In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy 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 fundamental groups.

** Intuition **

Start with a space (for example, a surface), 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 together 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 defined the fundamental group in 1895 in his paper "Analysis situs". The concept emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Poincaré, and Felix Klein. It describes the monodromy properties of complex-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 (also known as a continuous map) :$\backslash gamma\; \backslash colon,\; 1\backslash to\; X$ such that the starting point $\backslash gamma(0)$ and the end point $\backslash gamma(1)$ are both equal to $x\_0$. A ''homotopy'' is a continuous interpolation between two loops. More precisely, a homotopy between two loops $\backslash gamma,\; \backslash gamma\text{'}\; \backslash colon,\; 1\backslash to\; X$ (based at the same point $x\_0$) is a continuous map :$h\; \backslash colon,\; 1\backslash times,\; 1\backslash to\; X,$ such that * $h(0,\; t)\; =\; x\_0$ for all $t\; \backslash in,\; 1$ that is, the starting point of the homotopy is $x\_0$ for all ''t'' (which is often thought of as a time parameter). * $h(1,\; t)\; =\; x\_0$ for all $t\; \backslash in,\; 1$ that is, similarly the end point stays at $x\_0$ for all ''t''. * $h(r,\; 0)\; =\; \backslash gamma(r),\; h(r,\; 1)\; =\; \backslash gamma\text{'}(r)$ for all $r\; \backslash in,\; 1/math>.\; If\; such\; a\; homotopy\; \text{\'}\text{\'}h\text{\'}\text{\'}\; exists,$ \backslash gamma$and$ \backslash gamma\text{\'}$are\; said\; to\; be\; \text{\'}\text{\'}homotopic\text{\'}\text{\'}.\; The\; relation\; "$ \backslash gamma$is\; homotopic\; to$ \backslash gamma\text{\'}$"\; is\; anequivalence\; relationso\; that\; the\; set\; ofequivalence\; classes\; can\; be\; considered:\; :$ \backslash pi\_1(X,\; x\_0)\; :=\; \backslash \; /\; \backslash text$.\; This\; set\; (with\; the\; group\; structure\; described\; below)\; is\; called\; the\; \text{\'}\text{\'}fundamental\; group\text{\'}\text{\'}\; of\; the\; topological\; space\; \text{\'}\text{\'}X\text{\'}\text{\'}\; at\; the\; base\; point$ x\_0$.\; The\; purpose\; of\; considering\; the\; equivalence\; classes\; of\; loops\; up\; to\; homotopy,\; as\; opposed\; to\; the\; set\; of\; all\; loops\; (the\; so-calledloop\; spaceof\; \text{\'}\text{\'}X\text{\'}\text{\'})\; is\; that\; the\; latter,\; while\; being\; useful\; for\; various\; purposes,\; is\; a\; rather\; big\; and\; unwieldy\; object.\; By\; contrast\; the\; above\; quotient\; is,\; in\; many\; cases,\; more\; manageable\; and\; computable.$

Group structure

By the above definition, $\backslash pi\_1(X,\; x\_0)$ 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 $\backslash gamma\_0,\; \backslash gamma\_1$, their product is defined as the loop :$\backslash begin\; \backslash gamma\_0\; \backslash cdot\; \backslash gamma\_1\; \backslash colon,\; 1\&\backslash to\; X\; \backslash \backslash \; (\backslash gamma\_0\; \backslash cdot\; \backslash gamma\_1)(t)\; \&=\; \backslash begin\; \backslash gamma\_0(2t)\; \&\; 0\; \backslash leq\; t\; \backslash leq\; \backslash tfrac\; \backslash \backslash \; \backslash gamma\_1(2t\; -\; 1)\; \&\; \backslash tfrac\; \backslash leq\; t\; \backslash leq\; 1.\; \backslash end\; \backslash end$ Thus the loop $\backslash gamma\_0\; \backslash cdot\; \backslash gamma\_1$ first follows the loop $\backslash gamma\_0$ with "twice the speed" and then follows $\backslash gamma\_1$ with "twice the speed". The product of two homotopy classes of loops $gamma\_0/math>\; and$ gamma\_1/math>\; is\; then\; defined\; as$ gamma\_0\; \backslash cdot\; \backslash gamma\_1/math>.\; It\; can\; be\; shown\; that\; this\; product\; does\; not\; depend\; on\; the\; choice\; of\; representatives\; and\; therefore\; gives\; a\; well-defined\; operation\; on\; the\; set$ \backslash pi\_1(X,\; x\_0)$.\; This\; operation\; turns$ \backslash pi\_1(X,\; x\_0)$into\; a\; group.\; Itsneutral\; elementis\; the\; constant\; loop,\; which\; stays\; at$ x\_0$for\; all\; times\; \text{'}\text{'}t\text{'}\text{'}.\; The\; inverse\; of\; a\; (homotopy\; class\; of\; a)\; loop\; is\; the\; same\; loop,\; but\; traversed\; in\; the\; opposite\; direction.\; More\; formally,\; :$ \backslash gamma^\; (t)\; :=\; \backslash gamma(1-t)$.\; Given\; three\; based\; loops$ \backslash gamma\_0,\; \backslash gamma\_1,\; \backslash gamma\_2,$the\; product\; :$ (\backslash gamma\_0\; \backslash cdot\; \backslash gamma\_1)\; \backslash cdot\; \backslash gamma\_2$is\; the\; concatenation\; of\; these\; loops,\; traversing$ \backslash gamma\_0$and\; then$ \backslash gamma\_1$with\; quadruple\; speed,\; and\; then$ \backslash gamma\_2$with\; double\; speed.\; By\; comparison,\; :$ \backslash gamma\_0\; \backslash cdot\; (\backslash gamma\_1\; \backslash cdot\; \backslash gamma\_2)$traverses\; the\; same\; paths\; (in\; the\; same\; order),\; but$ \backslash gamma\_0$with\; double\; speed,\; and$ \backslash gamma\_1,\; \backslash gamma\_2$with\; quadruple\; speed.\; Thus,\; because\; of\; the\; differing\; speeds,\; the\; two\; paths\; are\; not\; identical.\; Theassociativityaxiom\; :$ gamma\_0\backslash cdot\; \backslash left(gamma\_1\backslash cdotgamma\_2right)\; =\; \backslash left(gamma\_0\backslash cdotgamma\_1right)\; \backslash cdotgamma\_2/math>\; therefore\; crucially\; depends\; on\; the\; fact\; that\; paths\; are\; considered\; up\; to\; homotopy.\; Indeed,\; both\; above\; composites\; are\; homotopic,\; for\; example,\; to\; the\; loop\; that\; traverses\; all\; three\; loops$ \backslash gamma\_0,\; \backslash gamma\_1,\; \backslash gamma\_2$with\; triple\; speed.\; The\; set\; of\; based\; loops\; up\; to\; homotopy,\; equipped\; with\; the\; above\; operation\; therefore\; does\; turn$ \backslash pi\_1(X,\; x\_0)$into\; a\; group.$$$$

