Fundamental group

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\pi_1(S^1)=?\, This article/section deals with mathematical concepts appropriate for a student in late university or graduate level.

One of the most important problems of topology (and algebraic topology in particular) is the precise and rigorous characterization of the "holes" in a space. This characterization is in terms of some groups known as the homotopy groups and homology groups. The most important of these is the 1st homotopy group, known as the fundamental group.

The concept of homotopy, on which the fundamental group is based, is a type of equivalence between paths. Two paths are said to be homotopic if one can be continuously deformed/moved/stretched into the other. This stretching can't pass over a hole, so a characterization of homotopic paths yields a characterization of holes.

Given a topological space X, the fundamental group is an associated algebraic object which describes the paths that are not homotopic, and hence describes the set of "holes" in the space.

The fundamental group is useful to distinguish between different topological spaces: for example, to prove rigorously that a surface with two holes cannot be smoothly deformed into a surface with three holes, one can check that these two spaces have different fundamental groups and so must be distinct. The fundamental group captures information about the set of holes in a space by looking at the set of loops drawn in the space.

Contents

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. A simply connected space is one in which the rubber band may always be contracted 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, and so the torus is not simply connected. A few other spaces which have the property of being simply connected are:

  • Euclidean space \mathbb R^n. This includes the cases familiar cases of the line (\mathbb R^1), the plane (\mathbb R^2), and 3-dimensional space (\mathbb R^3).
  • Convex subsets of Euclidean space. This includes, for example, the unit disk in the plane.
  • The sphere Sn, for n \geq 2. In the case n = 2, this is the familiar sphere in \mathbb R^3.
  • The space obtained by gluing two spheres S2 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 S1. A rubber band looped around a circle cannot be contracted to a point without ripping it.
  • The punctured plane \mathbb R^2 \setminus \{(0,0)\}.
  • 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 X", earlier represented by a rubber band, is now a continuous function \gamma : [0,1] \to X satisfying γ(0) = γ(1) = * of the unit interval into X. It is generally convenient to fix a "basepoint" * \in X, which may be any point in the topological space X, and assume that γ(0) = γ(1) = * . The choice of basepoint does not change the fundamental group (up to isomorphism). In terms of the analogy of a rubber band, the function γ is defined so that γ(t) is the point in X where the point a proportion t along the rubber band is initiall placed.

Two paths γ01 are said to be homotopic if there is a continuous function

H : [0,1] \times [0,1] \to X

which satisfies

H(t,0)=\gamma_0(t)\,, H(t,1)=\gamma_1(t)\,, H(0,s)=H(1,s)=*\,

The intuition is this: for every value of s \in [0,1], we get a path \gamma_s(t) : [0,1] \to X, and at s = 0 and s = 1, these are the two paths in question. Thus H provides a smoothly-varying set of paths in X, parametrized by a variable s, starting with γ0, and ending with γ1. This relation is often written as γ0˜γ1.

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

Problem: Prove that the relation of homotopy of paths is an equivalence relation on the set of paths.

Solution: First we need to check reflexivity, that γ˜γ for any path γ. A homotopy from γ to itself is provided by H(t,s) = γ(t) for all s. This clearly satisfies H(t,0) = H(t,1) = γ(t), as required. An equivalence relation must also satisfy reflexivity, so that γ0˜γ1 implies that γ1˜γ0. To prove that homotopy of loops satisfy this, suppose that γ0˜γ1. Then there is a homotopy H_0(t,s) : [0,1] \times [0,1] \to X satisfying the requisite conditions. A homotopy from γ1 to γ0 is provided by running H0 in reverse: set H(t,s) = H0(t,1 − s). The final requirement is that of transitivity, that γ0˜γ1 and γ1˜γ2 implies that γ0˜γ2. This is the intuitive fact that if one path may be deformed to another, and that one may be deformed some third path, then the original path may be deformed directly to the third path. To prove this, let H0 be a homotopy from γ0 to γ1, and H1 a homotopy from γ1 to γ2. Define a new homotopy by

H(t,s) =
\begin{cases}
  H_1(t,2s) & \mbox{if } 0 \leq s \leq 1/2, \\
  H_2(t,2s-1) & \mbox{if } 1/2 \leq s \leq 1
\end{cases}.

This is a homotopy from γ0 to γ2.

This exercise proves that homotopy is an equivalence relation on loops in X, whether or not X is simply connected. The elements of the fundamental group are defined to be homotopy equivalence class of loops in X based at * : thus two loops represent different elements of the fundamental group if and only if they are not homotopic to each other. In a simply connected space, all loops are homotopic and thus represent a single homotopy class, and so the fundamental group is the trivial group, with only one element.

