Fundamental group: Difference between revisions

Throughout this article, <math>X</math> is a topological space. A typical example is a surface such as the one depicted at the right. Moreover, <math>x_0</math> is a point in <math>X</math> called the ''base-point''. (As is explained below, its role is rather auxiliary.) The idea of the definition of the homotopy group is to measure how many (broadly speaking) curves on <math>X</math> can be deformed into each other. The precise definition depends on the notion of the homotopy of loops, which is explained first.

Given a topological space <math>X</math>, a ''[[Loop (topology)|loop]] based at <math>x_0</math>'' is defined to be a [[continuous function (topology)|continuous function]] (also known as a continuous map)

[[File:Homotopy_of_pointed_circle_maps.png|Homotopy of loops. Black loops are interpolation loops at time <math>t</math>.|thumb]]

<math display=block>h \colon [0, 1] \times [0, 1] \to X,</math>

* <math>h(0, t) = x_0</math> for all <math>t \in [0, 1]</math>, that is, the starting point of the homotopy is <math>x_0</math> for all <math>t</math> (which is often thought of as a time parameter).  

* <math>h(1, t) = x_0</math> for all <math>t \in [0, 1]</math>, that is, similarly the end point stays at <math>x_0</math> for all <math>t</math>.

* <math>h(r, 0) = \gamma(r)</math>, <math>h(r, 1) = \gamma'(r)</math> for all <math>r \in [0, 1]</math>.

If such a homotopy <math>h</math> exists, <math>\gamma</math> and <math>\gamma'</math> are said to be ''homotopic''. The relation "<math>\gamma</math> is homotopic to <math>\gamma'</math>" is an [[equivalence relation]] so that the set of equivalence classes can be considered:

<math display=block>\pi_1(X, x_0) := \{ \text{all loops }\gamma \text{ based at }x_0 \} / \text{homotopy}. </math>

This set (with the group structure described below) is called the ''fundamental group'' of the topological space <math>X</math> at the base point <math>x_0</math>. The purpose of considering the equivalence classes of loops [[up to]] homotopy, as opposed to the set of all loops (the so-called [[loop space]] of <math>X</math>) is that the latter, while being useful for various purposes, is a rather big and unwieldy object. By contrast the above [[quotient set|quotient]] is, in many cases, more  manageable and computable.

The product of two homotopy classes of loops <math>[\gamma_0]</math> and <math>[\gamma_1]</math> is then defined as <math>[\gamma_0 \cdot \gamma_1]</math>. It can be shown that this product does not depend on the choice of representatives and therefore gives a [[Equivalence relation#Well-definedness under an equivalence relation|well-defined]] operation on the set <math>\pi_1(X, x_0)</math>. This operation turns <math>\pi_1(X, x_0)</math> into a group. Its [[neutral element]] is the equivalence (homotopy) class of the constant loop, which stays at <math>x_0</math> for all times <math>t</math> (i.e. this class consists of all loops that can be continuously deformed into the constant loop; intuitively speaking of all the loops that do not "wrap around a hole"). The [[inverse element|inverse]] of a (homotopy class of a) loop is the same loop, but traversed in the opposite direction (which is in a different homotopy class). More formally,

Although the fundamental group in general depends on the choice of base point, it turns out that, up to [[isomorphism]], this choice makes no difference as long as the space <math>X</math> is [[Connected space#Path connectedness|path-connected]]: more precisely, one obtains an isomorphism by pre- and post-concatenating with a path between the two basepoints. This isomorphism is, in general, not unique: it depends on the choice of path up to homotopy. However changing the path results in changing the isomorphism between the two fundamental groups only by composition with an [[inner automorphism]]. It is therefore customary to write <math>\pi_1(X)</math> instead of <math>\pi_1(X, x_0)</math> when the choice of basepoint does not matter.

is not simply connected. Instead, each homotopy class consists of all loops that wind around the circle a given number of times (which can be positive or negative, depending on the direction of winding). The product of a loop that winds around <math>m</math> times and another that winds around <math>n</math> times is a loop that winds around <math>m + n</math> times. Therefore, the fundamental group of the circle is [[group isomorphism|isomorphic]] to <math>(\Z, +),</math> the additive group of [[Integer#Algebraic properties|integers]]. This fact can be used to give proofs of the [[Brouwer fixed point theorem]]<ref>{{harvtxt|May|1999|loc=Ch. 1, §6}}</ref> and the [[Borsuk–Ulam theorem]] in dimension 2.<ref>{{harvtxt|Massey|1991|loc=Ch. V, §9}}</ref>

[[File:Marked 2-rose.svg|thumb|class=skin-invert-image|The fundamental group of the figure eight is the [[free group]] on two generators {{mvar|a}} and {{mvar|b}}.]]