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 (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 $\backslash pi\_1(X)$ instead of $\backslash pi\_1(X,\; x\_0)$.

** Concrete examples **

This section lists some basic examples of fundamental groups. To begin with, in Euclidean space ($\backslash R^n$) or any convex subset of $\backslash R^n,$ 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 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 $S^2\; =\; \backslash left\backslash $ 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 $\backslash gamma$ such that there is a point $(x,\; y,\; z)\; \backslash in\; S^2$ that is in the image of $\backslash gamma.$ However, since there are loops such that $\backslash gamma(,\; 1=\; S^2$ (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 (also known as the 1-sphere) :$S^1\; =\; \backslash left\backslash $ 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 to $(\backslash Z,\; +),$ the additive group of integers. This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk–Ulam theorem in dimension 2.

The figure eight

The fundamental group of the 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 $\backslash gamma$ can be decomposed as :$\backslash gamma\; =\; a^\; b^\; \backslash cdots\; a^\; b^$ where ''a'' and ''b'' are the two loops winding around each half of the figure as depicted, and the exponents $n\_1,\; \backslash dots,\; n\_k,\; m\_1,\; \backslash dots,\; m\_k$ are integers. Unlike $\backslash pi\_1\backslash left(S^1\backslash right),$ the fundamental group of the figure eight is ''not'' abelian: the two ways of composing ''a'' and ''b'' are not homotopic to each other: :$\backslash cdot\backslash ne\backslash cdot$ 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 of two path connected spaces ''X'' and ''Y'' can be computed as the free product of the individual fundamental groups: :$\backslash pi\_1(X\; \backslash vee\; Y)\; \backslash cong\; \backslash pi\_1(X)\; *\; \backslash pi\_1(Y).$ 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 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 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 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 ''K'' embedded in $\backslash R^3.$ 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 to distinguish between knots: if $\backslash pi\_1\backslash left(\backslash R^3\; \backslash setminus\; K\backslash right)$ is not isomorphic to some other knot group $\backslash pi\_1\backslash left(\backslash R^3\; \backslash setminus\; K\text{'}\backslash right)$ of another knot ''K, then ''K'' can not be transformed into $K\text{'}.$ Thus the trefoil knot can not be continuously transformed into the circle (also known as the unknot), since the latter has knot group $\backslash Z$. 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 :$\backslash left\backslash langle\; A\_1,\; B\_1,\; \backslash ldots,\; A\_n,\; B\_n\; \backslash left|\; A\_1\; B\_1\; A\_1^\; B\_1^\; \backslash cdots\; A\_n\; B\_n\; A\_n^\; B\_n^\; \backslash right.\; \backslash right\backslash rangle.$ This includes the torus, being the case of genus 1, whose fundamental group is :$\backslash left\backslash langle\; A\_1,\; B\_1\; \backslash left|\; A\_1\; B\_1\; A\_1^\; B\_1^\; \backslash right.\; \backslash right\backslash rangle\; \backslash cong\; \backslash Z^2.$

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 $\backslash pi\_1(X)$ with another group structure: given two loops $\backslash gamma$ and $\backslash gamma\text{'}$ in ''X'', another loop $\backslash gamma\; \backslash star\; \backslash gamma\text{'}$ can defined by using the group multiplication in ''X'': :$\backslash left(\backslash gamma\; \backslash star\; \backslash gamma\text{'}\backslash right)(x)\; =\; \backslash gamma(x)\; \backslash cdot\; \backslash gamma\text{'}(x).$ This binary operation $\backslash star$ 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, $\backslash pi\_1(X)$ 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'', $X\; =\; \backslash Omega(Y),$ is abelian. Related ideas lead to Heinz Hopf's computation of the cohomology of a Lie group.