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 fundamental 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, in such a way 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 n times around clockwise as n, and n times counterclockwise as n, to describe a homotopy class of paths it is sufficient merely to give an integer counting the number of times the path goes around the circle!

The fundamental group π1(X) is defined to be the space of loops in a space, modulo the relation of homotopy. The preceding description indicates that \pi_1(S^1) \cong \Z, 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 π1(X) 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 γ1 and γ2, the product γ1 * γ2 is the path obtained by first following γ1, and then following γ2. More precisely, set

(\gamma_1 * \gamma_2)(t) = \begin{cases} \gamma_1(2t) & \mbox{if } 0 \leq t \leq 1/2 \\ \gamma_2(2t-1) & \mbox{if } 1/2 \leq t \leq 1 \end{cases}

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 * serves as an identity, and the inverse of a loop γ is obtained simply by following γ in the opposite direction:

− 1)(t) = γ(1 − t).

To see how this works in the preceding case of the circle, let γn be a path which goes around the circle n times. The composition γm * γn is the path which goes around m times, and then n more: this is exactly the loop γm + n. This indicates that the group operation on \pi_1(S^1) \cong \mathbb Z is just the usual operation of addition on \mathbb Z. To prove carefully that paths going around the circle different numbers of times are not homotopic to each other requires a bit more machinery than is developed in this article.

Examples

The fundamental groups of more complex spaces may be harder to describe. Consider the figure eight S^1 \vee S^1, 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 gluing 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

γ = a5b2a − 3,

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

\gamma = a^{i_1} b^{j_1} \cdots a^{i_n} b^{j_n},

where i_1,i_2,i_3,\ldots and j_1,j_2,j_3,\ldots 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 a,b with these operations is termed the free group on two generators, and usually denoted by F2 or \mathbb Z * \mathbb Z.

Problem: Describe the fundamental group of the topological space obtained by gluing three circles together at a single point. Solution: The fundamental group \pi_1(S_1 \vee S_1 \vee S_1) is described in a similar manner to that of the example. Let a, b, and c denote loops around each of the three circles. An element of the fundamental group is just a "word" on these three letters, describing the order in which a path goes around each of the three circles.

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

  • Torus: \pi_1(S^1 \times S^1) \cong \mathbb Z \oplus \mathbb Z. Let a be a loop going around the central hole, and b be a loop around the core of the torus. It's easy to check that ab is actually homotopic to ba, which means that to specific a loop on the torus, one needs only say how many times it goes around each of these, by specifying an ordered pair (m,n) of integers.
  • Mobius strip: \pi_1(M) \cong \mathbb Z. As in the circle, to specify a path on the Mobius strip up to homotopy, it suffices to say how many times the path goes around the hole.
  • Cylinder: \pi_1(S^1 \times \mathbb R) \cong \mathbb Z. This too works the same way as the circle: a homotopy class of paths is determined by the number of times it wraps around the hole.
  • The complement of a trefoil knot (overhand knot) in 3-dimensional space: \pi_1(S^3 \setminus K) is a non-abelian group, in contrast to the other examples given here.

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 (more precisely, that they are not homeomorphic or indeed homotopy equivalent). 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.

Properties

The fundamental group has a variety of convenient properties that make its application convenient. One is that it is a homeomorphism invariant: if two spaces are homeomorphic, then they have the same fundamental group. Additionally, if one is given a map f : X \to Y between two topological spaces X and Y, one can construct a map f_* : \pi_1(X) \to \pi_1(Y). They way to do this is easy: given a homotopy class \alpha \in \pi_1(X), choose a representative loop \gamma : [0,1] \to X. Composing γ with f, we obtain a loop f \circ \gamma  : [0,1] \to Y. This is just the image of the loop γ under f. Now define f_*(\alpha) \in \pi_1(Y) to be the equivalence class of f \circ \gamma. This construction is well-behaved with respect to the composition of maps: if f : X \to Y and g : Y \to Z are two maps, there are induced maps f_* : \pi_1(X) \to \pi_1(Y) and g_* \pi_1(Y) \to \pi_1(Z). On the other hand, there is a composition g \circ f : X \to Z, inducing (g \circ f)_* : \pi_1(X) \to \pi_1(Z). It turns out that (g \circ f)_* = g_* \circ f_*: this is an important property of the fundamental group, which, in the language of category theory, is called funtoriality.

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 S1 into a space, the n homotopy group πn(X) is defined as the set of homotopy classes of maps of Sn into a space. In contrast to the fundamental group, the homotopy groups πn(X) for n \geq 2, are abelian groups. In general, they are much more difficult to compute.

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