The fundamental group of a based topological space is defined as follows:
- As a set, it is the set of all homotopy classes of loops at in . The homotopy class of a loop is denoted . Note here that homotopy class of loop in particular means that at every stage of the homotopy, we must have a loop based at . In particular, it is not the same thing as the intersection with loops based at of homotopy classes of paths in .
- 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 are the two loops, then the composite of these is the loop given by for and for . 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 .
|Condition||How it is shown||Page detailing relevant homotopy|
|well defined||if and are homotopic to each other, and and are homotopic to each other, then and 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 , then for any loop , the composite is homotopic to , and so is the composite .||homotopy between loop and composite with constant loop|
|existence of inverses||the inverse of a loop is the loop , i.e., the same loop done backward. In other words the composite of and this loop is homotopic to the constant loop.||homotopy between constant loop and composite of loop with inverse|
|associativity||for loops , the composite is homotopic to the composite .||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.
This definition need not work for all topological spaces; however, it works for all compactly generated Hausdorff spaces and possibly many others.
The group structure arises as the structure induced on the quotient by the natural multiplication structure on . Because that is a H-space under the multiplication, 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 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 gives a topology on the fundamental group. It turns out, though, that if the path component of in is a locally path-connected space, then so is , 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|
|All loops based at , not up to homotopy||loop space of a based topological space|
|All loops based at , up to homotopies where the intermediaries must be loops but need not be based at||conjugacy class set of fundamental group||?||The set of conjugacy classes of . See also actions of the fundamental group|
|All loops based at , up to slice and rearrange as well as homotopies where the intermediates must be loops but need not be based at . In other words, homology classes||first homology group (if is path-connected)||or just||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||?||?|
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.|
|-sphere for||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||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||follows from Seifert-van Kampen theorem, for instance.|
Related properties of topological spaces
- A simply connected space is a topological space whose fundamental group is trivial
- For a H-space and hence in particular for any space that arises as a loop space, and for any topological monoid, the fundamental group is Abelian