** Functoriality **

If $f\backslash colon\; X\; \backslash to\; Y$ is a continuous map, $x\_0\; \backslash in\; X$ and $y\_0\; \backslash in\; Y$ with $f(x\_0)\; =\; y\_0,$ 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, called the induced homomorphism, is written as $\backslash pi(f)$ or, more commonly,
:$f\_*\; \backslash colon\; \backslash pi\_1(X,\; x\_0)\; \backslash to\; \backslash pi\_1(Y,\; y\_0).$
This mapping from continuous maps to group homomorphisms is compatible with composition of maps and identity morphisms. In the parlance of category theory, the formation of associating to a topological space its fundamental group is therefore a functor
:$\backslash begin\; \backslash pi\_1\; \backslash colon\; \backslash mathbf\_*\; \&\backslash to\; \backslash mathbf\; \backslash \backslash \; (X,\; x\_0)\; \&\backslash mapsto\; \backslash pi\_1(X,\; x\_0)\; \backslash end$
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:
:$X\; \backslash simeq\; Y\; \backslash Rightarrow\; \backslash pi\_1(X,\; x\_0)\; \backslash cong\; \backslash pi\_1(Y,\; y\_0).$
For example, the inclusion of the circle in the punctured plane
:$S^1\; \backslash subset\; \backslash mathbb^2\; \backslash setminus\; \backslash $
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
:$\backslash pi\_1\; (X\; \backslash times\; Y,\; (x\_0,\; y\_0))\; \backslash cong\; \backslash pi\_1(X,\; x\_0)\; \backslash times\; \backslash pi\_1(Y,\; y\_0).$

** 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.

** Relationship to first homology group **

The abelianization 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 $H\_1(X)$ 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
:$\backslash pi\_1(X)\; \backslash to\; H\_1(X)$
from the fundamental group of a topological space ''X'' to its first singular homology group $H\_1(X).$ 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 and its kernel is the commutator subgroup of the fundamental group, so that $H\_1(X)$ is isomorphic to the abelianization of the fundamental group.

Glueing topological spaces

Generalizing the statement above, for a family of path connected spaces $X\_i,$ the fundamental group $\backslash pi\_1\; \backslash left(\backslash bigvee\_\; X\_i\backslash right)$ is the free product 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 of the equator. In this case the theorem yields $\backslash pi\_1\backslash left(S^2\backslash right)$ 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 :$f:\; E\; \backslash to\; B$ 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 between the preimage of ''U'' and a disjoint union of copies of ''U'' (indexed by some set ''I''), :$\backslash varphi:\; \backslash bigsqcup\_\; U\; \backslash to\; f^(U)$ in such a way that $\backslash pi\; \backslash circ\; \backslash varphi$ is the standard projection map $\backslash bigsqcup\_\; U\; \backslash to\; U.$

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 :$p:\; \backslash widetilde\; \backslash to\; X$ of a topological space ''X'' is helpful in understanding its fundamental group in several ways: first, $\backslash pi\_1(X)$ identifies with the group of deck transformations, i.e., the group of homeomorphisms $\backslash varphi\; :\; \backslash widetilde\; \backslash to\; \backslash widetilde$ that commute with the map to ''X'', i.e., $p\; \backslash circ\; \backslash varphi\; =\; p.$ Another relation to the fundamental group is that $\backslash pi\_1(X,\; x)$ can be identified with the fiber $p^(x).$ For example, the map :$p:\; \backslash mathbb\; \backslash to\; S^1,\; t\; \backslash mapsto\; \backslash exp(2\; \backslash pi\; i\; t)$ (or, equivalently, $\backslash pi:\; \backslash mathbb\; \backslash to\; \backslash mathbb\; /\; \backslash mathbb,\backslash \; t\; \backslash mapsto/math>)\; is\; a\; universal\; covering.\; The\; deck\; transformations\; are\; the\; maps$ t\; \backslash mapsto\; t\; +\; n$for$ n\; \backslash in\; \backslash mathbb.$This\; is\; in\; line\; with\; the\; identification$ p^(1)\; =\; \backslash mathbb,$in\; particular\; this\; proves\; the\; above\; claim$ \backslash pi\_1\backslash left(S^1\backslash right)\; \backslash cong\; \backslash mathbb.$Any\; path\; connected,\; locallypath\; connectedand\; locally\; simply\; connected\; topological\; space\; \text{'}\text{'}X\text{'}\text{'}\; admits\; a\; universal\; covering.\; An\; abstract\; construction\; proceeds\; analogously\; to\; the\; fundamental\; group\; by\; taking\; pairs\; (\text{'}\text{'}x\text{'}\text{'},\gamma ),\; where\; \text{'}\text{'}x\text{'}\text{'}\; is\; a\; point\; in\; \text{'}\text{'}X\text{'}\text{'}\; and\; \gamma \; is\; a\; homotopy\; class\; of\; paths\; from\; \text{'}\text{'}x\text{'}\text{'}$_{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 is (isomorphic to) either $S^2,$ $\backslash mathbb,$ or the upper half plane. General Riemann surfaces then arise as quotients of group actions on these three surfaces.
The quotient of an action of a (discrete) group ''G'' on a simply connected space ''Y'' has fundamental group
:$\backslash pi\_1(Y/G)\; \backslash cong\; G.$
As an example, the real ''n''-dimensional real projective space $\backslash mathbb\backslash mathrm^n$ is obtained as the quotient of the ''n''-dimensional sphere $S^n$ by the antipodal action of the group $\backslash mathbb/2$ sending $x\; \backslash in\; S^n$ to $-x.$ As $S^n$ is simply connected for ''n'' ≥ 2, it is a universal cover of $\backslash mathbb\backslash mathrm^n$ in these cases, which implies $\backslash pi\_1\backslash left(\backslash mathbb\backslash mathrm^n\backslash right)\; \backslash cong\; \backslash mathbb/2$ for ''n'' ≥ 2.

