TheInfoList

OR:

In the mathematical field of
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
, the fundamental group of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
is the group of the equivalence classes under
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
fundamental groups. The fundamental group of a topological space $X$ is denoted by $\pi_1\left(X\right)$.

# Intuition

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
), and some point in it, and all the loops both starting and ending at this point— paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breaking. The set of all such loops with this method of combining and this equivalence between them is the fundamental group for that particular space.

# History

Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
defined the fundamental group in 1895 in his paper " Analysis situs". The concept emerged in the theory of
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
s, in the work of
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
, Poincaré, and
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
. It describes the monodromy properties of
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
-valued functions, as well as providing a complete topological classification of closed surfaces.

# Definition Throughout this article, ''X'' is a topological space. A typical example is a surface such as the one depicted at the right. Moreover, $x_0$ is a point in ''X'' 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 ''X'' can be deformed into each other. The precise definition depends on the notion of the homotopy of loops, which is explained first.

## Homotopy of loops

Given a topological space ''X'', a '' loop based at $x_0$'' is defined to be a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
(also known as a continuous map) : such that the starting point $\gamma\left(0\right)$ and the end point $\gamma\left(1\right)$ are both equal to $x_0$. A ''
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
'' is a continuous interpolation between two loops. More precisely, a homotopy between two loops (based at the same point $x_0$) is a continuous map : such that * $h\left(0, t\right) = x_0$ for all that is, the starting point of the homotopy is $x_0$ for all ''t'' (which is often thought of as a time parameter). * $h\left(1, t\right) = x_0$ for all that is, similarly the end point stays at $x_0$ for all ''t''. * $h\left(r, 0\right) = \gamma\left(r\right),\, h\left(r, 1\right) = \gamma\text{'}\left(r\right)$ for all
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
so that the set of equivalence classes can be considered: :$\pi_1\left(X, x_0\right) := \ / \text$. This set (with the group structure described below) is called the ''fundamental group'' of the topological space ''X'' at the base point $x_0$. The purpose of considering the equivalence classes of loops
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
homotopy, as opposed to the set of all loops (the so-called loop space of ''X'') is that the latter, while being useful for various purposes, is a rather big and unwieldy object. By contrast the above
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
is, in many cases, more manageable and computable.

## Group structure By the above definition, $\pi_1\left(X, x_0\right)$ is just a set. It becomes a group (and therefore deserves the name fundamental ''group'') using the concatenation of loops. More precisely, given two loops $\gamma_0, \gamma_1$, their product is defined as the loop : :$\left(\gamma_0 \cdot \gamma_1\right)\left(t\right) = \begin \gamma_0\left(2t\right) & 0 \leq t \leq \tfrac \\ \gamma_1\left(2t - 1\right) & \tfrac \leq t \leq 1. \end$ Thus the loop $\gamma_0 \cdot \gamma_1$ first follows the loop $\gamma_0$ with "twice the speed" and then follows $\gamma_1$ with "twice the speed". The product of two homotopy classes of loops
well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
operation on the set $\pi_1\left(X, x_0\right)$. This operation turns $\pi_1\left(X, x_0\right)$ into a group. Its
neutral element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
is the constant loop, which stays at $x_0$ for all times ''t''. The inverse of a (homotopy class of a) loop is the same loop, but traversed in the opposite direction. More formally, :$\gamma^\left(t\right) := \gamma\left(1-t\right)$. Given three based loops $\gamma_0, \gamma_1, \gamma_2,$ the product :$\left(\gamma_0 \cdot \gamma_1\right) \cdot \gamma_2$ is the concatenation of these loops, traversing $\gamma_0$ and then $\gamma_1$ with quadruple speed, and then $\gamma_2$ with double speed. By comparison, :$\gamma_0 \cdot \left(\gamma_1 \cdot \gamma_2\right)$ traverses the same paths (in the same order), but $\gamma_0$ with double speed, and $\gamma_1, \gamma_2$ with quadruple speed. Thus, because of the differing speeds, the two paths are not identical. The
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
axiom :

## Dependence of the base point

Although the fundamental group in general depends on the choice of base point, it turns out that, up to
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
(actually, even up to ''inner'' isomorphism), this choice makes no difference as long as the space ''X'' is path-connected. For path-connected spaces, therefore, many authors write $\pi_1\left(X\right)$ instead of $\pi_1\left(X, x_0\right).$

# Concrete examples This section lists some basic examples of fundamental groups. To begin with, in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
($\R^n$) or any convex subset of $\R^n,$ 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 In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within tha ...
– has a trivial fundamental group. Thus, the fundamental group does not distinguish between such spaces.

