l'École polytechnique, pp. 1–121), together with the concept of homotopy.<sup>[3](https://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/)</sup><sup> • </sup><sup>[4](https://henripoincarepapers.univ-nantes.fr/en/bibliohp/ajax.php?bibkey=hp1895ep)</sup> |
| Circle | \( \pi_1(S^1, x_0) \cong \mathbb{Z} \); the integer counts how many times, and in which direction, a loop winds around the circle.<sup>[5](https://www.math.columbia.edu/~mhaseliu/Docs/Intro.pdf)</sup><sup> • </sup><sup>[2](https://msl.cs.uiuc.edu/planning/node143.html)</sup> |
| Euclidean space | \( \pi_1(\mathbb{R}^n, 0) \) is trivial: every loop \( f \) is homotopic to the constant loop via \( H(s, t) = t \cdot f(s) \).<sup>[5](https://www.math.columbia.edu/~mhaseliu/Docs/Intro.pdf)</sup> |
| Torus | \( \pi_1(\mathbb{T}^n) = \mathbb{Z}^n \), the \( i \)-th component counting wraps around the \( i \)-th circle factor.<sup>[2](https://msl.cs.uiuc.edu/planning/node143.html)</sup> |
| Computation tool | The Seifert–van Kampen theorem computes \( \pi_1(X, x_0) \) as a pushout (amalgamated product) of the groups of two open sets covering \( X \), provided their intersection is path connected.<sup>[6](https://www.math.univ-paris13.fr/~harpaz/algebraic_topology.pdf)</sup> |
| Decidability | The word problem for finitely presented groups, posed by Dehn in 1911 and motivated by fundamental groups, was proved undecidable by Novikov and Boone in the 1950s.<sup>[7](https://kconrad.math.uconn.edu/blurbs/grouptheory/wordproblem.pdf)</sup> |

## Definition

Fix a base point \( x_0 \) in a topological space \( X \). A **loop** at \( x_0 \) is a continuous map \( f : I \to X \) from the unit interval with \( f(0) = f(1) = x_0 \). Two loops are identified when one can be continuously deformed into the other while keeping the endpoints fixed at \( x_0 \); homotopy is an equivalence relation on the set of loops, so the collection of classes is well defined.<sup>[5](https://www.math.columbia.edu/~mhaseliu/Docs/Intro.pdf)</sup>

The group operation is **concatenation**: for loops \( f \) and \( g \), the product \( f * g \) runs \( f \) at double speed on the first half of the interval and \( g \) at double speed on the second half,<sup>[5](https://www.math.columbia.edu/~mhaseliu/Docs/Intro.pdf)</sup><sup> • </sup><sup>[1](https://ncatlab.org/nlab/files/NicolaescuGeometryOfManifolds.pdf)</sup>

\[ (f * g)(t) = \begin{cases} f(2t) & 0 \le t \le 1/2, \\ g(2t - 1) & 1/2 \le t \le 1. \end{cases} \]

This operation descends to homotopy classes: if \( \alpha_0 \simeq \alpha_1 \) and \( \beta_0 \simeq \beta_1 \), then \( \alpha_0 * \beta_0 \simeq \alpha_1 * \beta_1 \). The class of the constant loop \( e_{x_0} \) is the identity, the reverse loop \( \gamma^-(s) = \gamma(1 - s) \) is the inverse, and the operation is associative up to homotopy, so \( (\Omega(X, x_0)/\simeq_{x_0}, \cdot) \) is a group.<sup>[1](https://ncatlab.org/nlab/files/NicolaescuGeometryOfManifolds.pdf)</sup> The class of a loop \( \gamma \) is written \( [\gamma] \).<sup>[1](https://ncatlab.org/nlab/files/NicolaescuGeometryOfManifolds.pdf)</sup>

## Dependence on the base point

The fundamental group sees only the path component containing the base point. If \( X \) is connected, all the groups \( \pi_1(X, x) \) for \( x \in X \) are isomorphic, since a path between two base points lets loops be transported from one to the other.<sup>[1](https://ncatlab.org/nlab/files/NicolaescuGeometryOfManifolds.pdf)</sup> For a space that is not path connected, the group can genuinely depend on the choice of base point.<sup>[8](https://euclid.nmu.edu/~joshthom/ma516/lectures/lecture14.html)</sup>

A space is **simply connected** when it is path connected and its fundamental group is trivial; equivalently, there is a unique homotopy class of paths connecting any two points.<sup>[9](https://drewkh.folk.ntnu.no/MA3408/the-fundamental-group.html)</sup> [Euclidean space](https://www.edgepedia.org/euclidean-space) \( \mathbb{R}^n \) is simply connected, since the straight-line homotopy \( H(s, t) = t \cdot f(s) \) shrinks any loop to the constant loop.<sup>[5](https://www.math.columbia.edu/~mhaseliu/Docs/Intro.pdf)</sup>