Lie groups

Let ''G'' be a connected, simply connected compact Lie group, for example, the special unitary group 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 (often semisimple), *''K'' is a maximal compact subgroup of ''G'' * Γ is a discrete countable 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 SL(2, Z). From the explicit realization, it also follows that the universal covering space of a path connected topological group ''H'' is again a path connected topological group ''G''. Moreover, the covering map is a continuous open homomorphism of ''G'' onto ''H'' with kernel Γ, a closed discrete normal subgroup of ''G'': :$1\; \backslash to\; \backslash Gamma\; \backslash to\; G\; \backslash to\; H\; \backslash to\; 1.$ 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 π_{1}(''H'') = Γ is an abelian group; 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 $f^(b)$ 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 $F\; =\; f^(b).$ The great importance of fibrations to the computation of homotopy groups stems from a long exact sequence :$\backslash dots\; \backslash to\; \backslash pi\_2(B)\; \backslash to\; \backslash pi\_1(F)\; \backslash to\; \backslash pi\_1(E)\; \backslash to\; \backslash pi\_1(B)\; \backslash to\; \backslash pi\_0(F)\; \backslash to\; \backslash pi\_0(E)$ provided that ''B'' is path-connected. The term $\backslash pi\_2(B)$ 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 $\backslash pi\_1.$ If ''E'' happens to be path-connected and simply connected, this sequence reduces to an isomorphism :$\backslash pi\_1(B)\; \backslash cong\; \backslash pi\_0(F)$ 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 :$\backslash pi\_1(E)\; \backslash cong\; \backslash pi\_1(B).$ What is more, the sequence can be continued at the left with the higher homotopy groups $\backslash pi\_n$ 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 $\backslash mathrm(n),$ with $n\; \backslash geq\; 2.$ This group acts transitively on the unit sphere $S^$ inside $\backslash mathbb\; C^n\; =\; \backslash mathbb\; R^.$ The stabilizer of a point in the sphere is isomorphic to $\backslash mathrm(n-1).$ It then can be shown that this yields a fiber sequence :$\backslash mathrm(n-1)\; \backslash to\; \backslash mathrm(n)\; \backslash to\; S^.$ Since $n\; \backslash geq\; 2,$ the sphere $S^$ has dimension at least 3, which implies :$\backslash pi\_1\backslash left(S^\backslash right)\; \backslash cong\; \backslash pi\_2\backslash left(S^\backslash right)\; =\; 1.$ The long exact sequence then shows an isomorphism :$\backslash pi\_1(\backslash mathrm(n))\; \backslash cong\; \backslash pi\_1(\backslash mathrm(n\; -\; 1)).$ Since $\backslash mathrm(1)$ is a single point, so that $\backslash pi\_1(\backslash mathrm(1))$ is trivial, this shows that $\backslash mathrm(n)$ 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 and the associated root system. Specifically, let $T$ be a maximal torus in a connected compact Lie group $K,$ and let $\backslash mathfrak\; t$ be the Lie algebra of $T.$ The exponential map :$\backslash exp\; :\; \backslash mathfrak\; t\; \backslash to\; T$ is a fibration and therefore its kernel $\backslash Gamma\; \backslash subset\; \backslash mathfrak\; t$ identifies with $\backslash pi\_1(T).$ The map :$\backslash pi\_1(T)\; \backslash to\; \backslash pi\_1(K)$ can be shown to be surjective with kernel given by the set ''I'' of integer linear combination of coroots. This leads to the computation :$\backslash pi\_1(K)\; \backslash cong\; \backslash Gamma\; /\; I.$ 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, its fundamental group can be described explicitly in terms of generators and relations. If ''X'' is a connected 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 π_{1}(|''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 π_{1}(|''X''|,''v'') and π_{1}(|''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 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 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 CW-complex of dimension 2 (or higher). As noted above, though, only free groups can occur as fundamental groups of 1-dimensional CW-complexes (that is, graphs).
*Every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold 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 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 if and only if there is no measurable cardinal.

** 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 $\backslash pi\_n(X)$, which are defined to consist of homotopy classes of (basepoint-preserving) maps from $S^n$ to ''X''. For example, the Hurewicz theorem implies that the ''n''-th homotopy group of the ''n''-sphere is (for all $n\; \backslash ge\; 1$) are :$\backslash pi\_n(S^n)\; =\; \backslash Z.$ As was mentioned in the above computation of $\backslash pi\_1$ 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 ''X'', endowed with the compact open topology, is known as the loop space, denoted $\backslash Omega\; X.$ The fundamental group of ''X'' is in bijection with the set of path components of its loop space: :$\backslash pi\_1(X)\; \backslash cong\; \backslash pi\_0(\backslash Omega\; X).$

Fundamental groupoid

The ''fundamental groupoid'' is a variant of the fundamental group that is useful in situations where the choice of a base point $x\_0\; \backslash in\; X$ is undesirable. It is defined by first considering the category of paths in $X,$ i.e., continuous functions :$\backslash gamma\; \backslash colon,\; r\backslash to\; X$, 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 $\backslash gamma,\; \backslash gamma\text{'}$ with the same endpoints and length ''r'', resp. ''r are considered equivalent if there exist real numbers $u,v\; \backslash geqslant\; 0$ such that $r\; +\; u\; =\; r\text{'}\; +\; v$ and $\backslash gamma\_u,\; \backslash gamma\text{'}\_v\; \backslash colon,\; r\; +\; u\backslash to\; X$ are homotopic relative to their end points, where $\backslash gamma\_u\; (t)\; =\; \backslash begin\; \backslash gamma(t),\; \&\; t\; \backslash in,\; r\backslash \backslash \; \backslash gamma(r),\; \&\; t\; \backslash in,\; r\; +\; u\backslash end$ The category of paths up to this equivalence relation is denoted $\backslash Pi\; (X).$ Each morphism in $\backslash Pi\; (X)$ is an isomorphism, with inverse given by the same path traversed in the opposite direction. Such a category is called a groupoid. It reproduces the fundamental group since :$\backslash pi\_1(X,\; x\_0)\; =\; \backslash mathrm\_(x\_0,\; x\_0)$. 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 spaces. Representations of the fundamental group have a very geometric significance: any ''local system'' (i.e., a sheaf $\backslash mathcal\; F$ 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 $\backslash mathcal\; F|\_U\; =\; \backslash Q^n$) gives rise to the so-called monodromy representation, a representation of the fundamental group on an ''n''-dimensional $\backslash Q$-vector space. Conversely, any such representation on a path-connected space ''X'' arises in this manner. This equivalence of categories between representations of $\backslash pi\_1(X)$ and local systems is used, for example, in the study of differential equations, such as the Knizhnik–Zamolodchikov equations.

