Fundamental group

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 $0\leq t\leq 1/2$ and $t\mapsto f_{2}(2t-1)$ for $1/2\leq t\leq 1$. 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

• 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

Aspects of the fundamental group

• Fundamental group functor
• Actions of the fundamental group