## The 2-sphere A path-connected space whose fundamental group is trivial is called simply connected. For example, the
2-sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
$S^2 = \left\$ depicted on the right, and also all the 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 $\gamma$ such that there is a point $\left(x, y, z\right) \in S^2$ that is in the image of $\gamma.$ However, since there are loops such that (constructed from the Peano curve, for example), a complete proof requires more careful analysis with tools from algebraic topology, such as the Seifert–van Kampen theorem or the cellular approximation theorem.

## The circle The
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
(also known as the 1-sphere) :$S^1 = \left\$ 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
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to $\left(\Z, +\right),$ the additive group of integers. This fact can be used to give proofs of the
Brouwer fixed point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simples ...
and the Borsuk–Ulam theorem in dimension 2.

## The figure eight The fundamental group of the figure eight is the
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
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 $\gamma$ can be decomposed as :$\gamma = a^ b^ \cdots a^ b^$ where ''a'' and ''b'' are the two loops winding around each half of the figure as depicted, and the exponents $n_1, \dots, n_k, m_1, \dots, m_k$ are integers. Unlike $\pi_1\left(S^1\right),$ the fundamental group of the figure eight is ''not'' abelian: the two ways of composing ''a'' and ''b'' are not homotopic to each other: : More generally, the fundamental group of a bouquet of ''r'' circles is the free group on ''r'' letters. The fundamental group of a
wedge sum In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the qu ...
of two path connected spaces ''X'' and ''Y'' can be computed as the
free product In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and i ...
of the individual fundamental groups: :$\pi_1\left(X \vee Y\right) \cong \pi_1\left(X\right) * \pi_1\left(Y\right).$ 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
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
graph , with a designated vertex ''v''0 in ''V''. The loops in ''G'' are the cycles that start and end at ''v''0. Let ''T'' be a
spanning tree In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is not ...
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 is a
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
, in which the number of generators is exactly the number of edges in ''E'' \ ''T''. This number equals . 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 , so the fundamental group of ''G'' is the free group with 9 generators. Note that ''G'' has 9 "holes", similarly to a bouquet of 9 circles, which has the same fundamental group.

## Knot groups '' Knot groups'' are by definition the fundamental group of the complement of a
knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ...
''K'' embedded in $\R^3.$ For example, the knot group of the trefoil knot is known to be the braid group $B_3,$ 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 In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...
to distinguish between knots: if $\pi_1\left(\R^3 \setminus K\right)$ is not isomorphic to some other knot group $\pi_1\left(\R^3 \setminus K\text{'}\right)$ of another knot ''K′'', then ''K'' can not be transformed into ''K′''. Thus the trefoil knot can not be continuously transformed into the circle (also known as the unknot), since the latter has knot group $\Z$. 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-''n'' orientable surface can be computed in terms of generators and relations as :$\left\langle A_1, B_1, \ldots, A_n, B_n \left, A_1 B_1 A_1^ B_1^ \cdots A_n B_n A_n^ B_n^ \right. \right\rangle.$ This includes the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not t ...
, being the case of genus 1, whose fundamental group is :$\left\langle A_1, B_1 \left, A_1 B_1 A_1^ B_1^ \right. \right\rangle \cong \Z^2.$