Étale fundamental group

In algebraic geometry, the so-called étale fundamental group is used as a replacement for the fundamental group. Since the Zariski topology on an algebraic variety or scheme ''X'' is much coarser than, say, the topology of open subsets in $\backslash R^n,$ it is no longer meaningful to consider continuous maps from an interval to ''X''. Instead, the approach developed by Grothendieck consists in constructing $\backslash pi\_1^\backslash text$ 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. Also, the étale fundamental group of a field is its (absolute) Galois group. On the other hand, for smooth varieties ''X'' over the complex numbers, the étale fundamental group retains much of the information inherent in the classical fundamental group: the former is the profinite completion of the latter.

Fundamental group of algebraic groups

The fundamental group of a root system 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 ''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, $\backslash pi\_1$ 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''.

See also

* Orbifold fundamental group

Notes

References

* * * * * * * * * * * * Peter Hilton and Shaun Wylie, ''Homology Theory'', Cambridge University Press (1967) arning: these authors use ''contrahomology'' for [[cohomology] * * * * * * [[Deane Montgomery and Leo Zippin, ''Topological Transformation Groups'', Interscience Publishers (1955) * * * * * * *

External links

* * Dylan G.L. Allegretti

''Simplicial Sets and van Kampen's Theorem''

A discussion of the fundamental groupoid of a topological space and the fundamental groupoid of a simplicial set

Animations to introduce fundamental group by Nicolas Delanoue

Sets of base points and fundamental groupoids: mathoverflow discussion

{{Commons category Category:Algebraic topology Category:Homotopy theory

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 (also known as a continuous map) :$\backslash gamma\; \backslash colon,\; 1\backslash to\; X$ such that the starting point $\backslash gamma(0)$ and the end point $\backslash gamma(1)$ are both equal to $x\_0$. A ''homotopy'' is a continuous interpolation between two loops. More precisely, a homotopy between two loops $\backslash gamma,\; \backslash gamma\text{'}\; \backslash colon,\; 1\backslash to\; X$ (based at the same point $x\_0$) is a continuous map :$h\; \backslash colon,\; 1\backslash times,\; 1\backslash to\; X,$ such that * $h(0,\; t)\; =\; x\_0$ for all $t\; \backslash in,\; 1$ that is, the starting point of the homotopy is $x\_0$ for all ''t'' (which is often thought of as a time parameter). * $h(1,\; t)\; =\; x\_0$ for all $t\; \backslash in,\; 1$ that is, similarly the end point stays at $x\_0$ for all ''t''. * $h(r,\; 0)\; =\; \backslash gamma(r),\; h(r,\; 1)\; =\; \backslash gamma\text{'}(r)$ for all $r\; \backslash in,\; 1/math>.\; If\; such\; a\; homotopy\; \text{\'}\text{\'}h\text{\'}\text{\'}\; exists,$ \backslash gamma$and$ \backslash gamma\text{\'}$are\; said\; to\; be\; \text{\'}\text{\'}homotopic\text{\'}\text{\'}.\; The\; relation\; "$ \backslash gamma$is\; homotopic\; to$ \backslash gamma\text{\'}$"\; is\; anequivalence\; relationso\; that\; the\; set\; ofequivalence\; classes\; can\; be\; considered:\; :$ \backslash pi\_1(X,\; x\_0)\; :=\; \backslash \; /\; \backslash text$.\; This\; set\; (with\; the\; group\; structure\; described\; below)\; is\; called\; the\; \text{\'}\text{\'}fundamental\; group\text{\'}\text{\'}\; of\; the\; topological\; space\; \text{\'}\text{\'}X\text{\'}\text{\'}\; at\; the\; base\; point$ x\_0$.\; The\; purpose\; of\; considering\; the\; equivalence\; classes\; of\; loops\; up\; to\; homotopy,\; as\; opposed\; to\; the\; set\; of\; all\; loops\; (the\; so-calledloop\; spaceof\; \text{\'}\text{\'}X\text{\'}\text{\'})\; is\; that\; the\; latter,\; while\; being\; useful\; for\; various\; purposes,\; is\; a\; rather\; big\; and\; unwieldy\; object.\; By\; contrast\; the\; above\; quotient\; is,\; in\; many\; cases,\; more\; manageable\; and\; computable.$

