This is also called the first homotopy group of .
The notion of fundamental group generalises in one direction to the fundamental groupoid , or in another direction to the homotopy groups for . Both of this is contained within the fundamental ∞-groupoid .
from the topological 1-simplex, such that .
such that .
This notion of based homotopy is an equivalence relation.
This is directly checked. It is also a special case of the general discussion at homotopy.
Given two loops , define their concatenation to be the loop
See also at path groupoid for similar constructions.
For a topological space and a point, the set of based homotopy equivalence classes of based loops in equipped with the group structure from prop. 2 is the fundamental group or first homotopy group of , denoted
Hence if we write for the based homotopy class of a loop , then then group operation is
For any two basepoints, there is by assumption a path connecting them, hence a continuous function
such that and .
It is immediate to check that this induces an isomorphism
Therefore one sometimes loosely speaks of ‘the’ fundamental group of a connected space. But beware that the isomorphism in the above construction is not unique. Therefore forming fundamental groups is not a functor on connected spaces.
The fundamental group is in general non-abelian (i.e. is not an abelian group), the Examples below. For a connected topological space , its abelianization is equivalent to the first singular homology group
See at singular homology – Relation to homotopy groups for more on this.
There is a relation to universal covers: Under suitable conditions the group of cover automorphisms of a universal cover is isomorphic to the fundamental group of the covered space. This is the topic of the étale fundamental group, also referred to at Chevalley fundamental group, see there for more.
In particular in algebraic geometry and arithmetic geometry this essentially identifies the concept of fundamental group with that of Galois groups. For this reason one also speaks of the algebraic fundamental group in this context. See at Galois theory for more on this.
See also at link between Galois theory and fundamental groups?.
In Grothendieck's Galois theory, the role of the basepoint is replaced by considering a ‘fibre functor’ or to , where is the category of coverings of the given space. This theory extends to other situations and the term algebraic fundamental group is used in particular for the case of schemes (of a suitable type); see (SGA1).
The definition of fundamental group in terms of homotopy classes of loops at a base point does not work well for the spaces that occur in algebraic geometry, nor for many spaces considered in analysis as there may be very few loops. For instance, for a scheme there are in general very few paths, and Grothendieck gave a definition of a fundamental group in SGA1 which is closely related to the Galois groups of number theory, but in cases where both the path-based group and this algebraic fundamental group make sense, the algebraic form tends to be related to the profinite completion of the topological fundamental group; see the example in that entry.
A similar type of construction gives the fundamental group of a topos. Other related forms include a Čech version of the fundamental group used in shape theory, and linked to Čech homology groups of a compact space.
The notion of fundamental group generalizes to that of fundamental groupoid in both the loop based theory and in Grothendieck's Galois theory as described in SGA1. In this form it has been used to give generalisations for simplicial profinite spaces in work by Quick and to pro-spaces in work of Isaksen.
In the context of proper homotopy theory there are two related fundamental groups for single ended spaces.
The fundamental group of the point is trivial: .
Discussion from the point of view of Galois theory is in
Isaksen’s work is
whilst Quick’s is in
Formalization in homotopy type theory is at