Fundamental group

The fundamental group is a basic construction of algebraic topology. It associates to any topological space <math>X</math> a group <math>\pi_1(X)</math> containing information about the space of "loops" in that space.

Simply Connected Spaces

Intuition

The easiest topological spaces to deal with, from the standpoint of the fundamental group, are the simply connected spaces. A space is termed simply connected if every loop in the space is homotopic to a constant loop. The intuition behind this definition is simple: imagine a rubber band is immersed in the topological space. In a simply connected space, the rubber band may contract to a point, while remaining entirely within the space. For example, no matter how a rubber band is arranged on a sphere, it may always be shrunk down while remaining on the sphere. In contrast, it is possible to place a rubber band on a torus in such a way that no such contraction is possible. A few other spaces which have the property of being simply connected are:

• Euclidean space <math>\mathbb R^n</math>. This includes the cases familiar cases of the line (<math>\mathbb R^1</math>), the plane (<math>\mathbb R^2</math>), and 3-dimensional space (<math>\mathbb R^3</math>).
• Convex subsets of Euclidean space. This includes, for example, the unit disk in the plane.
• The sphere <math>S^n</math>, for <math>n \geq 2</math>. In the case <math>n=2</math>, this is the familiar sphere in <math>\mathbb R^3</math>.
• The space obtained by gluing two spheres <math>S^2</math> together at a single point.
• The product of any two simply connected spaces is simply connected.

Other familiar spaces are not simply connected:

• The circle <math>S^1</math>. A rubber band looped around a circle cannot be contracted to a point without ripping it.
• The punctured plane <math>\mathbb R^2 \setminus \{(0,0)\}</math>.
• The torus (a hollow donut) is not simply connected. A rubber band looped around the central hole, or a rubber band about the inner circle, cannot be contracted.

As Homotopy

The notion of homotopy makes the above definition of simple connectedness precise. A "loop in <math>X</math>", earlier represented by a rubber band, is now a continuous function <math>\gamma : [0,1] \to X</math> satisfying <math>\gamma(0)=\gamma(1)=*</math> of the unit interval into <math>X</math>. It is generally convenient to fix a "basepoint" <math>* \in X</math>, which may be any point in the topological space <math>X</math>, and assume that <math>\gamma(0)=\gamma(1)=*</math>. The choice of basepoint does not change the fundamental group (up to isomorphism). In terms of the analogy of a rubber band, <math>\gamma(t)</math> is the point in <math>X</math> where the point a proportion <math>t</math> along the rubber band is initiall placed.

Two paths <math>\gamma_0,\gamma_1</math> are said to be homotopic if there is a continuous function

<math>H : [0,1] \times [0,1] \to X</math>

which satisfies

<math>H(t,0)=\gamma_1(t)</math>, <math>H(t,1)=\gamma_2(t)</math>, <math>H(0,s)=H(1,s)=*</math>

The intuition is this: for every value of <math>s \in [0,1]</math>, we get a path <math>\gamma_s(t) : [0,1] \to X</math>, and at <math>s=0</math> and <math>s=1</math>, these are the two paths in question. Thus <math>H</math> provides a smoothly-varying set of paths in <math>X</math>, parametrized by a variable <math>t</math>, starting with <math>\gamma_0</math>, and ending with <math>\gamma_1</math>.

The constant path is defined by <math>\gamma_c(t) = *</math>, for all <math>t</math>. A space is said to be simply-connected if every path based at <math>*</math> is homotopic to the constant path. This definition serves to make the intuition in terms of shrinking rubber bands precise.

The General Situation

In many topological spaces, there are loops which can not be contracted to a point: a few examples were given above. The topological group is a structure which serves to describe the set of loops in a space, where two loops that can be continuously deformed into each other are treated as equivalent.

As an example, consider loops in the circle. Let us pass to another analogy for now: a particle moves around in a topological space, and we wish to describe its path as simply as possible, and such that two paths that may be deformed into each other (i.e., are homotopic), are given the same description. One path might be described as "go around the circle once in a clockwise direction". It does not matter at what speed the particle moves, or whether it turns around before reversing itself and completing a cycle: such paths may all be deformed to one another, and to describe such a path up to homotopy it suffices to give the preceding description. Similarly, "go around the circle twice in a clockwise direction" describes another class of paths, not equivalent to going around once. Thus, to describe a class of loops in a circle, it suffices to say how many times a path goes around the circle. By writing a path going <math>n</math> times around clockwise as <math>n</math>, and <math>n</math> times counterclockwise as <math>-n</math>, to describe a class of paths it is sufficient merely to give an integer!

The fundamental group <math>\pi_1(X)</math> is defined to be the space of loops in a space, modulo the relation of homotopy. The preceding description indicates that <math>\pi_1(S^1) \cong \Z</math>, by an isomorphism which sends a class of paths to the integer describing the number of times these paths go around the circle.