Group structure

By the above definition, $\backslash pi\_1(X,\; x\_0)$ 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 $\backslash gamma\_0,\; \backslash gamma\_1$, their product is defined as the loop :$\backslash begin\; \backslash gamma\_0\; \backslash cdot\; \backslash gamma\_1\; \backslash colon,\; 1\&\backslash to\; X\; \backslash \backslash \; (\backslash gamma\_0\; \backslash cdot\; \backslash gamma\_1)(t)\; \&=\; \backslash begin\; \backslash gamma\_0(2t)\; \&\; 0\; \backslash leq\; t\; \backslash leq\; \backslash tfrac\; \backslash \backslash \; \backslash gamma\_1(2t\; -\; 1)\; \&\; \backslash tfrac\; \backslash leq\; t\; \backslash leq\; 1.\; \backslash end\; \backslash end$ Thus the loop $\backslash gamma\_0\; \backslash cdot\; \backslash gamma\_1$ first follows the loop $\backslash gamma\_0$ with "twice the speed" and then follows $\backslash gamma\_1$ with "twice the speed". The product of two homotopy classes of loops $gamma\_0/math>\; and$ gamma\_1/math>\; is\; then\; defined\; as$ gamma\_0\; \backslash cdot\; \backslash gamma\_1/math>.\; It\; can\; be\; shown\; that\; this\; product\; does\; not\; depend\; on\; the\; choice\; of\; representatives\; and\; therefore\; gives\; a\; well-defined\; operation\; on\; the\; set$ \backslash pi\_1(X,\; x\_0)$.\; This\; operation\; turns$ \backslash pi\_1(X,\; x\_0)$into\; a\; group.\; Itsneutral\; elementis\; the\; constant\; loop,\; which\; stays\; at$ x\_0$for\; all\; times\; \text{'}\text{'}t\text{'}\text{'}.\; The\; inverse\; of\; a\; (homotopy\; class\; of\; a)\; loop\; is\; the\; same\; loop,\; but\; traversed\; in\; the\; opposite\; direction.\; More\; formally,\; :$ \backslash gamma^\; (t)\; :=\; \backslash gamma(1-t)$.\; Given\; three\; based\; loops$ \backslash gamma\_0,\; \backslash gamma\_1,\; \backslash gamma\_2,$the\; product\; :$ (\backslash gamma\_0\; \backslash cdot\; \backslash gamma\_1)\; \backslash cdot\; \backslash gamma\_2$is\; the\; concatenation\; of\; these\; loops,\; traversing$ \backslash gamma\_0$and\; then$ \backslash gamma\_1$with\; quadruple\; speed,\; and\; then$ \backslash gamma\_2$with\; double\; speed.\; By\; comparison,\; :$ \backslash gamma\_0\; \backslash cdot\; (\backslash gamma\_1\; \backslash cdot\; \backslash gamma\_2)$traverses\; the\; same\; paths\; (in\; the\; same\; order),\; but$ \backslash gamma\_0$with\; double\; speed,\; and$ \backslash gamma\_1,\; \backslash gamma\_2$with\; quadruple\; speed.\; Thus,\; because\; of\; the\; differing\; speeds,\; the\; two\; paths\; are\; not\; identical.\; Theassociativityaxiom\; :$ gamma\_0\backslash cdot\; \backslash left(gamma\_1\backslash cdotgamma\_2right)\; =\; \backslash left(gamma\_0\backslash cdotgamma\_1right)\; \backslash cdotgamma\_2/math>\; therefore\; crucially\; depends\; on\; the\; fact\; that\; paths\; are\; considered\; up\; to\; homotopy.\; Indeed,\; both\; above\; composites\; are\; homotopic,\; for\; example,\; to\; the\; loop\; that\; traverses\; all\; three\; loops$ \backslash gamma\_0,\; \backslash gamma\_1,\; \backslash gamma\_2$with\; triple\; speed.\; The\; set\; of\; based\; loops\; up\; to\; homotopy,\; equipped\; with\; the\; above\; operation\; therefore\; does\; turn$ \backslash pi\_1(X,\; x\_0)$into\; a\; group.$$$$

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 (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 $\backslash pi\_1(X)$ instead of $\backslash pi\_1(X,\; x\_0)$.

The circle

The circle (also known as the 1-sphere) :$S^1\; =\; \backslash left\backslash $ 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 to $(\backslash Z,\; +),$ the additive group of integers. This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk–Ulam theorem in dimension 2.

The figure eight

The fundamental group of the 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 $\backslash gamma$ can be decomposed as :$\backslash gamma\; =\; a^\; b^\; \backslash cdots\; a^\; b^$ where ''a'' and ''b'' are the two loops winding around each half of the figure as depicted, and the exponents $n\_1,\; \backslash dots,\; n\_k,\; m\_1,\; \backslash dots,\; m\_k$ are integers. Unlike $\backslash pi\_1\backslash left(S^1\backslash right),$ the fundamental group of the figure eight is ''not'' abelian: the two ways of composing ''a'' and ''b'' are not homotopic to each other: :$\backslash cdot\backslash ne\backslash cdot$ 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 of two path connected spaces ''X'' and ''Y'' can be computed as the free product of the individual fundamental groups: :$\backslash pi\_1(X\; \backslash vee\; Y)\; \backslash cong\; \backslash pi\_1(X)\; *\; \backslash pi\_1(Y).$ 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.

Knot groups

''Knot groups'' are, by definition the fundamental group of the complement of a knot ''K'' embedded in $\backslash R^3.$ 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 to distinguish between knots: if $\backslash pi\_1\backslash left(\backslash R^3\; \backslash setminus\; K\backslash right)$ is not isomorphic to some other knot group $\backslash pi\_1\backslash left(\backslash R^3\; \backslash setminus\; K\text{'}\backslash right)$ of another knot ''K, then ''K'' can not be transformed into $K\text{'}.$ Thus the trefoil knot can not be continuously transformed into the circle (also known as the unknot), since the latter has knot group $\backslash Z$. 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 :$\backslash left\backslash langle\; A\_1,\; B\_1,\; \backslash ldots,\; A\_n,\; B\_n\; \backslash left|\; A\_1\; B\_1\; A\_1^\; B\_1^\; \backslash cdots\; A\_n\; B\_n\; A\_n^\; B\_n^\; \backslash right.\; \backslash right\backslash rangle.$ This includes the torus, being the case of genus 1, whose fundamental group is :$\backslash left\backslash langle\; A\_1,\; B\_1\; \backslash left|\; A\_1\; B\_1\; A\_1^\; B\_1^\; \backslash right.\; \backslash right\backslash rangle\; \backslash cong\; \backslash Z^2.$

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 $\backslash pi\_1(X)$ with another group structure: given two loops $\backslash gamma$ and $\backslash gamma\text{'}$ in ''X'', another loop $\backslash gamma\; \backslash star\; \backslash gamma\text{'}$ can defined by using the group multiplication in ''X'': :$\backslash left(\backslash gamma\; \backslash star\; \backslash gamma\text{'}\backslash right)(x)\; =\; \backslash gamma(x)\; \backslash cdot\; \backslash gamma\text{'}(x).$ This binary operation $\backslash star$ 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, $\backslash pi\_1(X)$ 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'', $X\; =\; \backslash Omega(Y),$ is abelian. Related ideas lead to Heinz Hopf's computation of the cohomology of a Lie group.