## Functoriality

A continuous map \( f : (X, x_0) \to (Y, y_0) \) sends a loop \( \gamma \) to the loop \( f \circ \gamma \), inducing a group homomorphism \( f_* : \pi_1(X, x_0) \to \pi_1(Y, y_0) \). These maps satisfy \( (g \circ f)_* = g_* \circ f_* \) and \( (\mathrm{Id})_* = \mathrm{Id} \), so \( \pi_1 \) is a functor from pointed topological spaces to groups. Moreover, homotopic maps (relative to the base point) induce the same homomorphism, which is what makes the group a homotopy invariant.<sup>[5](https://www.math.columbia.edu/~mhaseliu/Docs/Intro.pdf)</sup><sup> • </sup><sup>[1](https://ncatlab.org/nlab/files/NicolaescuGeometryOfManifolds.pdf)</sup> Nonisomorphic fundamental groups therefore rule out homotopy equivalence between spaces.<sup>[10](https://lawrencehook.github.io/apps-by-ai/notes/topology/fundamental-group/index.html)</sup>

## Computing fundamental groups

**Seifert–van Kampen theorem.** If \( X = U_1 \cup U_2 \) with \( U_1, U_2 \) open and \( A = U_1 \cap U_2 \) path connected, then the square of inclusions \( A \to U_1, U_2 \to X \) is sent by \( \pi_1 \) to a pushout square of groups; equivalently, \( \pi_1(X, x_0) \cong \pi_1(U_1, x_0) *_{\pi_1(A, x_0)} \pi_1(U_2, x_0) \), the amalgamated free product.<sup>[6](https://www.math.univ-paris13.fr/~harpaz/algebraic_topology.pdf)</sup><sup> • </sup><sup>[11](https://www.homepages.ucl.ac.uk/~ucahsgh/notesAT.pdf)</sup> The path-connectedness of the intersection is essential: for \( X = S^1 \) covered by \( S^1 \setminus \{1\} \) and \( S^1 \setminus \{-1\} \), all three pieces are simply connected, yet \( \pi_1(S^1) \) is nontrivial.<sup>[6](https://www.math.univ-paris13.fr/~harpaz/algebraic_topology.pdf)</sup>

**CW-complexes.** For a CW-complex \( X \) with skeletons \( X^n \), the inclusion \( X^2 \hookrightarrow X \) induces an isomorphism \( \pi_1(X^2, x_0) \to \pi_1(X, x_0) \): attaching a cell of dimension \( n \ge 3 \) does not change the fundamental group. The group of a connected one-dimensional complex is free, and attaching 2-cells adds one relation per cell, so \( \pi_1 \) of any CW-complex is computable from its 2-skeleton.<sup>[6](https://www.math.univ-paris13.fr/~harpaz/algebraic_topology.pdf)</sup> For example, van Kampen's theorem gives the torus the presentation \( \langle a, b : aba^{-1}b^{-1} = 1 \rangle \), which is \( \mathbb{Z}^2 \), and the orientable surface of genus \( g \) the presentation \( \langle a_1, b_1, \ldots, a_g, b_g : [a_1, b_1] \cdots [a_g, b_g] = 1 \rangle \).<sup>[12](https://math.ucsd.edu/~justin/Papers/knotes.pdf)</sup>

**Covering spaces.** The covering map \( p : \mathbb{R} \to S^1 \), \( p(t) = (\cos 2\pi t, \sin 2\pi t) \), is the universal cover of the circle, and the fiber \( p^{-1}(x_0) \), which is \( \mathbb{Z} \), inherits a group structure agreeing with \( \pi_1(S^1, x_0) \).<sup>[5](https://www.math.columbia.edu/~mhaseliu/Docs/Intro.pdf)</sup> More generally, a loop in the base of a covering map lifts to a unique path from any chosen preimage of the base point.<sup>[13](https://university.wyattau.com/mathematics/9-topology/9_introduction-to-algebraic-topology/)</sup>

## The knot group

For a knot \( K \subset \mathbb{R}^3 \), the **knot group** \( \pi(K) \) is the fundamental group of the complement \( \mathbb{R}^3 - K \); since the complement is path connected, the base point is irrelevant.<sup>[12](https://math.ucsd.edu/~justin/Papers/knotes.pdf)</sup> Equivalent knots have homeomorphic complements and hence isomorphic groups, so the knot group is a knot invariant, and the converse theorem of Gordon and Luecke (1987) states that knots with homeomorphic complements are equivalent, possibly up to mirror imaging.<sup>[12](https://math.ucsd.edu/~justin/Papers/knotes.pdf)</sup> The **Wirtinger presentation** computes \( \pi(K) \) from a diagram: one generator per arc and one relation per crossing, each relation conjugating the incoming label by the overpass label.<sup>[12](https://math.ucsd.edu/~justin/Papers/knotes.pdf)</sup> The abelianization of any knot group is \( \mathbb{Z} \), and the group of the unknot is the infinite cyclic group \( \mathbb{Z} \), the fundamental group of the circle.<sup>[14](https://www.phy.olemiss.edu/~luca/Topics/k/knot.html)</sup><sup> • </sup><sup>[15](https://math.uchicago.edu/~may/REU2014/REUPapers/Linov.pdf)</sup>