<math display=block>\gamma = a^{n_1} b^{m_1} \cdots a^{n_k} b^{m_k}</math>

where {{mvar|a}} and {{mvar|b}} are the two loops winding around each half of the figure as depicted, and the exponents <math>n_1, \dots, n_k, m_1, \dots, m_k</math> are integers. Unlike <math>\pi_1(S^1)</math>, the fundamental group of the figure eight is ''not'' [[abelian group|abelian]]: the two ways of composing <math>a</math> and <math>b</math> are not homotopic to each other:

<math display=block>[a] \cdot [b] \ne [b] \cdot [a]</math>

More generally, the fundamental group of a [[Rose (topology)|bouquet of <math>r</math> circles]] is the free group on <math>r</math> letters.

The fundamental group of a [[wedge sum]] of two [[path connected space]]s <math>X</math> and <math>Y</math> can be computed as the [[free product]] of the individual fundamental groups:

<math display=block>\pi_1(X \vee Y) \cong \pi_1(X) * \pi_1(Y)</math>

The fundamental group of the plane punctured at <math>n</math> points is also the free group with <math>n</math> generators. The <math>i</math>-th generator is the class of the loop that goes around the <math>i</math>-th puncture without going around any other punctures.

The fundamental group can be defined for discrete structures too. In particular, consider a [[Connectivity (graph theory)|connected]] [[graph (discrete mathematics)|graph]] <math>G = (V, E)</math>, with a designated vertex <math>v_0</math> in <math>V</math>. The loops in <math>G</math> are the [[cycle (graph theory)|cycles]] that start and end at <math>v_0</math>.<ref>{{Cite web|title=Meaning of Fundamental group of a graph|url=https://math.stackexchange.com/questions/515896/meaning-of-fundamental-group-of-a-graph|access-date=2020-07-28|website=Mathematics Stack Exchange}}</ref> Let <math>T</math> be a [[spanning tree]] of <math>G</math>. Every simple loop in <math>G</math> contains exactly one edge in <math>E \setminus T</math>; every loop in <math>G</math> is a concatenation of such simple loops. Therefore, the fundamental group of a [[graph (discrete mathematics)|graph]] is a [[free group]], in which the number of generators is exactly the number of edges in <math>E \setminus T</math>. This number equals <math>|E| - |V| + 1</math>.<ref>{{Cite web|last=Simon|first=J|date=2008|title=Example of calculating the fundamental group of a graph G|url=http://homepage.divms.uiowa.edu/~jsimon/COURSES/M201Fall08/HandoutsAndHomework/Graph1.pdf|archive-url=https://web.archive.org/web/20200728164140/http://homepage.divms.uiowa.edu/~jsimon/COURSES/M201Fall08/HandoutsAndHomework/Graph1.pdf|archive-date=2020-07-28|url-status=dead|access-date=2020-07-28}}</ref>

For example, suppose <math>G</math> has 16 vertices arranged in 4 rows of 4 vertices each, with edges connecting vertices that are adjacent horizontally or vertically. Then <math>G</math> has 24 edges overall, and the number of edges in each spanning tree is {{nowrap|16 − 1 {{=}} 15}}, so the fundamental group of <math>G</math> is the free group with 9 generators.<ref>{{Cite web|title=The Fundamental Groups of Connected Graphs - Mathonline|url=http://mathonline.wikidot.com/the-fundamental-groups-of-connected-graphs|access-date=2020-07-28|website=mathonline.wikidot.com}}</ref> Note that <math>G</math> has 9 "holes", similarly to a bouquet of 9 circles, which has the same fundamental group.

