Fundamental group

For the fundamental group of a factor see von Neumann algebra. In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the

In mathematics, the fundamental group is one of the basic concepts of algebraic topology. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic Associated with every point of a topological space there is a fundamental group that conveys information about the 1-dimensional structure of the portion of the space surrounding the given point. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element The fundamental group is the first homotopy group. In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional

1 Intuition and definition
2 Examples
3 Functoriality
4 Fibrations
5 Relationship to first homology group
6 Universal covering space
- 6.1 Examples
7 Edge-path group of a simplicial complex
8 Realizability
9 Related concepts
- 9.1 Fundamental groupoid
10 See also
11 Link
12 References

Intuition and definition

Before giving a precise definition of the fundamental group, we try to describe the general idea in non-mathematical terms. Take some space, and some point in it, and consider all the loops at this point — paths which start at this point, wander around as much as they like 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. The set of all the loops with this method of combining them is the fundamental group, except that for technical reasons it is necessary to consider two loops to be the same if one can be deformed into the other without breaking.

For the precise definition, let X be a topological space, and let x₀ be a point of X. We are interested in the set of continuous functions f : [0,1] → X with the property that f(0) = x₀ = f(1). In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function These functions are called loops with base point x₀. In Mathematics, a path in a Topological space X is a continuous map f from the Unit interval I = to Any two such loops, say f and g, are considered equivalent if there is a continuous function h : [0,1] × [0,1] → X with the property that, for all t in [0,1], h(t,0) = f(t), h(t,1) = g(t) and h(0,t) = x₀ = h(1,t). Such an h is called a homotopy from f to g, and the corresponding equivalence classes are called homotopy classes. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X

The product f ∗ g of two loops f and g is defined by setting (f ∗ g)(t) = f(2t) if t is in [0,1/2] and (f ∗ g)(t) = g(2t − 1) if t is in [1/2,1]. The loop f ∗ g thus 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

π 1 (X, x 0),

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 if the space X is path-connected. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece" 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. An isomorphism class is a collection of Mathematical objects Isomorphic with a certain mathematical object

Examples

In many spaces, such as Rⁿ, or any convex subset of Rⁿ, there is only one homotopy class of loops, and the fundamental group is therefore trivial, i. 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 e. ({0},+). A path-connected space with a trivial fundamental group is said to be simply connected. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be

A more interesting example is provided by the circle. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the It turns out that 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 $(\mathbb{Z} , +)$ , the additive group of integers. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk–Ulam theorem in dimension 2. In Mathematics, the Brouwer fixed point theorem is an important Fixed point theorem that applies to finite-dimensional spaces and which forms the basis for several The Borsuk–Ulam theorem states that any Continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of Antipodal points

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. The term winding number may also refer to the Rotation number of an Iterated map.

Unlike the homology groups and higher homotopy groups associated to a topological space, the fundamental group need not be abelian. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the For example, the fundamental group of a graph G is a free group. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be Here the rank of the free group is equal to 1 − χ(G): one minus the Euler characteristic of G, when G is connected. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant A somewhat more sophisticated example of a space with a non-abelian fundamental group is the complement of a trefoil knot in R³. In Knot theory, the trefoil knot is the simplest nontrivial knot.

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 homomorphism, called the induced homomorphism, is written as π(f) or, more commonly,

$f_*\colon \pi_1(X, x_0) \to \pi_1(Y,y_0).$

We thus obtain a functor from the category of topological spaces with base point to the category of groups. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, an induced homomorphism is a structure-preserving map between a pair of objects that is derived in a canonical way from another map between another pair of objects In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, the category of topological spaces, often denoted Top, is the category whose objects are Topological spaces and whose In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such

It turns out that this functor cannot distinguish maps which are homotopic relative 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_*. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical As a consequence, two homotopy equivalent path-connected spaces have isomorphic fundamental groups:

$X \simeq Y \Rightarrow \pi_1(X,x_0) \cong \pi_1(Y,y_0).$

The fundamental group functor takes products to products and coproducts to coproducts. In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as In Mathematics, one can often define a direct product of objectsalready known giving a new one In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological That is, if X and Y are path connected, then

$\pi_1 (X\times Y) \cong \pi_1(X) \times \pi_1(Y)$

and

$\pi_1 (X\vee Y) \cong \pi_1(X) * \pi_1(Y).$

(In the latter formula, $\vee$ denotes the wedge sum of topological spaces, and * the free product of groups. In Topology, the wedge sum (sometimes wedge product, though not to be confused with the Exterior product, which also shares this terminology is a "one-point In Abstract algebra, the free product of groups constructs a group from two or more given ones ) 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 pushouts along inclusions to pushouts. In Mathematics, the Seifert – van Kampen theorem of Algebraic topology, sometimes just called van Kampen's theorem, expresses the In Category theory, a branch of Mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square) is the

