Fundamental group
Encyclopedia
In mathematics
, more specifically algebraic topology
, the fundamental group (defined by Henri Poincaré
in his article Analysis Situs, published in 1895) is a group
associated to any given pointed topological space
that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. Intuitively, it records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest of the homotopy group
s.
Fundamental groups can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the associated universal covering space. Its abelianisation can be identified with the first homology group of the space. When the topological space is homeomorphic to a simplicial complex
, its fundamental group can be described explicitly in terms of generators and relations.
Historically, the concept of fundamental group first emerged in the theory of Riemann surface
s, in the work of Bernhard Riemann
, Henri Poincaré
and Felix Klein
, where it describes the monodromy
properties of complex functions, as well as providing a complete topological classification of closed surfaces.
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.
with base point x0.
Now the fundamental group of X with base point x is this set modulo homotopy
equipped with the group multiplication defined by (f ∗ g)(t) := f(2t) if 0 ≤ t ≤ 1/2 and (f ∗ g)(t) := g(2t − 1) if 1/2 ≤ t ≤ 1. Thus the loop f ∗ g first follows the loop f with "twice the speed" and then follows g with twice the speed. The product of two homotopy classes of loops [f] and [g] is then defined as [f ∗ g], and it can be shown that this product does not depend on the choice of representatives.
With the above product, the set of all homotopy classes of loops with base point x0 forms the fundamental group of X at the point x0 and is denoted
or simply π(X, x0). The identity element is the constant map at the basepoint, and the inverse of a loop f is the loop g defined by g(t) = f(1 − t). That is, g follows f backwards.
Although the fundamental group in general depends on the choice of base point, it turns out that, up to
isomorphism
, this choice makes no difference as long as the space X is path-connected. For path-connected spaces, therefore, we can write π1(X) instead of π1(X, x0) without ambiguity whenever we care about the isomorphism class
only.
In Euclidean space Rn, or any convex subset
of Rn, there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element. A path-connected space with a trivial fundamental group is said to be simply connected
.
Infinite cyclic fundamental group.
The circle
. Each homotopy class consists of all loops which 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 which winds around m times and another that winds around n times is a loop which winds around m + n times. So the fundamental group of the circle is isomorphic to
, the additive group of integer
s. This fact can be used to give proofs of the Brouwer fixed point theorem
and the Borsuk–Ulam theorem
in dimension 2.
Since the fundamental group is a homotopy invariant, the theory of the winding number
for the complex plane minus one point is the same as for the circle.
Free groups of higher rank: Graphs.
Unlike the homology groups and higher homotopy groups associated to a topological space, the fundamental group need not be abelian
. For example, the fundamental group of the figure eight is the free group
on two letters. More generally, the fundamental group of any graph
is a free group
. If the graph G is connected, then the rank of the free group is equal to 1 − χ(G): one minus the Euler characteristic
of G.
Knot theory. A somewhat more sophisticated example of a space with a non-abelian fundamental group is the complement of a trefoil knot
in R3.
, called the induced homomorphism
, is written as π(f) or, more commonly,
We thus obtain a functor
from the category of topological spaces with base point to the category of groups
.
It turns out that this functor cannot distinguish maps which are homotopic relative to the base point: if f and g : X → Y are continuous maps with f(x0) = g(x0) = y0, and f and g are homotopic relative to {x0}, then f* = g*. As a consequence, two homotopy equivalent path-connected spaces have isomorphic fundamental groups:
As an important special case, if X is path-connected then any two basepoints give isomorphic fundamental groups, with isomorphism given by a choice of path between the given basepoints.
The fundamental group functor takes products
to products
and coproduct
s to coproducts. That is, if X and Y are path connected, then
and
(In the latter formula,
denotes the wedge sum
of topological spaces, and * the free product
of groups.) Both formulas generalize to arbitrary products. Furthermore the latter formula is a special case of the Seifert–van Kampen theorem
which states that the fundamental group functor takes pushout
s along inclusions to pushouts.
,
Here the total space
E is a sort of "twisted product" of the base space
B and the fiber
F.
In general the fundamental groups of B, E and F
are terms in a long exact sequence involving higher homotopy groups
. When all the spaces are connected, this has the following consequences for the fundamental groups:
The latter is often applied to the situation E = path space
of B, F = loop space of B or B = classifying space
BG of a topological group
G, E = universal G-bundle EG.
is the commutator subgroup
of π1(X, x0), and H1(X) is therefore isomorphic to the abelianization of π1(X, x0). This is a special case of the Hurewicz theorem
of algebraic topology.
freely by deck transformations with quotient space
X. This space can be constructed analogously to the fundamental group by taking pairs (x, γ),
where x is a point in X and γ is a homotopy class of paths from x0 to x and the action of π(X, x0) is by concatenation of paths. It is uniquely determined as a covering space.