## Topological groups

The fundamental group of a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
''X'' (with respect to the base point being the neutral element) is always commutative. In particular, the fundamental group of a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
is commutative. In fact, the group structure on ''X'' endows $\pi_1\left(X\right)$ with another group structure: given two loops $\gamma$ and $\gamma\text{'}$ in ''X'', another loop $\gamma \star \gamma\text{'}$ can defined by using the group multiplication in ''X'': :$\left(\gamma \star \gamma\text{'}\right)\left(x\right) = \gamma\left(x\right) \cdot \gamma\text{'}\left(x\right).$ This binary operation $\star$ 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. An inspection of the proof shows that, more generally, $\pi_1\left(X\right)$ is abelian for any
H-space In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed. Definition An H-space consists of a topological space , together wit ...
''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'', $X = \Omega\left(Y\right),$ is abelian. Related ideas lead to Heinz Hopf's computation of the cohomology of a Lie group.

# Functoriality

If $f\colon X \to Y$ is a continuous map, $x_0 \in X$ and $y_0 \in Y$ with $f\left(x_0\right) = y_0,$ then every loop in ''X'' with base point $x_0$ can be composed with ''f'' to yield a loop in ''Y'' with base point $y_0.$ This operation is compatible with the homotopy equivalence relation and with composition of loops. The resulting
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) ...
, called the induced homomorphism, is written as $\pi\left(f\right)$ or, more commonly, :$f_* \colon \pi_1\left(X, x_0\right) \to \pi_1\left(Y, y_0\right).$ This mapping from continuous maps to group homomorphisms is compatible with composition of maps and identity morphisms. In the parlance of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, the formation of associating to a topological space its fundamental group is therefore a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
:$\begin \pi_1 \colon \mathbf_* &\to \mathbf \\ \left(X, x_0\right) &\mapsto \pi_1\left(X, x_0\right) \end$ from the 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 ''f'', ''g'' : ''X'' → ''Y'' are continuous maps with ''f''(''x''0) = ''g''(''x''0) = ''y''0, and ''f'' and ''g'' are homotopic relative to , then ''f'' = ''g''. As a consequence, two homotopy equivalent path-connected spaces have isomorphic fundamental groups: :$X \simeq Y \implies \pi_1\left(X, x_0\right) \cong \pi_1\left(Y, y_0\right).$ For example, the inclusion of the circle in the punctured plane :$S^1 \subset \mathbb^2 \setminus \$ is a homotopy equivalence and therefore yields an isomorphism of their fundamental groups. The fundamental group functor takes products to products and coproducts to coproducts. That is, if ''X'' and ''Y'' are path connected, then :$\pi_1 \left(X \times Y, \left(x_0, y_0\right)\right) \cong \pi_1\left(X, x_0\right) \times \pi_1\left(Y, y_0\right)$ and if they are also locally contractible, then :$\pi_1\left(X \vee Y\right) \cong \pi_1\left(X\right)*\pi_1\left(Y\right).$ (In the latter formula, $\vee$ denotes the
wedge sum In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the qu ...
of pointed topological spaces, and $*$ the
free product In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and i ...
of groups.) The latter formula is a special case of the Seifert–van Kampen theorem, which states that the fundamental group functor takes pushouts along inclusions to pushouts.

# Abstract results

As was mentioned above, computing the fundamental group of even relatively simple topological spaces tends to be not entirely trivial, but requires some methods of
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
.

## Relationship to first homology group

The
abelianization In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
of the fundamental group can be identified with the first homology group of the space. A special case of the Hurewicz theorem asserts that the first singular homology group $H_1\left(X\right)$ is, colloquially speaking, the closest approximation to the fundamental group by means of an abelian group. In more detail, mapping the homotopy class of each loop to the homology class of the loop gives a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) ...
:$\pi_1\left(X\right) \to H_1\left(X\right)$ from the fundamental group of a topological space ''X'' to its first singular homology group $H_1\left(X\right).$ 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 ''X'' is path-connected, this homomorphism is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
and its kernel is the
commutator subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest norma ...
of the fundamental group, so that $H_1\left(X\right)$ is isomorphic to the
abelianization In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
of the fundamental group.