Fibrations

Main article: Fibration

A generalization of a product of spaces is given by a fibration,

$F \rightarrow E \rightarrow B.$

Here the total space E is a sort of "twisted product" of the base space B and the fiber F. In Mathematics, especially Algebraic topology, a fibration is a continuous mapping pE\to B\ satisfying the In Mathematics, especially Algebraic topology, a fibration is a continuous mapping pE\to B\ satisfying the In Mathematics, especially Algebraic topology, a fibration is a continuous mapping pE\to B\ satisfying the In Mathematics, especially Algebraic topology, a fibration is a continuous mapping pE\to B\ satisfying the In Mathematics, especially Algebraic topology, a fibration is a continuous mapping pE\to B\ satisfying the In general the fundamental groups of B, E and F are terms in a long exact sequence involving higher homotopy groups. In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional 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
π₁(B) and π₁(F) are isomorphic if E is contractible

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. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is 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 π(X,x₀) to the homology group H₁(X). If X is path-connected, then this homomorphism is surjective and its kernel is the commutator subgroup of π(X,x₀), and H₁(X) is therefore isomorphic to the abelianization of π(X,x₀). In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every 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 In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup This is a special case of the Hurewicz theorem of algebraic topology. In Mathematics, the Hurewicz theorem is a basic result of Algebraic topology, connecting Homotopy theory with Homology theory via a map known

Universal covering space

Main article: 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) acts freely by deck transformations with quotient space X. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying 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 group SU_n, and let Γ be a finite subgroup of G. In Mathematics, a compact ( topological, often understood group is a Topological group whose Topology is Compact. Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n Then the homogeneous space X=G/Γ has fundamental group Γ, which acts by right multiplication on the universal covering space G. In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group 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 Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a Lie algebra is semisimple if it is a Direct sum of Simple Lie algebras i In Mathematics, the term torsion-free may refer to several unrelated notions In abstract algebra a group is torsion -free if the only element of finite

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). In Mathematics, a Topological space X is contractible if the Identity map on X is Null-homotopic, i 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 In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

As an example take G=SL₂(R), K=SO₂ and Γ any torsion-free congruence subgroup of the modular group SL₂(Z). In Mathematics, a congruence subgroup of a Matrix group with Integer entries is a Subgroup defined by congruence conditions on the entries In Mathematics, the modular group Γ is a fundamental object of study in Number theory, Geometry, algebra, and many other areas of advanced

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. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the Moreover the covering map is a continuous open homomorphism of G onto H with kernel Γ, a closed discrete normal subgroup of G:

$1 \rightarrow \Gamma \rightarrow G \rightarrow H \rightarrow 1.$

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 Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the In particular π₁(H) = Γ is an Abelian group; this can also easily be seen directly without using covering spaces. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the The group G is called the universal covering group of H. In Mathematics, a covering group of a Topological group H is a Covering space G of H such that G is a topological

Edge-path group of a simplicial complex

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. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments 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. In Mathematics, a simplicial set is a construction in categorical Homotopy theory which is a purely algebraic model of the notion of a " Well-behaved Since it depends only on the 2-skeleton X² of X (i. This article is not about the Topological skeleton concept of Computer graphics In Mathematics, particularly in Algebraic topology 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. In Mathematics, one method of defining a group is by a presentation. 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 edges of X not occurring in T and relations the edge-equivalences corresponding to triangles in X containing one or more edge not in T. This article is not about the Topological skeleton concept of Computer graphics In Mathematics, particularly in Algebraic topology A similar result holds if T is replaced by any simply connected—in particular contractible—subcomplex of X. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be In Mathematics, a Topological space X is contractible if the Identity map on X is Null-homotopic, i 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. In Mathematics, one method of defining a group is by a presentation. It is also one of the classical methods used for topological surfaces, which are classified by their fundamental groups. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold.

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 Cech and Leray and explicitly appeared as a remark in a paper by Weil (1960); various other authors such as L. Jean Leray ( 7 November 1906 &ndash 10 November 1998) was a French Mathematician, who worked on both Partial differential André Weil should not be confused with two other mathematicians with similar names Hermann Weyl (1885-1955 who made substantial contributions 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. In Mathematics, the nerve of an open covering is a construction in Topology, of an Abstract simplicial complex from an Open covering of a

Realizability

Every group can be realized as the fundamental group of a connected CW-complex of dimension 2 (or higher). In Topology, a CW complex is a type of Topological space introduced by J 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). In Mathematics, one method of defining a group is by a presentation. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. 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. In 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

Related concepts

The fundamental group measures the 1-dimensional hole structure of a space. For studying "higher-dimensional holes", the homotopy groups are used. In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional 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. In Mathematics, the space of loops or (free loop space of a Topological space X is the space of loops from the Unit circle