Fundamental group

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Template:Group associated to based topospaces

Definition

Basic definition

The fundamental group of a based topological space (X,x_0) is defined as follows:

  • As a set, it is the set of all homotopy classes of loops at x_0 in X. The homotopy class of a loop f is denoted [f]. Note here that homotopy class of loop in particular means that at every stage of the homotopy, we must have a loop based at x_0. In particular, it is not the same thing as the intersection with loops based at x_0 of homotopy classes of paths in X.
  • The group structure is obtained as follows: the composite of two loops is obtained by first traversing the first loop, and then traversing the second loop. Explicitly, if f_1, f_2:[0,1] \to X are the two loops, then the composite f_1 * f_2 of these is the loop given by t \mapsto f_1(2t) for 0 \le t \le 1/2 and t \mapsto f_2(2t - 1) for 1/2 \le t \le 1. Continuity of this new loop follows from the gluing lemma for closed subsets.

When the topological space is path-connected, the fundamental groups at any two basepoints are isomorphic. Further information: Actions of the fundamental group

Proof that this gives a group structure

All loops here are based at x_0.

Condition How it is shown Page detailing relevant homotopy
well defined if f_1 and g_1 are homotopic to each other, and f_2 and g_2 are homotopic to each other, then f_1 * f_2 and g_1 * g_2 are homotopic to each other. homotopy between composites of homotopic loops
existence of identity element the identity element is the homotopy class of the constant loop that stays at the base point. In other words, if we denote this loop by e, then for any loop f, the composite e * f is homotopic to f, and so is the composite f * e. homotopy between loop and composite with constant loop
existence of inverses the inverse of a loop f is the loop t \mapsto f(1 - t), i.e., the same loop done backward. In other words the composite of f and this loop is homotopic to the constant loop. homotopy between constant loop and composite of loop with inverse
associativity for loops f_1, f_2, f_3, the composite f_1 * (f_2 * f_3) is homotopic to the composite (f_1 * f_2) * f_3. homotopy between composites associated in different ways

One nice thing about all these homotopies is that they do not depend on additional properties of the ambient space, and the homotopies do not use any points of the space other than those used in the original loops.

Alternative definitions

This definition need not work for all topological spaces; however, it works for all compactly generated Hausdorff spaces and possibly many others.

The fundamental group of a based topological space (X,x_0) is defined as the space of path components of the loop space of (X,x_0), i.e.:

\! \pi_1(X,x_0) = \pi_0(\Omega(X,x_0))

The group structure arises as the structure induced on the quotient by the natural multiplication structure on \Omega(X,x_0). Because that is a H-space under the multiplication, \pi_0 of the space gets a monoid structure. It turns out that this monoid structure is also a group.

This definition can be reconciled with the usual definition as follows: paths in the loop space of a based topological space are the same thing as homotopies of based loops in the original space. Thus, the set of path components of the loop space based at a point is the same thing as the set of homotopy classes of loops based at the point. Further, the composition used to give a H-space structure to \Omega(X,x_0) descends precisely to the group multiplication we use to define the fundamental group.

Topology on the fundamental group

The definition of the fundamental group as the space of path components of \Omega(X,x_0) gives a topology on the fundamental group. It turns out, though, that if the path component of x_0 in X is a locally path-connected space, then so is \Omega(X,x_0), in which case the fundamental group has a discrete topology. Since the spaces we typically study (such as manifolds) are locally path-connected, the fundamental group is habitually viewed as a discrete group.

Omission of basepoint

For a path-connected space, the fundamental groups at two different points are isomorphic. Moreover, any path between these two points yields an isomorphism, and any two such isomorphisms have quotient equal to an inner automorphism of the fundamental group at one point. When viewing the fundamental group as an abstract group, it is thus often customary to omit the basepoint for a path-connected space.

More generally, for a topological space where all the path components are homeomorphic (or even more generally, where they are all homotopy-equivalent spaces), the isomorphism class of the fundamental group is independent of the choice of basepoint. In particular, this is true for homogeneous spaces such as topological groups.

For a path-connected space with abelian fundamental group, the fundamental groups at any two points are canonically identified and thus it makes even more sense to omit the basepoint.

Relation with other constructs

Construct Name Symbol Relation with fundamental group \pi_1(X,x_0)
All loops based at x_0, not up to homotopy loop space of a based topological space \Omega(X,x_0) \pi_1(X,x_0) = \pi_0(\Omega(X,x_0))
All loops based at x_0, up to homotopies where the intermediaries must be loops but need not be based at x_0 conjugacy class set of fundamental group  ? The set of conjugacy classes of \pi_1(X,x_0). See also actions of the fundamental group
All loops based at x_0, up to slice and rearrange as well as homotopies where the intermediates must be loops but need not be based at x_0. In other words, homology classes first homology group (if X is path-connected) H_1(X,x_0) or just H_1(X) For a path-connected space, the first homology group is the abelianization of the fundamental group.
All paths up to homotopies that preserve endpoints of paths fundamental groupoid  ?  ?

Examples

Most of the examples of spaces below are topological spaces that are either path-connected or where all the path components are homeomorphic. Thus, the value of the fundamental group is independent of the choice of basepoint.

Topological space Fundamental group Conjugacy class set of fundamental group Abelianization of fundamental group (first based homology group) Explanation/comment
empty space doesn't make sense, no basepoint doesn't make sense, no basepoint doesn't make sense, no basepoint
one-point space trivial group one-point set trivial group the only loop is a constant loop
discrete space trivial group one-point set trivial group each path component is a one-point space
closed unit interval trivial group one-point set trivial group the space is a contractible space
open unit interval trivial group one-point set trivial group the space is a contractible space
unit circle group of integers group of integers group of integers the fundamental group is abelian, hence its conjugacy class set and abelianization can be identified with it. Also, the generator of the fundamental group is the identity map from the unit circle to itself.
Euclidean space trivial group one-point set trivial group the space is a contractible space
open unit disk in Euclidean space trivial group one-point set trivial group the space is a contractible space
closed unit disk in Euclidean space trivial group one-point set trivial group the space is a contractible space
2-sphere trivial group one-point set trivial group although the space is not contractible, any loop can still be homotoped to the constant loop. One way of seeing this is via the Seifert-van Kampen theorem.
n-sphere for n > 1 trivial group one-point set trivial group although the space is not contractible, any loop can still be homotoped to the constant loop. One way of seeing this is via the Seifert-van Kampen theorem.
2-torus \mathbb{Z} \times \mathbb{Z} \mathbb{Z} \times \mathbb{Z} \mathbb{Z} \times \mathbb{Z} fundamental group of a product of topological spaces is direct product of fundamental groups.
wedge of two circles (also called figure of eight) free group:F2 conjugacy class set of free group:F2 \mathbb{Z} \times \mathbb{Z} follows from Seifert-van Kampen theorem, for instance.

Related properties of topological spaces

Aspects of the fundamental group