## Algorithmic limits

Because fundamental groups of reasonable spaces are finitely presented, questions about spaces translate into decision problems about such groups. Dehn posed the word problem (deciding whether a word in the generators represents the identity) and the conjugacy problem in 1911, and Tietze posed the isomorphism problem in 1908; both men were topologists, and much early work on finitely presented groups was motivated by fundamental groups.<sup>[7](https://kconrad.math.uconn.edu/blurbs/grouptheory/wordproblem.pdf)</sup> Novikov and Boone independently proved in the 1950s that there is a finitely presented group for which the word problem is undecidable, so no algorithm solves these questions in general.<sup>[7](https://kconrad.math.uconn.edu/blurbs/grouptheory/wordproblem.pdf)</sup> The Adyan–Rabin theorem extends this to recognition: if a property is a [Markov property](https://www.edgepedia.org/markov-property), such as being trivial or being abelian, no algorithm decides from a finite presentation whether the group has it.<sup>[16](https://pagesperso.g-scop.grenoble-inp.fr/~lazarusf/Enseignement/compuTopo7.pdf)</sup> Positive results exist for restricted classes: the word and conjugacy problems are solvable for [surface groups](https://www.edgepedia.org/surface-group), and for fundamental groups of closed, orientable 3-manifolds, though none of those classes of groups is algorithmically recognizable from presentations alone.<sup>[16](https://pagesperso.g-scop.grenoble-inp.fr/~lazarusf/Enseignement/compuTopo7.pdf)</sup>

## History

[Henri Poincaré](https://www.edgepedia.org/henri-poincare) introduced the fundamental group of a variety, and the concept of homotopy, in his 1895 memoir *Analysis situs* in the Journal de l'École polytechnique, the work that initiated algebraic topology.<sup>[3](https://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/)</sup><sup> • </sup><sup>[4](https://henripoincarepapers.univ-nantes.fr/en/bibliohp/ajax.php?bibkey=hp1895ep)</sup><sup> • </sup><sup>[17](https://ncatlab.org/nlab/show/Analysis%2BSitus)</sup> The group's elements are the "unshrinkable loops" of a space: classes of closed curves, each carrying a description of a motion of a point around the curve, that cannot be contracted to a point within the space.<sup>[1](https://ncatlab.org/nlab/files/NicolaescuGeometryOfManifolds.pdf)</sup>

## References

1. [The Fundamental group and covering spaces (Nicolaescu, Geometry of Manifolds, ch. 6)](https://ncatlab.org/nlab/files/NicolaescuGeometryOfManifolds.pdf)
2. [The fundamental group, Motion Planning notes, University of Illinois](https://msl.cs.uiuc.edu/planning/node143.html)
3. [Topology history, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/)
4. [Analysis situs, Henri Poincaré, Journal de l'École polytechnique (1895), bibliographic record](https://henripoincarepapers.univ-nantes.fr/en/bibliohp/ajax.php?bibkey=hp1895ep)
5. [Overview and fundamental groups, Columbia University lecture notes](https://www.math.columbia.edu/~mhaseliu/Docs/Intro.pdf)
6. [Basic Notions in Algebraic Topology, Université Paris 13](https://www.math.univ-paris13.fr/~harpaz/algebraic_topology.pdf)
7. [Keith Conrad, Decision problems about groups](https://kconrad.math.uconn.edu/blurbs/grouptheory/wordproblem.pdf)
8. [The Fundamental Group of X relative to x0, MA516 lecture notes, Northern Michigan University](https://euclid.nmu.edu/~joshthom/ma516/lectures/lecture14.html)
9. [The Fundamental Group, MA3408 notes, NTNU](https://drewkh.folk.ntnu.no/MA3408/the-fundamental-group.html)
10. [The fundamental group, Notes by Lawrence Hook](https://lawrencehook.github.io/apps-by-ai/notes/topology/fundamental-group/index.html)
11. [Algebraic topology notes, University College London](https://www.homepages.ucl.ac.uk/~ucahsgh/notesAT.pdf)
12. [Knots Knots, UC San Diego](https://math.ucsd.edu/~justin/Papers/knotes.pdf)
13. [Introduction to Algebraic Topology, Wyatt Au](https://university.wyattau.com/mathematics/9-topology/9_introduction-to-algebraic-topology/)
14. [Topics: Knot Theory, University of Mississippi](https://www.phy.olemiss.edu/~luca/Topics/k/knot.html)
15. [An Introduction to Knot Theory and the Knot Group, University of Chicago REU](https://math.uchicago.edu/~may/REU2014/REUPapers/Linov.pdf)
16. [Undecidability in Topology, F. Lazarus, G-SCOP Grenoble](https://pagesperso.g-scop.grenoble-inp.fr/~lazarusf/Enseignement/compuTopo7.pdf)
17. [Analysis Situs, nLab](https://ncatlab.org/nlab/show/Analysis%2BSitus)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Algebraic topology*

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