## Glueing topological spaces

Generalizing the statement above, for a family of path connected spaces $X_i,$ the fundamental group $\pi_1 \left(\bigvee_ X_i\right)$ is the
free product In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and i ...
of the fundamental groups of the $X_i.$ This fact is a special case of the Seifert–van Kampen theorem, which allows to compute, more generally, fundamental groups of spaces that are glued together from other spaces. For example, the 2-sphere $S^2$ can be obtained by glueing two copies of slightly overlapping half-spheres along a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
of the
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can a ...
. In this case the theorem yields $\pi_1\left(S^2\right)$ is trivial, since the two half-spheres are contractible and therefore have trivial fundamental group. The fundamental groups of surfaces, as mentioned above, can also be computed using this theorem. In the parlance of category theory, the theorem can be concisely stated by saying that the fundamental group functor takes pushouts (in the category of topological spaces) along inclusions to pushouts (in the category of groups).

## Coverings Given a topological space ''B'', a continuous map :$f: E \to B$ is called a ''covering'' or ''E'' is called a '' covering space'' of ''B'' if every point ''b'' in ''B'' admits an open neighborhood ''U'' such that there is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
between the
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through ...
of ''U'' and a
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (t ...
of copies of ''U'' (indexed by some set ''I''), :$\varphi: \bigsqcup_ U \to f^\left(U\right)$ in such a way that $\pi \circ \varphi$ is the standard projection map $\bigsqcup_ U \to U.$

### Universal covering

A covering is called a universal covering if ''E'' is, in addition to the preceding condition, simply connected. It is universal in the sense that all other coverings can be constructed by suitably identifying points in ''E''. Knowing a universal covering :$p: \widetilde \to X$ of a topological space ''X'' is helpful in understanding its fundamental group in several ways: first, $\pi_1\left(X\right)$ identifies with the group of deck transformations, i.e., the group of
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
s $\varphi : \widetilde \to \widetilde$ that commute with the map to ''X'', i.e., $p \circ \varphi = p.$ Another relation to the fundamental group is that $\pi_1\left(X, x\right)$ can be identified with the fiber $p^\left(x\right).$ For example, the map :$p: \mathbb \to S^1,\, t \mapsto \exp\left(2 \pi i t\right)$ (or, equivalently, 0 to ''x''. The passage from a topological space to its universal covering can be used in understanding the geometry of ''X''. For example, the uniformization theorem shows that any simply connected
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
is (isomorphic to) either $S^2,$ $\mathbb,$ or the upper half plane. General Riemann surfaces then arise as quotients of
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
s on these three surfaces. The
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of a
free action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of a
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
group ''G'' on a simply connected space ''Y'' has fundamental group :$\pi_1\left(Y/G\right) \cong G.$ As an example, the real ''n''-dimensional real
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
$\mathbb\mathrm^n$ is obtained as the quotient of the ''n''-dimensional unit sphere $S^n$ by the antipodal action of the group $\mathbb/2$ sending $x \in S^n$ to $-x.$ As $S^n$ is simply connected for ''n'' ≥ 2, it is a universal cover of $\mathbb\mathrm^n$ in these cases, which implies $\pi_1\left(\mathbb\mathrm^n\right) \cong \mathbb/2$ for ''n'' ≥ 2.

### Lie groups

Let ''G'' be a connected, simply connected compact Lie group, for example, the
special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the speci ...
SU(''n''), and let Γ be a finite subgroup of ''G''. Then the homogeneous space ''X'' = ''G''/Γ has fundamental group Γ, which acts by right multiplication on the universal covering space ''G''. Among the many variants of this construction, one of the most important is given by locally symmetric spaces ''X'' = Γ \''G''/''K'', where *''G'' is a non-compact simply connected, connected
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
(often
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ' ...
), *''K'' is a maximal compact subgroup of ''G'' * Γ is a discrete
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
torsion-free subgroup of ''G''. In this case the fundamental group is Γ and the universal covering space ''G''/''K'' is actually contractible (by the Cartan decomposition for Lie groups). As an example take ''G'' = SL(2, R), ''K'' = SO(2) and Γ any torsion-free congruence subgroup of the
modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractiona ...
SL(2, Z). From the explicit realization, it also follows that the universal covering space of a path connected
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
''H'' is again a path connected topological group ''G''. Moreover, the covering map is a continuous
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (Y ...
homomorphism of ''G'' onto ''H'' with kernel Γ, a closed discrete
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of ''G'': :$1 \to \Gamma \to G \to H \to 1.$ Since ''G'' is a connected group with a continuous action by conjugation on a discrete group Γ, it must act trivially, so that Γ has to be a subgroup of the center of ''G''. In particular π1(''H'') = Γ is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
; this can also easily be seen directly without using covering spaces. The group ''G'' is called the '' universal covering group'' of ''H''. As the universal covering group suggests, there is an analogy between the fundamental group of a topological group and the center of a group; this is elaborated at Lattice of covering groups.

