The fundamental group

In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant, topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space \(X\) is denoted by \(\pi_1(X)\).

Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point, paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breaking. The set of all such loops with this method of combining and this equivalence between them is the fundamental group for that particular space.

Henri Poincaré defined the fundamental group in 1895 in his paper "Analysis situs". The concept emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Poincaré, and Felix Klein. It describes the monodromy properties of complex-valued functions, as well as providing a complete topological classification of closed surfaces.

Throughout this article, \(X\) is a topological space. A typical example is a surface such as the one depicted at the right. Moreover, \(x_0\) is a point in \(X\) called the base-point. (As is explained below, its role is rather auxiliary.) The idea of the definition of the homotopy group is to measure how many (broadly speaking) curves on \(X\) can be deformed into each other. The precise definition depends on the notion of the homotopy of loops, which is explained first.

Given a topological space \(X\), a loop based at \(x_0\) is defined to be a continuous function (also known as a continuous map)

\(\gamma \colon [0, 1] \to X\)

such that the starting point \(\gamma(0)\) and the end point \(\gamma(1)\) are both equal to \(x_0\).

A homotopy is a continuous interpolation between two loops. More precisely, a homotopy between two loops \(\gamma, \gamma' \colon [0, 1] \to X\) (based at the same point \(x_0\)) is a continuous map \[h \colon [0, 1] \times [0, 1] \to X,\] such that

• \(h(0, t) = x_0\) for all \(t \in [0, 1]\), that is, the starting point of the homotopy is \(x_0\) for all \(t\) (which is often thought of as a time parameter).
• \(h(1, t) = x_0\) for all \(t \in [0, 1]\), that is, similarly the end point stays at \(x_0\) for all \(t\).
• \(h(r, 0) = \gamma(r)\), \(h(r, 1) = \gamma'(r)\) for all \(r \in [0, 1]\).

If such a homotopy \(h\) exists, \(\gamma\) and \(\gamma'\) are said to be homotopic. The relation "\(\gamma\) is homotopic to \(\gamma'\)" is an equivalence relation so that the set of equivalence classes can be considered: \[\pi_1(X, x_0) := \{ \text{all loops }\gamma \text{ based at }x_0 \} / \text{homotopy}.\] This set (with the group structure described below) is called the fundamental group of the topological space \(X\) at the base point \(x_0\). The purpose of considering the equivalence classes of loops up to homotopy, as opposed to the set of all loops (the so-called loop space of \(X\)) is that the latter, while being useful for various purposes, is a rather big and unwieldy object. By contrast the above quotient is, in many cases, more manageable and computable.

By the above definition, \(\pi_1(X, x_0)\) is just a set. It becomes a group (and therefore deserves the name fundamental group) using the concatenation of loops. More precisely, given two loops \(\gamma_0, \gamma_1\), their product is defined as the loop

\(\gamma_0 \cdot \gamma_1 \colon [0, 1] \to X\)

\((\gamma_0 \cdot \gamma_1)(t) = \begin{cases} \gamma_0(2t) & 0 \leq t \leq \tfrac{1}{2} \\ \gamma_1(2t - 1) & \tfrac{1}{2} \leq t \leq 1. \end{cases}\)

Thus the loop \(\gamma_0 \cdot \gamma_1\) first follows the loop \(\gamma_0\) with "twice the speed" and then follows \(\gamma_1\) with "twice the speed".

The product of two homotopy classes of loops \([\gamma_0]\) and \([\gamma_1]\) is then defined as \([\gamma_0 \cdot \gamma_1]\). It can be shown that this product does not depend on the choice of representatives and therefore gives a well-defined operation on the set \(\pi_1(X, x_0)\). This operation turns \(\pi_1(X, x_0)\) into a group. Its neutral element is the equivalence (homotopy) class of the constant loop, which stays at \(x_0\) for all times \(t\) (i.e. this class consists of all loops that can be continuously deformed into the constant loop; intuitively speaking of all the loops that do not "wrap around a hole"). The inverse of a (homotopy class of a) loop is the same loop, but traversed in the opposite direction (which is in a different homotopy class). More formally,

\(\gamma^{-1}(t) := \gamma(1-t).\)

Given three based loops \(\gamma_0, \gamma_1, \gamma_2,\) the product

\((\gamma_0 \cdot \gamma_1) \cdot \gamma_2\)

is the concatenation of these loops, traversing \(\gamma_0\) and then \(\gamma_1\) with quadruple speed, and then \(\gamma_2\) with double speed. By comparison,

\(\gamma_0 \cdot (\gamma_1 \cdot \gamma_2)\)

traverses the same paths (in the same order), but \(\gamma_0\) with double speed, and \(\gamma_1, \gamma_2\) with quadruple speed. Thus, because of the differing speeds, the two paths are not identical. The associativity axiom

\([\gamma_0] \cdot \left([\gamma_1] \cdot [\gamma_2]\right) = \left([\gamma_0] \cdot [\gamma_1]\right) \cdot [\gamma_2]\)

therefore crucially depends on the fact that paths are considered up to homotopy. Indeed, both above composites are homotopic, for example, to the loop that traverses all three loops \(\gamma_0, \gamma_1, \gamma_2\) with triple speed. The set of based loops up to homotopy, equipped with the above operation therefore does turn \(\pi_1(X, x_0)\) into a group.