SUn, 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
In this case the fundamental group is Γ and the universal covering space G/K is actually contractible (by the Cartan decomposition
for Lie group
s).
As an example take G = SL2(R), K = SO2 and Γ any torsion-free congruence subgroup
of the modular group
SL2(Z).
An even simpler example is given by G = R (so that K is trivial) and Γ = Z: in this case X=R/Z = S1.
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:
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
; 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.
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
X2 of X (i.e. the vertices, edges and triangles of X), the groups π1(|X|,v) and π1(|X2|, 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
surface
s, 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 γu from v to u. The points (w,γ) and (u, γu) 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 Čech
and Leray
and explicitly appeared as a remark in a paper by ; various other authors such as L. Calabi, W-T. Wu and N. 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
.
s are used. The elements of the n-th homotopy group of X are homotopy classes of (basepoint-preserving) maps from Sn to X.
The set of loops at a particular base point can be studied without regarding homotopic loops as equivalent. This larger object is the loop space.
For topological groups, a different group multiplication may be assigned to the set of loops in the space, with pointwise multiplication rather than concatenation. The resulting group is the loop group
.
, the fundamental groupoid of the space.
There are also similar notions of fundamental group for algebraic varieties (the étale fundamental group
) and for orbifold
s (the orbifold fundamental group).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, more specifically algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which 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 up to homotopy equivalence.Although algebraic topology...
, the fundamental group (defined by Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
in his article Analysis Situs, published in 1895) is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
associated to any given pointed topological space
Pointed space
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. Intuitively, it records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest of the homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s.
Fundamental groups can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the associated universal covering space. Its abelianisation can be identified with the first homology group of the space. When the topological space is homeomorphic to a simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
, its fundamental group can be described explicitly in terms of generators and relations.
Historically, the concept of fundamental group first emerged in the theory of Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s, in the work of Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
, Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
and Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
, where it describes the monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
properties of complex functions, as well as providing a complete topological classification of closed surfaces.
Intuition
Start with a space (e.g. 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.
Definition
Let X be a topological space, and let x0 be a point of X. We are interested in the following set of continuous functions called loopsPath (topology)
In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
with base point x0.
Now the fundamental group of X with base point x is this set modulo homotopy
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
equipped with the group multiplication defined by (f ∗ g)(t) := f(2t) if 0 ≤ t ≤ 1/2 and (f ∗ g)(t) := g(2t − 1) if 1/2 ≤ t ≤ 1. Thus the loop f ∗ g first follows the loop f with "twice the speed" and then follows g with twice the speed. The product of two homotopy classes of loops [f] and [g] is then defined as [f ∗ g], and it can be shown that this product does not depend on the choice of representatives.
With the above product, the set of all homotopy classes of loops with base point x0 forms the fundamental group of X at the point x0 and is denoted
or simply π(X, x0). The identity element is the constant map at the basepoint, and the inverse of a loop f is the loop g defined by g(t) = f(1 − t). That is, g follows f backwards.
Although the fundamental group in general depends on the choice of base point, it turns out that, up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
, this choice makes no difference as long as the space X is path-connected. For path-connected spaces, therefore, we can write π1(X) instead of π1(X, x0) without ambiguity whenever we care about the isomorphism class
Isomorphism class
An isomorphism class is a collection of mathematical objects isomorphic to each other.Isomorphism classes are often defined if the exact identity of the elements of the set is considered irrelevant, and the properties of the structure of the mathematical object are studied. Examples of this are...
only.
Examples
Trivial fundamental group.In Euclidean space Rn, or any convex subset
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
of Rn, there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element. A path-connected space with a trivial fundamental group is said to be simply connected
Simply connected space
In topology, a topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question .If a space is not simply connected, it is convenient...
.
Infinite cyclic fundamental group.
The circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
. Each homotopy class consists of all loops which 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 which winds around m times and another that winds around n times is a loop which winds around m + n times. So the fundamental group of the circle is isomorphic to
, the additive group of integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s. This fact can be used to give proofs of the Brouwer fixed point theorem
Brouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
and the Borsuk–Ulam theorem
Borsuk–Ulam theorem
In mathematics, the Borsuk–Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point....
in dimension 2.