[[File:Trefoil_knot_left.svg|thumb|class=skin-invert-image|A [[trefoil knot]]]]

''[[Knot group]]s'' are by definition the fundamental group of the [[complement (set theory)|complement]] of a [[knot (mathematics)|knot]] <math>K</math> embedded in <math>\R^3</math>. For example, the knot group of the [[trefoil knot]] is known to be the [[braid group]] <math>B_3</math>, which gives another example of a non-abelian fundamental group. The [[Wirtinger presentation]] explicitly describes knot groups in terms of [[generators and relations]] based on a diagram of the knot. Therefore, knot groups have some usage in [[knot theory]] to distinguish between knots: if <math>\pi_1(\R^3 \setminus K)</math> is not isomorphic to some other knot group <math>\pi_1(\R^3 \setminus K')</math> of another knot <math>K'</math>, then <math>K</math> cannot be transformed into <math>K'</math>. Thus the trefoil knot cannot be continuously transformed into the circle (also known as the [[unknot]]), since the latter has knot group <math>\Z</math>. There are, however, knots that cannot be deformed into each other, but have isomorphic knot groups.

<math display=block>\left\langle A_1, B_1, \ldots, A_n, B_n \left| A_1 B_1 A_1^{-1} B_1^{-1} \cdots A_n B_n A_n^{-1} B_n^{-1} \right. \right\rangle.</math>

<math display=block>\left\langle A_1, B_1 \left| A_1 B_1 A_1^{-1} B_1^{-1} \right. \right\rangle \cong \Z^2.</math>

The fundamental group of a [[topological group]] <math>X</math> (with respect to the base point being the neutral element) is always commutative. In particular, the fundamental group of a [[Lie group]] is commutative. In fact, the group structure on <math>X</math> endows <math>\pi_1(X)</math> with another group structure: given two loops <math>\gamma</math> and <math>\gamma'</math> in <math>X</math>, another loop <math>\gamma \star \gamma'</math> can defined by using the group multiplication in <math>X</math>:  

An inspection of the proof shows that, more generally, <math>\pi_1(X)</math> is abelian for any [[H-space]] <math>X</math>, i.e., the multiplication need not have an inverse, nor does it have to be associative. For example, this shows that the fundamental group of a [[loop space]] of another topological space <math>Y</math>, <math>X = \Omega(Y),</math> is abelian. Related ideas lead to [[Heinz Hopf]]'s computation of the [[Hopf algebra#Cohomology of Lie groups|cohomology of a Lie group]].

from the [[category of pointed spaces|category of topological spaces together with a base point]] to the [[category of groups]]. It turns out that this functor does not distinguish maps that are [[homotopic]] relative to the base point: if <math>f,g:X\to Y</math> are continuous maps with <math>f(x_0) = g(x_0) = y_0</math>, and <math>f</math> and <math>g</math> are homotopic relative to <math>\{x_0\}</math>, then <math>f_* = g_*</math>. As a consequence, two [[homotopy equivalent]] path-connected spaces have isomorphic fundamental groups:

The fundamental group functor takes [[product topology|products]] to [[direct product|products]] and [[wedge sum|coproducts]] to [[free product of groups|coproducts]]. That is, if <math>X</math> and <math>Y</math> are path connected, then

from the fundamental group of a topological space <math>X</math> to its first singular homology group <math>H_1(X).</math> This homomorphism is not in general an isomorphism since the fundamental group may be non-abelian, but the homology group is, by definition, always abelian. This difference is, however, the only one: if <math>X</math> is path-connected, this homomorphism is [[surjective]] and its [[Kernel (algebra)|kernel]] is the [[commutator subgroup]] of the fundamental group, so that <math>H_1(X)</math> is isomorphic to the [[abelianization]] of the fundamental group.<ref>{{harvtxt|Fulton|1995|loc=Prop. 12.22}}</ref>

[[File:Covering_map.svg|thumb|The map <math>\mathbb{R} \times [0,1] \to S^1 \times [0,1]</math> is a covering: the preimage of <math>U</math> (highlighted in gray) is a disjoint union of copies of <math>U</math>. Moreover, it is a universal covering since <math>\mathbb{R} \times [0,1]</math> is contractible and therefore simply connected.]]