Group Structure

The preceding definition is incomplete, because it does not specify the operation on <math>\pi_1(X)</math> which gives these classes of paths the structure of not just a set, but a group. The definition of the group operation is simple: given two paths <math>\gamma_1</math> and <math>\gamma_2</math>, the sum <math>\gamma_1 * \gamma_2</math> is the path obtained by first following <math>\gamma_1</math>, and then following <math>\gamma_2</math>. More precisely, set

<math>(\gamma_1 * \gamma_2)(t) = \begin{cases} \gamma_1(2t) & \textrm{if $0 \leq t \leq 1/2$} \\ \gamma_2(2t-1) & \textrm{if $1/2 \leq t \leq 1$} \end{cases}</math>

For this to be a group, it is necessary to see that there is an identity element with respect to the operation, and that every loop has an inverse. It is clear that the constant loop at <math>*</math> serves as an identity, and the inverse of a loop <math>\gamma</math> is obtained simply by following <math>\gamma</math> in the opposite direction:

<math>(\gamma^{-1})(t) = \gamma(1-t)</math>.

To see how this works in the preceding case of the circle, let <math>\gamma^n</math> be a path which goes around the circle <math>n</math> times. The composition <math>\gamma^m * \gamma^n</math> is the path which goes around <math>m</math> times, and then <math>n</math> more: this is exactly the loop <math>\gamma^{m+n}</math>. This indicates that the group operation on <math>\pi_1(S^1) \cong \mathbb Z</math> is just the usual operation of addition on <math>\mathbb Z</math>.

Examples

The fundamental groups of more complex spaces may be harder to describe. Consider the figure eight <math>S^1 \vee S^1</math>, the space obtained by gluing two circles together at a point. What is required to describe a class of paths in this space? A path could be specified by saying "starting at the glue point, go around the left circle 5 times, then the right circle 2 times, then the left circle -3 times (i.e., three times counterclockwise)", etc. This might be written as

<math>\gamma = a^5 b^2 a^{-3}</math>,

where <math>a</math> denotes a path which goes around the left circle once clockwise, and <math>b</math> a path which goes around the right circle once clockwise. More generally, any loop in the figure-eight can be specified by a "word"

<math>\gamma = a^{i_1} b^{j_1} \cdots a^{i_n} b^{j_n}</math>,

where <math>i_1,i_2,i_3,\ldots</math> and <math>j_1,j_2,j_3,\ldots</math> are sequences of integers. The multiplication of two loops written in this form is obtained by concatenating the two words describing them. The inverse of a word is obtained by reversing it and switching the signs on all of the exponents. The group of such words on two letters <math>a,b</math> with these operations is termed the free group on two generators, and usually denoted by <math>F_2</math> or <math>\mathbb Z * \mathbb Z</math>.

The fundamental groups of a few other familiar topological spaces are given by:

• Torus: <math>\pi_1(S^1 \times S^1) \cong \mathbb Z \oplus \mathbb Z</math>.
• Mobius strip: <math>\pi_1(M) \cong \mathbb Z</math>.
• Cylinder: <math>\pi_1(S^1 \times \mathbb R) \cong \mathbb Z</math>.
• The complement of a trefoil knot in 3-dimensional space: <math>\pi_1(S^3 \setminus K)</math> is a non-abelian group, in contrast to the other examples given here.
• Klein bottle: <math>\langle a,b : aba^{-1}b =e\rangle</math>

These examples show that a wide range of groups are in fact the fundamental groups of some topological space. In fact, every finitely presented group is the fundamental group of a topological space! This observation makes possible the study of many aspects in group theory using techniques from topology.

Methods of Computation

The main theorem for computing the fundamental group of topological spaces is the Seifert-van Kampen theorem. This theorem gives a means to describe the fundamental group of a space obtained by gluing two other spaces together along subsets of these two spaces. The computation of the fundamental group of the figure eight, described above, is the most elementary application of the Seifert-van Kampen theorem.

The theory of covering spaces provides another powerful tool for the computation of fundamental groups.

Applications

The fundamental group is a basic object in the study of algebraic topology. In its most basic form, it provides a way to prove that two topological spaces are not the same thing. For example, the torus and the 2-sphere have different fundamental groups, and this proves that they not the same space. Intuitively, the "hole" in a torus makes it different from a sphere with no holes. This fact, while apparently obvious, is difficult to prove without the methods of algebraic topology, including the fundamental group.

The applications of the fundamental group within algebraic topology are many, and it is one of the most basic tools in the field.

Higher Homotopy Groups

The fundamental group is the first homotopy group. Higher homotopy groups are defined by generalizing the construction of the fundamental group: while the fundamental group is defined as the set of homotopy classes of maps from <math>S^1</math> into a space, the <math>n</math> homotopy group <math>\pi_n(X)</math> is defined as the set of homotopy classes of maps of <math>S^n</math> into a space. In contrast to the fundamental group, the homotopy groups <math>\pi_n(X)</math> for <math>n \geq 2</math>, are abelian groups. In general, they are much more difficult to compute.