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 have isomorphic fundamental groups. The fundamental group of a topological space is denoted by.

Intuition

Start with a space, 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.

History

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.

Definition

Throughout this article, is a topological space. A typical example is a surface such as the one depicted at the right. Moreover, is a point in called the base-point. The idea of the definition of the homotopy group is to measure how many curves on can be deformed into each other. The precise definition depends on the notion of the homotopy of loops, which is explained first.

Homotopy of loops

Given a topological space, a loop based at is defined to be a continuous function
such that the starting point and the end point are both equal to.
A homotopy is a continuous interpolation between two loops. More precisely, a homotopy between two loops is a continuous map
such that

for all that is, the starting point of the homotopy is for all .
for all that is, similarly the end point stays at for all t.
for all.

If such a homotopy exists, and are said to be homotopic. The relation " is homotopic to " is an equivalence relation so that the set of equivalence classes can be considered:
This set is called the fundamental group of the topological space at the base point. The purpose of considering the equivalence classes of loops up to homotopy, as opposed to the set of all loops 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.

Group structure

By the above definition, is just a set. It becomes a group using the concatenation of loops. More precisely, given two loops, their product is defined as the loop
Thus the loop first follows the loop with "twice the speed" and then follows with "twice the speed".
The product of two homotopy classes of loops and is then defined as. 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. This operation turns into a group. Its neutral element is the equivalence class of the constant loop, which stays at for all times . The inverse of a loop is the same loop, but traversed in the opposite direction. More formally,
Given three based loops the product
is the concatenation of these loops, traversing and then with quadruple speed, and then with double speed. By comparison,
traverses the same paths, but with double speed, and with quadruple speed. Thus, because of the differing speeds, the two paths are not identical. The associativity axiom
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 with triple speed. The set of based loops up to homotopy, equipped with the above operation therefore does turn into a group.

Dependence of the base point

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 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 instead of when the choice of basepoint does not matter.

Concrete examples

This section lists some basic examples of fundamental groups. To begin with, in Euclidean space or any convex subset of 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.

The 2-sphere

A path-connected space whose fundamental group is trivial is called simply connected.
For example, the 2-sphere 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 such that there is a point that is in the image of However, since there are loops such that , 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

The circle
is not simply connected. Instead, each homotopy class consists of all loops that wind around the circle a given number of times. The product of a loop that winds around times and another that winds around times is a loop that winds around times. Therefore, the fundamental group of the circle is isomorphic to 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 figure eight

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, any loop can be decomposed as
where a and b are the two loops winding around each half of the figure as depicted, and the exponents are integers. Unlike the fundamental group of the figure eight is not abelian: the two ways of composing and are not homotopic to each other:
More generally, the fundamental group of a bouquet of circles is the free group on letters.
The fundamental group of a wedge sum of two path connected spaces and can be computed as the free product of the individual fundamental groups:
This generalizes the above observations since the figure eight is the wedge sum of two circles.
The fundamental group of the plane punctured at points is also the free group with generators. The -th generator is the class of the loop that goes around the -th puncture without going around any other punctures.

Graphs

The fundamental group can be defined for discrete structures too. In particular, consider a connected graph, with a designated vertex in. The loops in are the cycles that start and end at. Let be a spanning tree of. Every simple loop in contains exactly one edge in ; every loop in 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. This number equals.
For example, suppose has 16 vertices arranged in 4 rows of 4 vertices each, with edges connecting vertices that are adjacent horizontally or vertically. Then has 24 edges overall, and the number of edges in each spanning tree is, so the fundamental group of is the free group with 9 generators. Note that has 9 "holes", similarly to a bouquet of 9 circles, which has the same fundamental group.

Knot groups

Knot groups are by definition the fundamental group of the complement of a knot embedded in For example, the knot group of the trefoil knot is known to be the braid group 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 is not isomorphic to some other knot group of another knot, then can not be transformed into. Thus the trefoil knot can not be continuously transformed into the circle, since the latter has knot group. There are, however, knots that can not be deformed into each other, but have isomorphic knot groups.

Oriented surfaces

The fundamental group of a genus-n orientable surface can be computed in terms of generators and relations as
This includes the torus, being the case of genus 1, whose fundamental group is

Topological groups

The fundamental group of a topological group is always commutative. In particular, the fundamental group of a Lie group is commutative. In fact, the group structure on endows with another group structure: given two loops and in, another loop can defined by using the group multiplication in :
This binary operation on the set of all loops is a priori independent from the one described above. However, the Eckmann–Hilton argument shows that it does in fact agree with the above concatenation of loops, and moreover that the resulting group structure is abelian.
An inspection of the proof shows that, more generally, is abelian for any H-space, i.e., the multiplication need not have an inverse, nor does it have to be associative. For example, this shows that the fundamental group of a loop space of another topological space, is abelian. Related ideas lead to Heinz Hopf's computation of the cohomology of a Lie group.