Editing Fundamental group (section)

== Related concepts ==

===Higher homotopy groups===
Roughly speaking, the fundamental group detects the 1-dimensional hole structure of a space, but not higher-dimensional holes such as for the 2-sphere. Such "higher-dimensional holes" can be detected using the higher [[homotopy group]]s <math>\pi_n(X)</math>, which are defined to consist of homotopy classes of (basepoint-preserving) maps from <math>S^n</math> to ''X''. For example, the [[Hurewicz theorem]] implies that for all <math>n \ge 1</math> the [[homotopy groups of spheres|''n''-th homotopy group of the ''n''-sphere]] is
:<math>\pi_n(S^n) = \Z.</math><ref>{{harvtxt|Hatcher|2002|loc=§4.1}}</ref> 
As was mentioned in the above computation of <math>\pi_1</math> of classical Lie groups, higher homotopy groups can be relevant even for computing fundamental groups.

===Loop space===
The set of based loops (as is, i.e. not taken up to homotopy) in a [[pointed space]] ''X'', endowed with the [[compact open topology]], is known as the [[loop space]], denoted <math>\Omega X.</math> The fundamental group of ''X'' is in [[bijection]] with the set of [[path component]]s of its loop space:<ref>{{harvtxt|Adams|1978|loc=p. 5}}</ref>
:<math>\pi_1(X) \cong \pi_0(\Omega X).</math>

===Fundamental groupoid===
The ''[[fundamental groupoid]]'' is a variant of the fundamental group that is useful in situations where the choice of a base point <math>x_0 \in X</math> is undesirable. It is defined by first considering the [[category (mathematics)|category]] of [[Moore path|path]]s in <math>X,</math> i.e., continuous functions
:<math>\gamma \colon [0, r] \to X</math>,
where ''r'' is an arbitrary non-negative real number. Since the length ''r'' is variable in this approach, such paths can be concatenated as is (i.e., not up to homotopy) and therefore yield a category.<ref>{{harvtxt|Brown|loc=§6.1|2006}}</ref> Two such paths <math>\gamma, \gamma'</math> with the same endpoints and length ''r'', resp. ''r''' are considered equivalent if there exist real numbers <math>u,v \geqslant 0</math> such that <math>r + u = r' + v</math> and <math> \gamma_u, \gamma'_v \colon [0, r + u] \to X</math> are homotopic relative to their end points, where <math> \gamma_u (t) = \begin{cases} \gamma(t), & t \in [0, r] \\ \gamma(r), & t \in [r, r + u]. \end{cases} </math><ref>{{harvtxt|Brown|2006|loc=§6.2}}</ref><ref>{{harvtxt|Crowell|Fox|1963}} use a different definition by reparametrizing the paths to length ''1''.</ref>

The category of paths up to this equivalence relation is denoted <math>\Pi (X).</math> Each morphism in <math>\Pi (X)</math> is an [[isomorphism]], with inverse given by the same path traversed in the opposite direction. Such a category is called a [[groupoid]]. It reproduces the fundamental group since
:<math>\pi_1(X, x_0) = \mathrm{Hom}_{\Pi(X)}(x_0, x_0)</math>.

More generally, one can consider the fundamental groupoid on a set ''A'' of base points, chosen according to the geometry of the situation; for example, in the case of the circle, which can be represented as the [[union (set theory)|union]] of two connected open sets whose intersection has two components, one can choose one base point in each component. The [[Seifert–van Kampen theorem|van Kampen theorem]] admits a version for fundamental groupoids which gives, for example, another way to compute the fundamental group(oid) of <math>S^1.</math><ref>{{harvtxt|Brown|2006|loc=§6.7}}</ref>

===Local systems===

Generally speaking, [[Group representation|representation]]s may serve to exhibit features of a group by its actions on other mathematical objects, often [[vector space]]s. Representations of the fundamental group have a very geometric significance: any ''[[local system]]'' (i.e., a [[sheaf (mathematics)|sheaf]] <math>\mathcal F</math> on ''X'' with the property that locally in a sufficiently small neighborhood ''U'' of any point on ''X'', the restriction of ''F'' is a [[constant sheaf]] of the form <math>\mathcal F|_U = \Q^n</math>) gives rise to the so-called [[monodromy representation]], a representation of the fundamental group on an ''n''-[[dimension (vector space)|dimensional]] <math>\Q</math>-vector space. [[Converse (logic)|Conversely]], any such representation on a path-connected space ''X'' arises in this manner.<ref>{{harvtxt|El Zein|Suciu|Tosun|Uludağ|2010|loc=p. 117, Prop. 1.7}}</ref> This [[equivalence of categories]] between representations of <math>\pi_1(X)</math> and local systems is used, for example, in the study of [[differential equation]]s, such as the [[Knizhnik–Zamolodchikov equations]].

===Étale fundamental group===

In [[algebraic geometry]], the so-called [[étale fundamental group]] is used as a replacement for the fundamental group.<ref>{{harvtxt|Grothendieck|Raynaud|2003}}.</ref> Since the [[Zariski topology]] on an [[algebraic variety]] or [[scheme (mathematics)|scheme]] ''X'' is much [[comparison of topologies|coarser]] than, say, the [[topological space|topology]] of open subsets in <math>\R^n,</math> it is no longer meaningful to consider continuous maps from an [[interval (mathematics)|interval]] to ''X''. Instead, the approach developed by [[Grothendieck]] consists in constructing <math>\pi_1^\text{et}</math> by considering all [[finite morphism|finite]] [[étale morphism|étale covers]] of ''X''. These serve as an algebro-geometric analogue of coverings with finite fibers.

This yields a theory applicable in situations where no great generality classical topological intuition whatsoever is available, for example for varieties defined over a [[finite field]]. Also, the étale fundamental group of a [[field (mathematics)|field]] is its ([[absolute Galois group|absolute]]) [[Galois group]]. On the other hand, for smooth varieties ''X'' over the complex numbers, the étale fundamental group retains much of the information inherent in the classical fundamental group: the former is the [[profinite completion]] of the latter.<ref>{{harvtxt|Grothendieck|Raynaud|2003|loc=Exposé XII, Cor. 5.2}}.</ref>

===Fundamental group of algebraic groups===
The fundamental group of a [[root system]] is defined in analogy to the computation for Lie groups.<ref>{{harvtxt|Humphreys|loc=§13.1|1972}}</ref> This allows to define and use the fundamental group of a semisimple [[linear algebraic group]] ''G'', which is a useful basic tool in the classification of linear algebraic groups.<ref>{{harvtxt|Humphreys|loc=§31.1|2004}}</ref>

===Fundamental group of simplicial sets===
The homotopy relation between 1-simplices of a [[simplicial set]] ''X'' is an equivalence relation if ''X'' is a [[Kan complex]] but not necessarily so in general.<ref>{{harvtxt|Goerss|Jardine|1999|loc=§I.7}}</ref> Thus, <math>\pi_1</math> of a Kan complex can be defined as the set of homotopy classes of 1-simplices. The fundamental group of an arbitrary simplicial set ''X'' are defined to be the homotopy group of its [[topological realization]], <math>|X|,</math> i.e., the topological space obtained by gluing topological simplices as prescribed by the simplicial set structure of ''X''.<ref>{{harvtxt|Goerss|Jardine|1999|loc=§I.11}}</ref>