Since the fundamental group is a homotopy invariant, the theory of the winding number
Winding number
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
for the complex plane minus one point is the same as for the circle.
Free groups of higher rank: Graphs.
Unlike the homology groups and higher homotopy groups associated to a topological space, the fundamental group need not be abelian
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
. For example, the fundamental group of the figure eight is the free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
on two letters. More generally, the fundamental group of any graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
is a free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
. If the graph G is connected, then the rank of the free group is equal to 1 − χ(G): one minus the Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
of G.
Knot theory. A somewhat more sophisticated example of a space with a non-abelian fundamental group is the complement of a trefoil knot
Trefoil knot
In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop...
in R3.
Functoriality
If f : X → Y is a continuous map, x0 ∈ X and y0 ∈ Y with f(x0) = y0, then every loop in X with base point x0 can be composed with f to yield a loop in Y with base point y0. This operation is compatible with the homotopy equivalence relation and with composition of loops. The resulting group homomorphismGroup homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
, called the induced homomorphism
Induced homomorphism (fundamental group)
In mathematics, especially in the area of topology known as algebraic topology, the induced homomorphism is a group homomorphism related to the study of the fundamental group.-Definition:...
, is written as π(f) or, more commonly,
We thus obtain a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
from the category of topological spaces with base point to the category of groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
.
It turns out that this functor cannot distinguish maps which are homotopic relative to the base point: if f and g : X → Y are continuous maps with f(x0) = g(x0) = y0, and f and g are homotopic relative to {x0}, then f* = g*. As a consequence, two homotopy equivalent path-connected spaces have isomorphic fundamental groups:
As an important special case, if X is path-connected then any two basepoints give isomorphic fundamental groups, with isomorphism given by a choice of path between the given basepoints.
The fundamental group functor takes products
Product (category theory)
In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
to products
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
and coproduct
Coproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
s to coproducts. That is, if X and Y are path connected, then
and
(In the latter formula,
denotes the wedge sumWedge 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 the wedge sum of X and Y is the quotient of the disjoint union of X and Y by the identification x0 ∼ y0:X\vee Y = \;/ \sim,\,where ∼ is the...
of topological spaces, and * the free product
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
of groups.) Both formulas generalize to arbitrary products. Furthermore the latter formula is a special case of the Seifert–van Kampen theorem
Seifert–van Kampen theorem
In mathematics, the Seifert-van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space X, in terms of the fundamental groups of two open, path-connected subspaces U and V that cover X...
which states that the fundamental group functor takes pushout
Pushout (category theory)
In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain: it is the colimit of the span X \leftarrow Z \rightarrow Y.The pushout is the...
s along inclusions to pushouts.
Fibrations
A generalization of a product of spaces is given by a fibrationFibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
,
Here the total space
Fibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
E is a sort of "twisted product" of the base space
Fibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
B and the fiber
Fibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
F.
In general the fundamental groups of B, E and F
are terms in a long exact sequence involving higher homotopy groups
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
. When all the spaces are connected, this has the following consequences for the fundamental groups:
- π1(B) and π1(E) are isomorphic if F is simply connected
- πn+1(B) and πn(F) are isomorphic if E is contractible
The latter is often applied to the situation E = path space
Path space
In mathematics, the term path space refers to any topological space of paths from one specified set into another. In particular, it may refer to* the classical Wiener space of continuous paths;* the Skorokhod space of càdlàg paths....
of B, F = loop space of B or B = classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
BG of a topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
G, E = universal G-bundle EG.
Relationship to first homology group
The fundamental groups of a topological space X are related to its first singular homology group, because a loop is also a singular 1-cycle. Mapping the homotopy class of each loop at a base point x0 to the homology class of the loop gives a homomorphism from the fundamental group π1(X, x0) to the homology group H1(X). If X is path-connected, then this homomorphism is surjective and its kernelKernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
is the commutator subgroup
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....
of π1(X, x0), and H1(X) is therefore isomorphic to the abelianization of π1(X, x0). This is a special case of the Hurewicz theorem
Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism...
of algebraic topology.
Universal covering space
If X is a topological space that is path connected, locally path connected and locally simply connected, then it has a simply connected universal covering space on which the fundamental group π(X,x0) actsGroup action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
freely by deck transformations with quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
X. This space can be constructed analogously to the fundamental group by taking pairs (x, γ),
where x is a point in X and γ is a homotopy class of paths from x0 to x and the action of π(X, x0) is by concatenation of paths. It is uniquely determined as a covering space.