## Fibrations

'' Fibrations'' provide a very powerful means to compute homotopy groups. A fibration ''f'' the so-called ''total space'', and the base space ''B'' has, in particular, the property that all its fibers $f^\left(b\right)$ are homotopy equivalent and therefore can not be distinguished using fundamental groups (and higher homotopy groups), provided that ''B'' is path-connected. Therefore, the space ''E'' can be regarded as a " twisted product" of the base space ''B'' and the
fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...
$F = f^\left(b\right).$ The great importance of fibrations to the computation of homotopy groups stems from a long exact sequence :$\dots \to \pi_2\left(B\right) \to \pi_1\left(F\right) \to \pi_1\left(E\right) \to \pi_1\left(B\right) \to \pi_0\left(F\right) \to \pi_0\left(E\right)$ provided that ''B'' is path-connected. The term $\pi_2\left(B\right)$ is the second homotopy group of ''B'', which is defined to be the set of homotopy classes of maps from $S^2$ to ''B'', in direct analogy with the definition of $\pi_1.$ If ''E'' happens to be path-connected and simply connected, this sequence reduces to an isomorphism :$\pi_1\left(B\right) \cong \pi_0\left(F\right)$ which generalizes the above fact about the universal covering (which amounts to the case where the fiber ''F'' is also discrete). If instead ''F'' happens to be connected and simply connected, it reduces to an isomorphism :$\pi_1\left(E\right) \cong \pi_1\left(B\right).$ What is more, the sequence can be continued at the left with the higher homotopy groups $\pi_n$ of the three spaces, which gives some access to computing such groups in the same vein.

### Classical Lie groups

Such fiber sequences can be used to inductively compute fundamental groups of compact classical Lie groups such as the
special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the speci ...
$\mathrm\left(n\right),$ with $n \geq 2.$ This group acts transitively on the unit sphere $S^$ inside $\mathbb C^n = \mathbb R^.$ The stabilizer of a point in the sphere is isomorphic to $\mathrm\left(n-1\right).$ It then can be shown that this yields a fiber sequence :$\mathrm\left(n-1\right) \to \mathrm\left(n\right) \to S^.$ Since $n \geq 2,$ the sphere $S^$ has dimension at least 3, which implies :$\pi_1\left(S^\right) \cong \pi_2\left(S^\right) = 1.$ The long exact sequence then shows an isomorphism :$\pi_1\left(\mathrm\left(n\right)\right) \cong \pi_1\left(\mathrm\left(n - 1\right)\right).$ Since $\mathrm\left(1\right)$ is a single point, so that $\pi_1\left(\mathrm\left(1\right)\right)$ is trivial, this shows that $\mathrm\left(n\right)$ is simply connected for all $n.$ The fundamental group of noncompact Lie groups can be reduced to the compact case, since such a group is homotopic to its maximal compact subgroup. These methods give the following results: A second method of computing fundamental groups applies to all connected compact Lie groups and uses the machinery of the
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefor ...
and the associated
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation ...
. Specifically, let $T$ be a maximal torus in a connected compact Lie group $K,$ and let $\mathfrak t$ be the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of $T.$ The exponential map :$\exp : \mathfrak t \to T$ is a fibration and therefore its kernel $\Gamma \subset \mathfrak t$ identifies with $\pi_1\left(T\right).$ The map :$\pi_1\left(T\right) \to \pi_1\left(K\right)$ can be shown to be surjective with kernel given by the set ''I'' of integer linear combination of coroots. This leads to the computation :$\pi_1\left(K\right) \cong \Gamma / I.$ This method shows, for example, that any connected compact Lie group for which the associated root system is of type $G_2$ is simply connected. Thus, there is (up to isomorphism) only one connected compact Lie group having Lie algebra of type $G_2$; this group is simply connected and has trivial center.