Given a topological space <math>B</math>, a [[continuous function (topology)|continuous map]]  

is called a ''covering'' or <math>E</math> is called a ''[[covering space]]'' of <math>B</math> if every point <math>b</math> in <math>B</math> admits an [[open neighborhood]] <math>U</math> such that there is a [[homeomorphism]] between the [[preimage]] of <math>U</math> and a [[disjoint union]] of copies of <math>U</math> (indexed by some set <math>I</math>),

in such a way that <math>f \circ \varphi</math> is the standard projection map <math>\bigsqcup_{i \in I} U \to U.</math><ref>{{harvtxt|Hatcher|2002|loc=§1.3}}</ref>

A covering is called a [[universal covering]] if <math>E</math> is, in addition to the preceding condition, simply connected.<ref>{{harvtxt|Hatcher|2002|loc=p. 65}}</ref> It is universal in the sense that all other coverings can be constructed by suitably identifying points in <math>E</math>. Knowing a universal covering  

of a topological space <math>X</math> is helpful in understanding its fundamental group in several ways: first, <math>\pi_1(X)</math> identifies with the group of [[deck transformations]], i.e., the group of [[homeomorphism]]s <math>\varphi : \widetilde{X} \to \widetilde{X}</math> that commute with the map to <math>X</math>, i.e., <math>p \circ \varphi = p.</math>

Any path connected, [[Locally_connected_space#Definitions|locally path connected]] and [[locally simply connected]] topological space <math>X</math> admits a universal covering.<ref>{{harvtxt|Hatcher|2002|loc=Proposition 1.36}}</ref> An abstract construction proceeds analogously to the fundamental group by taking pairs <math>(x,\gamma)</math>, where <math>x</math> is a point in <math>X</math> and <math>\gamma</math> is a homotopy class of paths from <math>x_0</math> to <math>x</math>. The passage from a topological space to its universal covering can be used in understanding the geometry of <math>X</math>. For example, the [[uniformization theorem]] shows that any simply connected [[Riemann surface]] is (isomorphic to) either <math>S^2,</math> <math>\mathbb{C},</math> or the [[upper half-plane]].<ref>{{harvtxt|Forster|1981|loc=Theorem 27.9}}</ref> General Riemann surfaces then arise as quotients of [[group action]]s on these three surfaces.

The [[quotient topology|quotient]] of a [[Group action#Remarkable properties of actions|free action]] of a [[discrete topology|discrete]] group <math>G</math> on a simply connected space <math>Y</math> has fundamental group  

As an example, the real <math>n</math>-dimensional real [[projective space]] <math>\mathbb{R}\mathrm{P}^n</math> is obtained as the quotient of the <math>n</math>-dimensional unit sphere <math>S^n</math> by the antipodal action of the group <math>\mathbb{Z}/2</math> sending <math>x \in S^n</math> to <math>-x.</math> As <math>S^n</math> is simply connected for <math>n \geq 2</math>, it is a universal cover of <math>\mathbb{R}\mathrm{P}^n</math> in these cases, which implies <math>\pi_1(\mathbb{R}\mathrm{P}^n) \cong \mathbb{Z}/2</math> for <math>n \geq 2</math>.

Let <math>G</math> be a connected, simply connected [[compact Lie group]], for example, the [[special unitary group]] <math>\text{SU}(n)</math>, and let <math>\Gamma</math> be a finite subgroup of <math>G</math>. Then the [[homogeneous space]] <math>X=G/\Gamma</math> has fundamental group <math>\Gamma</math>, which acts by right multiplication on the universal covering space <math>G</math>. Among the many variants of this construction, one of the most important is given by [[locally symmetric space]]s <math>X = \Gamma \setminus G / K</math>, where

*<math>G</math> is a non-compact simply connected, connected [[Lie group]] (often [[semisimple Lie group|semisimple]]),

*<math>K</math> is a maximal compact subgroup of <math>G</math>

* <math>\Gamma</math> is a discrete [[countable set|countable]] [[torsion-free group|torsion-free]] subgroup of <math>G</math>.

In this case the fundamental group is <math>\Gamma</math> and the universal covering space <math>G/K</math> is actually [[contractible]] (by the [[Cartan decomposition]] for Lie groups).

As an example take <math>G = \text{SL}(2,\R)</math>, <math>K = \text{SO}(2)</math> and <math>\Gamma</math> any torsion-free [[congruence subgroup]] of the [[modular group]] <math>\text{SL}(2,\Z)</math>.