Although the fundamental group in general depends on the choice of base point, it turns out that, up to isomorphism, this choice makes no difference as long as the space \(X\) is path-connected: more precisely, one obtains an isomorphism by pre- and post-concatenating with a path between the two basepoints. This isomorphism is, in general, not unique: it depends on the choice of path up to homotopy. However changing the path results in changing the isomorphism between the two fundamental groups only by composition with an inner automorphism. It is therefore customary to write \(\pi_1(X)\) instead of \(\pi_1(X, x_0)\) when the choice of basepoint does not matter.

This section lists some basic examples of fundamental groups. To begin with, in Euclidean space (\(\R^n\)) or any convex subset of \(\R^n,\) there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element. More generally, any star domain (and yet more generally, any contractible space) has a trivial fundamental group. Thus, the fundamental group does not distinguish between such spaces.

A path-connected space whose fundamental group is trivial is called simply connected. For example, the 2-sphere \(S^2 = \left\{(x, y, z) \in \R^3 \mid x^2 + y^2 + z^2 = 1\right\}\) depicted on the right, and also all the higher-dimensional spheres, are simply-connected. The figure illustrates a homotopy contracting one particular loop to the constant loop. This idea can be adapted to all loops \(\gamma\) such that there is a point \((x, y, z) \in S^2\) that is not in the image of \(\gamma.\) However, since there are loops such that \(\gamma([0, 1]) = S^2\) (constructed from the Peano curve, for example), a complete proof requires more careful analysis with tools from algebraic topology, such as the Seifert, van Kampen theorem or the cellular approximation theorem.

The circle (also known as the 1-sphere)

\(S^1 = \left\{(x, y) \in \R^2 \mid x^2 + y^2 = 1\right\}\)

is not simply connected. Instead, each homotopy class consists of all loops that wind around the circle a given number of times (which can be positive or negative, depending on the direction of winding). The product of a loop that winds around \(m\) times and another that winds around \(n\) times is a loop that winds around \(m + n\) times. Therefore, the fundamental group of the circle is isomorphic to \((\Z, +),\) the additive group of integers. This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk-Ulam theorem in dimension 2.

The fundamental group of the figure eight is the free group on two letters. The idea to prove this is as follows: choosing the base point to be the point where the two circles meet (dotted in black in the picture at the right), any loop \(\gamma\) can be decomposed as \[\gamma = a^{n_1} b^{m_1} \cdots a^{n_k} b^{m_k}\] where a and b are the two loops winding around each half of the figure as depicted, and the exponents \(n_1, \dots, n_k, m_1, \dots, m_k\) are integers. Unlike \(\pi_1(S^1)\), the fundamental group of the figure eight is not abelian: the two ways of composing \(a\) and \(b\) are not homotopic to each other: \[[a] \cdot [b] \ne [b] \cdot [a]\]

More generally, the fundamental group of a bouquet of \(r\) circles is the free group on \(r\) letters.

The fundamental group of a wedge sum of two path connected spaces \(X\) and \(Y\) can be computed as the free product of the individual fundamental groups: \[\pi_1(X \vee Y) \cong \pi_1(X) * \pi_1(Y)\] This generalizes the above observations since the figure eight is the wedge sum of two circles.

The fundamental group of the plane punctured at \(n\) points is also the free group with \(n\) generators. The \(i\)-th generator is the class of the loop that goes around the \(i\)-th puncture without going around any other punctures.

The fundamental group can be defined for discrete structures too. In particular, consider a connected graph \(G = (V, E)\), with a designated vertex \(v_0\) in \(V\). The loops in \(G\) are the cycles that start and end at \(v_0\). Let \(T\) be a spanning tree of \(G\). Every simple loop in \(G\) contains exactly one edge in \(E \setminus T\); every loop in \(G\) is a concatenation of such simple loops. Therefore, the fundamental group of a graph is a free group, in which the number of generators is exactly the number of edges in \(E \setminus T\). This number equals \(|E| - |V| + 1\).

For example, suppose \(G\) has 16 vertices arranged in 4 rows of 4 vertices each, with edges connecting vertices that are adjacent horizontally or vertically. Then \(G\) has 24 edges overall, and the number of edges in each spanning tree is 16 − 1 = 15, so the fundamental group of \(G\) is the free group with 9 generators. Note that \(G\) has 9 "holes", similarly to a bouquet of 9 circles, which has the same fundamental group.

Knot groups are by definition the fundamental group of the complement of a knot \(K\) embedded in \(\R^3\). For example, the knot group of the trefoil knot is known to be the braid group \(B_3\), which gives another example of a non-abelian fundamental group. The Wirtinger presentation explicitly describes knot groups in terms of generators and relations based on a diagram of the knot. Therefore, knot groups have some usage in knot theory to distinguish between knots: if \(\pi_1(\R^3 \setminus K)\) is not isomorphic to some other knot group \(\pi_1(\R^3 \setminus K')\) of another knot \(K'\), then \(K\) cannot be transformed into \(K'\). Thus the trefoil knot cannot be continuously transformed into the circle (also known as the unknot), since the latter has knot group \(\Z\). There are, however, knots that cannot be deformed into each other, but have isomorphic knot groups.

The fundamental group of a genus-n orientable surface can be computed in terms of generators and relations as \[\left\langle A_1, B_1, \ldots, A_n, B_n \left| A_1 B_1 A_1^{-1} B_1^{-1} \cdots A_n B_n A_n^{-1} B_n^{-1} \right. \right\rangle.\]

This includes the torus, being the case of genus 1, whose fundamental group is \[\left\langle A_1, B_1 \left| A_1 B_1 A_1^{-1} B_1^{-1} \right. \right\rangle \cong \Z^2.\]