# Edge-path group of a simplicial complex

When the topological space is homeomorphic to a
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial s ...
, its fundamental group can be described explicitly in terms of generators and relations. If ''X'' is a
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
simplicial complex, an ''edge-path'' in ''X'' is defined to be a chain of vertices connected by edges in ''X''. 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 ''X''. If ''v'' is a fixed vertex in ''X'', an ''edge-loop'' at ''v'' is an edge-path starting and ending at ''v''. The edge-path group ''E''(''X'', ''v'') is defined to be the set of edge-equivalence classes of edge-loops at ''v'', with product and inverse defined by concatenation and reversal of edge-loops. The edge-path group is naturally isomorphic to π1(, ''X'' , , ''v''), the fundamental group of the geometric realisation , ''X'' , of ''X''. Since it depends only on the 2-skeleton ''X'' 2 of ''X'' (that is, the vertices, edges, and triangles of ''X''), the groups π1(, ''X'' , ,''v'') and π1(, ''X'' 2, , ''v'') are isomorphic. The edge-path group can be described explicitly in terms of generators and relations. If ''T'' is a maximal spanning tree in the
1-skeleton In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of ) of dimensions In other wo ...
of ''X'', then ''E''(''X'', ''v'') is canonically isomorphic to the group with generators (the oriented edge-paths of ''X'' not occurring in ''T'') and relations (the edge-equivalences corresponding to triangles in ''X''). A similar result holds if ''T'' is replaced by any simply connected—in particular contractible—subcomplex of ''X''. This often gives a practical way of computing fundamental groups and can be used to show that every
finitely presented group In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
arises as the fundamental group of a finite simplicial complex. It is also one of the classical methods used for topological surfaces, which are classified by their fundamental groups. The ''universal covering space'' of a finite connected simplicial complex ''X'' can also be described directly as a simplicial complex using edge-paths. Its vertices are pairs (''w'',γ) where ''w'' is a vertex of ''X'' and γ is an edge-equivalence class of paths from ''v'' to ''w''. The ''k''-simplices containing (''w'',γ) correspond naturally to the ''k''-simplices containing ''w''. Each new vertex ''u'' of the ''k''-simplex gives an edge ''wu'' and hence, by concatenation, a new path γ''u'' from ''v'' to ''u''. The points (''w'',γ) and (''u'', γ''u'') 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 ''X''. 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; various other authors such as Lorenzo Calabi, Wu Wen-tsün, and Nodar Berikashvili have also published proofs. In the simplest case of a compact space ''X'' with a finite open covering in which all non-empty finite 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 the covering.

# Realizability

*Every group can be realized as the fundamental group of a
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
CW-complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
of dimension 2 (or higher). As noted above, though, only
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
s can occur as fundamental groups of 1-dimensional CW-complexes (that is, graphs). *Every
finitely presented group In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
can be realized as the fundamental group of a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
, connected,
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
of dimension 4 (or higher). But there are severe restrictions on which groups occur as fundamental groups of low-dimensional manifolds. For example, no
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
of rank 4 or higher can be realized as the fundamental group of a manifold of dimension 3 or less. It can be proved that every group can be realized as the fundamental group of a compact
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bico ...
there is no
measurable cardinal In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivisi ...
.