Glueing topological spaces

Generalizing the statement above, for a family of path connected spaces $X\_i,$ the fundamental group $\backslash pi\_1\; \backslash left(\backslash bigvee\_\; X\_i\backslash right)$ is the free product 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 of the equator. In this case the theorem yields $\backslash pi\_1\backslash left(S^2\backslash right)$ 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 :$f:\; E\; \backslash to\; B$ 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 between the preimage of ''U'' and a disjoint union of copies of ''U'' (indexed by some set ''I''), :$\backslash varphi:\; \backslash bigsqcup\_\; U\; \backslash to\; f^(U)$ in such a way that $\backslash pi\; \backslash circ\; \backslash varphi$ is the standard projection map $\backslash bigsqcup\_\; U\; \backslash to\; U.$

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 :$p:\; \backslash widetilde\; \backslash to\; X$ of a topological space ''X'' is helpful in understanding its fundamental group in several ways: first, $\backslash pi\_1(X)$ identifies with the group of deck transformations, i.e., the group of homeomorphisms $\backslash varphi\; :\; \backslash widetilde\; \backslash to\; \backslash widetilde$ that commute with the map to ''X'', i.e., $p\; \backslash circ\; \backslash varphi\; =\; p.$ Another relation to the fundamental group is that $\backslash pi\_1(X,\; x)$ can be identified with the fiber $p^(x).$ For example, the map :$p:\; \backslash mathbb\; \backslash to\; S^1,\; t\; \backslash mapsto\; \backslash exp(2\; \backslash pi\; i\; t)$ (or, equivalently, $\backslash pi:\; \backslash mathbb\; \backslash to\; \backslash mathbb\; /\; \backslash mathbb,\backslash \; t\; \backslash mapsto/math>)\; is\; a\; universal\; covering.\; The\; deck\; transformations\; are\; the\; maps$ t\; \backslash mapsto\; t\; +\; n$for$ n\; \backslash in\; \backslash mathbb.$This\; is\; in\; line\; with\; the\; identification$ p^(1)\; =\; \backslash mathbb,$in\; particular\; this\; proves\; the\; above\; claim$ \backslash pi\_1\backslash left(S^1\backslash right)\; \backslash cong\; \backslash mathbb.$Any\; path\; connected,\; locallypath\; connectedand\; locally\; simply\; connected\; topological\; space\; \text{'}\text{'}X\text{'}\text{'}\; admits\; a\; universal\; covering.\; An\; abstract\; construction\; proceeds\; analogously\; to\; the\; fundamental\; group\; by\; taking\; pairs\; (\text{'}\text{'}x\text{'}\text{'},\gamma ),\; where\; \text{'}\text{'}x\text{'}\text{'}\; is\; a\; point\; in\; \text{'}\text{'}X\text{'}\text{'}\; and\; \gamma \; is\; a\; homotopy\; class\; of\; paths\; from\; \text{'}\text{'}x\text{'}\text{'}$

Lie groups

Let ''G'' be a connected, simply connected compact Lie group, for example, the special unitary group 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 (often semisimple), *''K'' is a maximal compact subgroup of ''G'' * Γ is a discrete countable 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 SL(2, Z). From the explicit realization, it also follows that the universal covering space of a path connected topological group ''H'' is again a path connected topological group ''G''. Moreover, the covering map is a continuous open homomorphism of ''G'' onto ''H'' with kernel Γ, a closed discrete normal subgroup of ''G'': :$1\; \backslash to\; \backslash Gamma\; \backslash to\; G\; \backslash to\; H\; \backslash to\; 1.$ 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 π

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 $f^(b)$ 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 $F\; =\; f^(b).$ The great importance of fibrations to the computation of homotopy groups stems from a long exact sequence :$\backslash dots\; \backslash to\; \backslash pi\_2(B)\; \backslash to\; \backslash pi\_1(F)\; \backslash to\; \backslash pi\_1(E)\; \backslash to\; \backslash pi\_1(B)\; \backslash to\; \backslash pi\_0(F)\; \backslash to\; \backslash pi\_0(E)$ provided that ''B'' is path-connected. The term $\backslash pi\_2(B)$ 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 $\backslash pi\_1.$ If ''E'' happens to be path-connected and simply connected, this sequence reduces to an isomorphism :$\backslash pi\_1(B)\; \backslash cong\; \backslash pi\_0(F)$ 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 :$\backslash pi\_1(E)\; \backslash cong\; \backslash pi\_1(B).$ What is more, the sequence can be continued at the left with the higher homotopy groups $\backslash pi\_n$ 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 $\backslash mathrm(n),$ with $n\; \backslash geq\; 2.$ This group acts transitively on the unit sphere $S^$ inside $\backslash mathbb\; C^n\; =\; \backslash mathbb\; R^.$ The stabilizer of a point in the sphere is isomorphic to $\backslash mathrm(n-1).$ It then can be shown that this yields a fiber sequence :$\backslash mathrm(n-1)\; \backslash to\; \backslash mathrm(n)\; \backslash to\; S^.$ Since $n\; \backslash geq\; 2,$ the sphere $S^$ has dimension at least 3, which implies :$\backslash pi\_1\backslash left(S^\backslash right)\; \backslash cong\; \backslash pi\_2\backslash left(S^\backslash right)\; =\; 1.$ The long exact sequence then shows an isomorphism :$\backslash pi\_1(\backslash mathrm(n))\; \backslash cong\; \backslash pi\_1(\backslash mathrm(n\; -\; 1)).$ Since $\backslash mathrm(1)$ is a single point, so that $\backslash pi\_1(\backslash mathrm(1))$ is trivial, this shows that $\backslash mathrm(n)$ 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 and the associated root system. Specifically, let $T$ be a maximal torus in a connected compact Lie group $K,$ and let $\backslash mathfrak\; t$ be the Lie algebra of $T.$ The exponential map :$\backslash exp\; :\; \backslash mathfrak\; t\; \backslash to\; T$ is a fibration and therefore its kernel $\backslash Gamma\; \backslash subset\; \backslash mathfrak\; t$ identifies with $\backslash pi\_1(T).$ The map :$\backslash pi\_1(T)\; \backslash to\; \backslash pi\_1(K)$ can be shown to be surjective with kernel given by the set ''I'' of integer linear combination of coroots. This leads to the computation :$\backslash pi\_1(K)\; \backslash cong\; \backslash Gamma\; /\; I.$ 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, its fundamental group can be described explicitly in terms of generators and relations. If ''X'' is a connected 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 π