From the explicit realization, it also follows that the universal covering space of a path connected [[topological group]] <math>H</math> is again a path connected topological group <math>G</math>.  Moreover, the covering map is a continuous [[open map|open]] homomorphism of <math>G</math> onto <math>H</math> with kernel <math>\Gamma</math>, a closed discrete [[normal subgroup]] of <math>G</math>:

Since <math>G</math> is a connected group with a continuous action by conjugation on a discrete group <math>\Gamma</math>, it must act trivially, so that <math>\Gamma</math> has to be a subgroup of the [[center (group theory)|center]] of <math>G</math>.  In particular <math>\pi_1(H) = \Gamma</math> is an [[abelian group]]; this can also easily be seen directly without using covering spaces.  The group <math>G</math> is called the ''[[universal covering group]]'' of <math>H</math>.

''[[Fibrations]]'' provide a very powerful means to compute homotopy groups. A fibration <math>f</math> the so-called ''total space'', and the base space <math>B</math> has, in particular, the property that all its fibers <math>f^{-1}(b)</math> are homotopy equivalent and therefore can not be distinguished using fundamental groups (and higher homotopy groups), provided that <math>B</math> is path-connected.<ref>{{harvtxt|Hatcher|2002|loc=Prop. 4.61}}</ref> Therefore, the space <math>E</math> can be regarded as a "[[Dehn twist|twisted]] product" of the [[fibration|base space]] <math>B</math> and the [[Fiber (algebraic geometry)|fiber]] <math>F = f^{-1}(b).</math> The great importance of fibrations to the computation of homotopy groups stems from a [[Homotopy group#Long exact sequence of a fibration|long exact sequence]]

provided that <math>B</math> is path-connected.<ref>{{harvtxt|Hatcher|2002|loc=Theorem 4.41}}</ref> The term <math>\pi_2(B)</math> is the second [[homotopy group]] of <math>B</math>, which is defined to be the set of homotopy classes of maps from <math>S^2</math> to <math>B</math>, in direct analogy with the definition of <math>\pi_1.</math>

If <math>E</math> happens to be path-connected and simply connected, this sequence reduces to an isomorphism

which generalizes the above fact about the universal covering (which amounts to the case where the fiber <math>F</math> is also discrete). If instead <math>F</math> happens to be connected and simply connected, it reduces to an isomorphism

can be shown to be surjective<ref>{{harvtxt|Bump|2013|loc=Prop. 23.7}}</ref> with kernel given by the set <math>I</math> of integer linear combination of [[coroot]]s. This leads to the computation  

If <math>X</math> is a [[connected space|connected]] simplicial complex, an ''edge-path'' in <math>X</math> is defined to be a chain of vertices connected by edges in <math>X</math>.  Two edge-paths are said to be ''edge-equivalent'' if one can be obtained from the other by successively switching between an edge and the two opposite edges of a triangle in <math>X</math>.  If <math>v</math> is a fixed vertex in <math>X</math>, an ''edge-loop'' at <math>v</math> is an edge-path starting and ending at <math>v</math>.  The '''edge-path group''' <math>E(X,v)</math> is defined to be the set of edge-equivalence classes of edge-loops at <math>v</math>, with product and inverse defined by concatenation and reversal of edge-loops.

The edge-path group is naturally isomorphic to <math>\pi_1(|X|,v)</math>, the fundamental group of the [[Simplicial set|geometric realisation]] <math>|X|</math> of <math>X</math>.<ref>{{cite book|last1=Singer|first1=Isadore|author-link1=Isadore Singer|last2=Thorpe|first2=John A.|title=Lecture notes on elementary topology and geometry|url=https://archive.org/details/lecturenotesonel00sing_949|url-access=limited|date=1967|publisher=Springer-Verlag|isbn=0-387-90202-3|page=[https://archive.org/details/lecturenotesonel00sing_949/page/n101 98]}}</ref>  Since it depends only on the [[n-skeleton|2-skeleton]] <math>X^2</math> of <math>X</math> (that is, the vertices, edges, and triangles of <math>X</math>), the groups <math>\pi_1(|X|,v)</math> and <math>\pi_1(|X^2|,v)</math> are isomorphic.