# Related concepts

## Higher homotopy groups

Roughly speaking, the fundamental group detects the 1-dimensional hole structure of a space, but not holes in higher dimensions such as for the 2-sphere. Such "higher-dimensional holes" can be detected using the higher homotopy groups $\pi_n\left(X\right)$, which are defined to consist of homotopy classes of (basepoint-preserving) maps from $S^n$ to ''X''. For example, the Hurewicz theorem implies that for all $n \ge 1$ the ''n''-th homotopy group of the ''n''-sphere is :$\pi_n\left(S^n\right) = \Z.$ As was mentioned in the above computation of $\pi_1$ 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 In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...
''X'', endowed with the compact open topology, is known as the loop space, denoted $\Omega X.$ The fundamental group of ''X'' is in
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
with the set of path components of its loop space: :$\pi_1\left(X\right) \cong \pi_0\left(\Omega X\right).$

## Fundamental groupoid

The '' fundamental groupoid'' is a variant of the fundamental group that is useful in situations where the choice of a base point $x_0 \in X$ is undesirable. It is defined by first considering the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
of paths in $X,$ i.e., continuous functions :, 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. Two such paths $\gamma, \gamma\text{'}$ with the same endpoints and length ''r'', resp. ''r are considered equivalent if there exist real numbers $u,v \geqslant 0$ such that $r + u = r\text{'} + v$ and are homotopic relative to their end points, where The category of paths up to this equivalence relation is denoted $\Pi \left(X\right).$ Each morphism in $\Pi \left(X\right)$ is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, 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 :$\pi_1\left(X, x_0\right) = \mathrm_\left(x_0, x_0\right)$. 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 of two connected open sets whose intersection has two components, one can choose one base point in each component. The van Kampen theorem admits a version for fundamental groupoids which gives, for example, another way to compute the fundamental group(oid) of $S^1.$

## Local systems

Generally speaking, representations may serve to exhibit features of a group by its actions on other mathematical objects, often
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s. Representations of the fundamental group have a very geometric significance: any '' local system'' (i.e., a sheaf $\mathcal F$ 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 $\mathcal F, _U = \Q^n$) gives rise to the so-called monodromy representation, a representation of the fundamental group on an ''n''- dimensional $\Q$-vector space. Conversely, any such representation on a path-connected space ''X'' arises in this manner. This equivalence of categories between representations of $\pi_1\left(X\right)$ and local systems is used, for example, in the study of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, such as the Knizhnik–Zamolodchikov equations.

## Étale fundamental group

In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the so-called étale fundamental group is used as a replacement for the fundamental group. Since the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is ...
on an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
or scheme ''X'' is much coarser than, say, the
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of open subsets in $\R^n,$ it is no longer meaningful to consider continuous maps from an interval to ''X''. Instead, the approach developed by Grothendieck consists in constructing $\pi_1^\text$ by considering all finite étale covers of ''X''. These serve as an algebro-geometric analogue of coverings with finite fibers. This yields a theory applicable in situation where no great generality classical topological intuition whatsoever is available, for example for varieties defined over a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
. Also, the étale fundamental group of a field is its (
absolute Absolute may refer to: Companies * Absolute Entertainment, a video game publisher * Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK * Absolute Software Corporation, specializes in security and data risk manage ...
)
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
. 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.

## Fundamental group of algebraic groups

The fundamental group of a
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation ...
is defined, in analogy to the computation for Lie groups. This allows to define and use the fundamental group of a semisimple
linear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n ...
''G'', which is a useful basic tool in the classification of linear algebraic groups.

## 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. Thus, $\pi_1$ 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, $, X, ,$ i.e., the topological space obtained by glueing topological simplices as prescribed by the simplicial set structure of ''X''.

* Orbifold fundamental group * Fundamental group scheme

# References

* * * * * * * * * * * * Peter Hilton and Shaun Wylie, ''Homology Theory'', Cambridge University Press (1967) arning:_these_authors_use_''contrahomology''_for_cohomology.html" ;"title="cohomology.html" ;"title="arning: these authors use ''contrahomology'' for cohomology">arning: these authors use ''contrahomology'' for cohomology">cohomology.html" ;"title="arning: these authors use ''contrahomology'' for cohomology">arning: these authors use ''contrahomology'' for cohomology* * * * * * Deane Montgomery and Leo Zippin, ''Topological Transformation Groups'', Interscience Publishers (1955) * * * * * * *