Editing Fundamental group (section)

== Concrete examples ==
[[File:Star_domain.svg|thumb|left|A star domain is simply connected since any loop can be contracted to the center of the domain, denoted <math>x_0</math>.]]
This section lists some basic examples of fundamental groups. To begin with, in [[Euclidean space]] (<math>\R^n</math>) or any [[convex set|convex subset]] of <math>\R^n,</math> there is only one homotopy class of loops, and the fundamental group is therefore the [[trivial group]] with one element.  More generally, any [[star domain]] – and yet more generally, any [[contractible space]] – has a trivial fundamental group. Thus, the fundamental group does not distinguish between such spaces.

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=== The 2-sphere ===
[[File:P1S2all.jpg|upright=1.8|thumb|A loop on a [[2-sphere]] (the surface of a ball) being contracted to a point]]
A path-connected space whose fundamental group is trivial is called [[simply connected space|simply connected]]. 
For example, the [[2-sphere]] <math>S^2 = \left\{(x, y, z) \in \R^3 \mid x^2 + y^2 + z^2 = 1\right\}</math> depicted on the right, and also all the [[n-sphere|higher-dimensional spheres]], are simply-connected. The figure illustrates a homotopy contracting one particular loop to the constant loop. This idea can be adapted to all loops <math>\gamma</math> such that there is a point <math>(x, y, z) \in S^2</math> that is {{em|not}} in the image of <math>\gamma.</math> However, since there are loops such that <math>\gamma([0, 1]) = S^2</math> (constructed from the [[Peano curve]], for example), a complete [[mathematical proof|proof]] requires more careful analysis with tools from algebraic topology, such as the [[Seifert–van Kampen theorem]] or the [[cellular approximation theorem]].

===The circle===
[[File:Fundamental_group_of_the_circle.svg|Elements of the homotopy group of the circle|thumb]]
The [[circle]] (also known as the 1-sphere)
:<math>S^1 = \left\{(x, y) \in \R^2 \mid x^2 + y^2 = 1\right\}</math>
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 ''m'' times and another that winds around ''n'' times is a loop that winds around ''m'' + ''n'' 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>

===The figure eight===
[[File:Wedge_of_Two_Circles.png|thumb|The fundamental group of the figure eight is the [[free group]] on two generators ''a'' and ''b''.]]
The fundamental group of the [[Rose (topology)|figure eight]] is the [[free group]] on two letters. The idea to prove this is as follows: choosing the base point to be the point where the two circles meet (dotted in black in the picture at the right), any loop <math>\gamma</math> can be decomposed as
:<math>\gamma = a^{n_1} b^{m_1} \cdots a^{n_k} b^{m_k}</math>
where ''a'' and ''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 ''a'' and ''b'' are not homotopic to each other:
:<math>[a] \cdot [b] \ne [b] \cdot [a].</math>

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

The fundamental group of a [[wedge sum]] of two [[path connected space]]s ''X'' and ''Y'' can be computed as the [[free product]] of the individual fundamental groups:
:<math>\pi_1(X \vee Y) \cong \pi_1(X) * \pi_1(Y).</math>
This generalizes the above observations since the figure eight is the wedge sum of two circles.

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

=== Graphs ===

The fundamental group can be defined for discrete structures too. In particular, consider a [[Connectivity (graph theory)|connected]] [[graph (discrete mathematics)|graph]] {{nowrap|''G'' {{=}} (''V'', ''E'')}}, with a designated vertex ''v''<sub>0</sub> in ''V''. The loops in ''G'' are the [[cycle (graph theory)|cycles]] that start and end at ''v''<sub>0</sub>.<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 ''T'' be a [[spanning tree]] of ''G''. Every simple loop in ''G'' contains exactly one edge in ''E'' \ ''T''; every loop in ''G'' 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 ''E'' \ ''T''. This number equals {{nowrap|{{!}}''E''{{!}} − {{!}}''V''{{!}} + 1}}.<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 ''G'' has 16 vertices arranged in 4 rows of 4 vertices each, with edges connecting vertices that are adjacent horizontally or vertically. Then ''G'' has 24 edges overall, and the number of edges in each spanning tree is {{nowrap|16 − 1 {{=}} 15}}, so the fundamental group of ''G'' 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 ''G'' has 9 "holes", similarly to a bouquet of 9 circles, which has the same fundamental group.

===Knot groups===
[[File:Trefoil_knot_left.svg|thumb|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> can not be transformed into <math>K'</math>. Thus the trefoil knot can not 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 can not be deformed into each other, but have isomorphic knot groups.

===Oriented surfaces===
The fundamental group of a [[genus (mathematics)#Orientable surface|genus-''n'' orientable surface]] can be computed in terms of [[generators and relations]] as
:<math>\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>

This includes the [[torus (mathematics)|torus]], being the case of genus 1, whose fundamental group is
:<math>\left\langle A_1, B_1 \left| A_1 B_1 A_1^{-1} B_1^{-1} \right. \right\rangle \cong \Z^2.</math>

===Topological groups===

The fundamental group of a [[topological group]] ''X'' (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 ''X'' endows <math>\pi_1(X)</math> with another group structure: given two loops <math>\gamma</math> and <math>\gamma'</math> in ''X'', another loop <math>\gamma \star \gamma'</math> can defined by using the group multiplication in ''X'': 
:<math>(\gamma \star \gamma')(x) = \gamma(x) \cdot \gamma'(x).</math>
This binary operation <math>\star</math> on the set of all loops is ''a priori'' independent from the one described above. However, the [[Eckmann–Hilton argument]] shows that it does in fact agree with the above concatenation of loops, and moreover that the resulting group structure is abelian.<ref>{{harvtxt|Strom|2011|loc=Problem 9.30, 9.31}}, {{harvtxt|Hall|2015|loc=Exercise 13.7}}</ref><ref>Proof: Given two loops <math>\alpha, \beta: [0, 1] \to  G</math> in <math>\pi_1(G),</math> define the mapping <math>A\colon [0, 1] \times [0, 1] \to G</math> by <math>A(s, t) = \alpha(s)\cdot\beta(t),</math> multiplied pointwise in <math>G.</math> Consider the homotopy family of paths in the rectangle from <math>(s, t) = (0, 0)</math> to <math>(1, 1)</math> that starts with the horizontal-then-vertical path, moves through various diagonal paths, and ends with the vertical-then-horizontal path. Composing this family with <math>A</math> gives a homotopy <math>\alpha * \beta \sim \beta * \alpha,</math> which shows the fundamental group is abelian.</ref>

An inspection of the proof shows that, more generally, <math>\pi_1(X)</math> is abelian for any [[H-space]] ''X'', 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 ''Y'', <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]].