The edge-path group can be described explicitly in terms of [[generators and relations]].  If <math>T</math> is a [[spanning tree|maximal spanning tree]] in the [[n-skeleton|1-skeleton]] of <math>X</math>, then <math>E(X,v)</math> is canonically isomorphic to the group with generators (the oriented edge-paths of <math>X</math> not occurring in <math>T</math>) and relations (the edge-equivalences corresponding to triangles in <math>X</math>).  A similar result holds if <math>T</math> is replaced by any [[simply connected]]&mdash;in particular [[contractible]]&mdash;subcomplex of <math>X</math>. This often gives a practical way of computing fundamental groups and can be used to show that every [[finitely presented group]] arises as the fundamental group of a finite simplicial complex.  It is also one of the classical methods used for [[Surface (topology)|topological surfaces]], which are classified by their fundamental groups.

The ''universal covering space'' of a finite connected simplicial complex <math>X</math> can also be described directly as a simplicial complex using edge-paths.  Its vertices are pairs <math>(w,\gamma)</math> where <math>w</math> is a vertex of <math>X</math> and γ is an edge-equivalence class of paths from <math>v</math> to <math>w</math>.  The <math>k</math>-simplices containing <math>(w,\gamma)</math> correspond naturally to the <math>k</math>-simplices containing <math>w</math>.  Each new vertex <math>u</math> of the <math>k</math>-simplex gives an edge <math>w u</math> and hence, by concatenation, a new path <math>\gamma_u</math> from <math>v</math> to <math>u</math>.  The points <math>(w,\gamma)</math> and <math>(u,\gamma_u)</math> are the vertices of the "transported" simplex in the universal covering space.  The edge-path group acts naturally by concatenation, preserving the simplicial structure, and the quotient space is just <math>X</math>.

It is well known that this method can also be used to compute the fundamental group of an arbitrary topological space.  This was doubtless known to [[Eduard Čech]] and [[Jean Leray]] and explicitly appeared as a remark in a paper by [[André Weil]];<ref>[[André Weil]], ''On discrete subgroups of Lie groups'', [[Annals of Mathematics]] '''72''' (1960), 369-384.</ref> various other authors such as Lorenzo Calabi, [[Wu Wenjun|Wu Wen-tsün]], and Nodar  Berikashvili have also published proofs.  In the simplest case of a compact space <math>X</math> with a finite open covering in which all [[empty set|non-empty]] finite [[intersection (set theory)|intersections]] of open sets in the covering are contractible, the fundamental group can be identified with the edge-path group of the simplicial complex corresponding to the [[Nerve of an open covering|nerve of the covering]].

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 <math>X</math>. For example, the [[Hurewicz theorem]] implies that for all <math>n \ge 1</math> the [[homotopy groups of spheres|<math>n</math>-th homotopy group of the ''n''-sphere]] is

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

where <math>r</math> is an arbitrary non-negative real number. Since the length <math>r</math> 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 <math>r</math>, resp. <math>r</math>' 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>

More generally, one can consider the fundamental groupoid on a set <math>A</math> 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>

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 <math>X</math> with the property that locally in a sufficiently small neighborhood <math>U</math> of any point on <math>X</math>, the restriction of <math>F</math> 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 <math>n</math>-[[dimension (vector space)|dimensional]] <math>\Q</math>-vector space. [[Converse (logic)|Conversely]], any such representation on a path-connected space <math>X</math> 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]].

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]] <math>X</math> 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 <math>X</math>. 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 <math>X</math>. 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 <math>X</math> 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>

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]] <math>G</math>, which is a useful basic tool in the classification of linear algebraic groups.<ref>{{harvtxt|Humphreys|loc=§31.1|2004}}</ref>

The homotopy relation between 1-simplices of a [[simplicial set]] <math>X</math> is an equivalence relation if <math>X</math> 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 <math>X</math> 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 <math>X</math>.<ref>{{harvtxt|Goerss|Jardine|1999|loc=§I.11}}</ref>

* {{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | last2=Raynaud | first2=Michèle |author-link2=Michèle Raynaud | title=Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques '''3''') | orig-year=1971 | arxiv=math.AG/0206203 | publisher=[[Société Mathématique de France]] | location=Paris | isbn=978-2-85629-141-2 | year=2003 | pages=xviii+327, see Exp. V, IX, X}}

* {{Citation|isbn=0-521-79540-0|title=Algebraic Topology|last1=Hatcher|first1=Allen|author-link=Allen Hatcher|year=2002|url=https://pi.math.cornell.edu/~hatcher/AT/ATpage.html|publisher=Cambridge University Press}}