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 $\backslash pi\_n(X)$, which are defined to consist of homotopy classes of (basepoint-preserving) maps from $S^n$ to ''X''. For example, the Hurewicz theorem implies that the ''n''-th homotopy group of the ''n''-sphere is (for all $n\; \backslash ge\; 1$) are :$\backslash pi\_n(S^n)\; =\; \backslash Z.$ As was mentioned in the above computation of $\backslash pi\_1$ 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 ''X'', endowed with the compact open topology, is known as the loop space, denoted $\backslash Omega\; X.$ The fundamental group of ''X'' is in bijection with the set of path components of its loop space: :$\backslash pi\_1(X)\; \backslash cong\; \backslash pi\_0(\backslash Omega\; X).$

Fundamental groupoid

The ''fundamental groupoid'' is a variant of the fundamental group that is useful in situations where the choice of a base point $x\_0\; \backslash in\; X$ is undesirable. It is defined by first considering the category of paths in $X,$ i.e., continuous functions :$\backslash gamma\; \backslash colon,\; r\backslash to\; X$, 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 $\backslash gamma,\; \backslash gamma\text{'}$ with the same endpoints and length ''r'', resp. ''r are considered equivalent if there exist real numbers $u,v\; \backslash geqslant\; 0$ such that $r\; +\; u\; =\; r\text{'}\; +\; v$ and $\backslash gamma\_u,\; \backslash gamma\text{'}\_v\; \backslash colon,\; r\; +\; u\backslash to\; X$ are homotopic relative to their end points, where $\backslash gamma\_u\; (t)\; =\; \backslash begin\; \backslash gamma(t),\; \&\; t\; \backslash in,\; r\backslash \backslash \; \backslash gamma(r),\; \&\; t\; \backslash in,\; r\; +\; u\backslash end$ The category of paths up to this equivalence relation is denoted $\backslash Pi\; (X).$ Each morphism in $\backslash Pi\; (X)$ is an isomorphism, with inverse given by the same path traversed in the opposite direction. Such a category is called a groupoid. It reproduces the fundamental group since :$\backslash pi\_1(X,\; x\_0)\; =\; \backslash mathrm\_(x\_0,\; x\_0)$. 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 spaces. Representations of the fundamental group have a very geometric significance: any ''local system'' (i.e., a sheaf $\backslash mathcal\; F$ 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 $\backslash mathcal\; F|\_U\; =\; \backslash Q^n$) gives rise to the so-called monodromy representation, a representation of the fundamental group on an ''n''-dimensional $\backslash Q$-vector space. Conversely, any such representation on a path-connected space ''X'' arises in this manner. This equivalence of categories between representations of $\backslash pi\_1(X)$ and local systems is used, for example, in the study of differential equations, such as the Knizhnik–Zamolodchikov equations.

Étale fundamental group

In algebraic geometry, the so-called étale fundamental group is used as a replacement for the fundamental group. Since the Zariski topology on an algebraic variety or scheme ''X'' is much coarser than, say, the topology of open subsets in $\backslash R^n,$ it is no longer meaningful to consider continuous maps from an interval to ''X''. Instead, the approach developed by Grothendieck consists in constructing $\backslash pi\_1^\backslash text$ 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. Also, the étale fundamental group of a field is its (absolute) Galois group. On the other hand, for smooth varieties ''X'' over the complex numbers, the étale fundamental group retains much of the information inherent in the classical fundamental group: the former is the profinite completion of the latter.

Fundamental group of algebraic groups

The fundamental group of a root system 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 ''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, $\backslash pi\_1$ 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''.

See also

* Orbifold fundamental group

Notes

References

* * * * * * * * * * * * Peter Hilton and Shaun Wylie, ''Homology Theory'', Cambridge University Press (1967) arning: these authors use ''contrahomology'' for [[cohomology] * * * * * * [[Deane Montgomery and Leo Zippin, ''Topological Transformation Groups'', Interscience Publishers (1955) * * * * * * *

External links

* * Dylan G.L. Allegretti

''Simplicial Sets and van Kampen's Theorem''

A discussion of the fundamental groupoid of a topological space and the fundamental groupoid of a simplicial set

Animations to introduce fundamental group by Nicolas Delanoue

Sets of base points and fundamental groupoids: mathoverflow discussion

{{Commons category Category:Algebraic topology Category:Homotopy theory