Examples
Let G be a connected, simply connected compact Lie group, for example the special unitary groupSpecial unitary group
The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
SUn, and let Γ be a finite subgroup of G. Then the homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
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 groupLie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
(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
Cartan decomposition
The Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition of matrices.- Cartan involutions on Lie algebras :...
for Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s).
As an example take G = SL2(R), K = SO2 and Γ any torsion-free congruence subgroup
Congruence subgroup
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2x2 integer matrices of determinant 1, such that the off-diagonal entries are even.An importance class of congruence...
of the modular group
Modular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
SL2(Z).
An even simpler example is given by G = R (so that K is trivial) and Γ = Z: in this case X=R/Z = S1.
From the explicit realization, it also follows that the universal covering space of a path connected topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
H is again a path connected topological group G. Moreover the covering map is a continuous open homomorphism of G onto H with kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
Γ, a closed discrete normal subgroup of G:
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
Center (group theory)
In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
of G. In particular π1(H) = Γ is an Abelian group
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
; this can also easily be seen directly without using covering spaces. The group G is called the universal covering group
Universal covering group
In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p : G → H is a continuous group homomorphism. The map p is called the covering homomorphism...
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.
Edge-path group of a simplicial complex
If X is a connectedConnected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
, 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
N-skeleton
In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex refers to the subspace Xn that is the union of the simplices of X of dimensions m ≤ n...
X2 of X (i.e. the vertices, edges and triangles of X), the groups π1(|X|,v) and π1(|X2|, v) are isomorphic.
The edge-path group can be described explicitly in terms of generators and relations. If T is a maximal spanning tree
Spanning tree
Spanning tree can refer to:* Spanning tree , a tree which contains every vertex of a more general graph* Spanning tree protocol, a protocol for finding spanning trees in bridged networks...
in the 1-skeleton
N-skeleton
In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex refers to the subspace Xn that is the union of the simplices of X of dimensions m ≤ n...
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
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
s, 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 γu from v to u. The points (w,γ) and (u, γu) 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 Čech
Eduard Cech
Eduard Čech was a Czech mathematician born in Stračov, Bohemia . His research interests included projective differential geometry and topology. In 1921–1922 he collaborated with Guido Fubini in Turin...
and Leray
Leray
Leray or LeRay may refer to*Danielle Leray*David Leray*Francis Xavier Leray*Jean Leray , French mathematician*Marie-Pierre Leray...
and explicitly appeared as a remark in a paper by ; various other authors such as L. Calabi, W-T. Wu and N. 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
Nerve of an open covering
In mathematics, the nerve of an open covering is a construction in topology, of an abstract simplicial complex from an open covering of a topological space X.The notion of nerve was introduced by Pavel Alexandrov....
.
Realizability
- Every group can be realized as the fundamental group of a connectedConnected spaceIn topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
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 compactCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, 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 groupFree abelian groupIn abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
of rank 4 or higher can be realized as the fundamental group of a manifold of dimension 3 or less.
Related concepts
The fundamental group measures the 1-dimensional hole structure of a space. For studying "higher-dimensional holes", the homotopy groupHomotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s are used. The elements of the n-th homotopy group of X are homotopy classes of (basepoint-preserving) maps from Sn to X.
The set of loops at a particular base point can be studied without regarding homotopic loops as equivalent. This larger object is the loop space.
For topological groups, a different group multiplication may be assigned to the set of loops in the space, with pointwise multiplication rather than concatenation. The resulting group is the loop group
Loop group
In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. Specifically, letLG \,denote the space of continuous mapsS^1 \to G...
.
Fundamental groupoid
Rather than singling out one point and considering the loops based at that point up to homotopy, one can also consider all paths in the space up to homotopy (fixing the initial and final point). This yields not a group but a groupoidGroupoid
In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...
, the fundamental groupoid of the space.
See also
- Homotopy groupHomotopy groupIn mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
, generalization of fundamental group
There are also similar notions of fundamental group for algebraic varieties (the étale fundamental group
Étale fundamental group
The étale fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.-Topological analogue:In algebraic topology, the fundamental group\pi_1...
) and for orbifold
Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
s (the orbifold fundamental group).
External links
- Dylan G.L. Allegretti, Simplicial Sets and van Kampen's Theorem (An elementary discussion of the fundamental groupoid of a topological space and the fundamental groupoid of a simplicial set).
- Animations to introduce to the fundamental group by Nicolas Delanoue