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Encyclopedia > Fundamental group

In mathematics, the fundamental group is one of the basic concepts of algebraic topology. 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. The fundamental group is the first homotopy group. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. ...

Contents

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). These functions are called loops with base point x0. 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. The product fg of two loops f and g is defined by setting (fg)(t) = f(2t) if t is in [0,1/2] and (fg)(t) = g(2t − 1) if t is in [1/2,1]. The loop fg 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 [fg], and it can be shown that this product does not depend on the choice of representatives. With this 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 π1(X,x0), 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. In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X f : I → X. The initial point of the path is f(0) and the terminal point is f(1). ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...


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. For path-connected spaces, therefore, we can write π(X) instead of π(X,x0) without ambiguity whenever we care about the isomorphy class only. Look up Up to on Wiktionary, the free dictionary 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 mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...


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.e. ({0},+). A path-connected space with a trivial fundamental group is said to be simply connected. Look up Convex set in Wiktionary, the free dictionary. ... In topology, a geometrical object or space is called simply connected if it is path-connected and every path between two points can be continuously transformed into every other. ...


A more interesting example is provided by the circle. 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. This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk–Ulam theorem in dimension 2. Circle illustration In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... The integers are commonly denoted by the above symbol. ... In mathematics, the Brouwer fixed point theorem states that every continuous function from the closed unit ball D n to itself has a fixed point. ... The Borsuk–Ulam theorem states that any continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. ...


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. A point z0 and a curve C In mathematics, the winding number is a topological invariant playing a leading role in complex analysis. ...


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 a graph G is a free group. Here the rank of the free group is equal to 1 − χ(G): one minus the Euler characteristic of G. A somewhat more sophisticated example of a space with a non-Abelian fundamental group is the complement of a trefoil knot in R3. In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... The Cayley graph of the free group on two generators a and b 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... It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ... Categories: Stub | Knot theory ...


Functoriality

If f : XY is a continuous map, x0X and y0Y 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 the composition of loops, and we get a group homomorphism from π(X,x0) to π(Y,y0). This homomorphism is written as π(f) or f*. We thus obtain a functor from the category of topological spaces with base point to the category of groups. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...


It turns out that this functor cannot distinguish maps which are homotopic relative the base point: if f and g : XY 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. An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...


The fundamental group functor takes products to products and coproducts to coproducts. That is, if X and Y are path connected, then π1(X×Y)=π1(X)×π1(Y) and π1(XveeY)=π1(X)*π1(Y). (In the latter formula, vee 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 pushouts along inclusions to pushouts. In topology, the wedge sum is a one-point union of a family of topological spaces. ... In abstract algebra, the free product of groups constructs a group from two or more given ones. ... In mathematics, the Seifert–van Kampen theorem of algebraic topology, sometimes just called Van Kampens theorem, explains the structure of the fundamental group of a topological space X, in terms of those of two overlapping subspaces U and V, under certain hypothesis about connectedness. ... In category theory, a branch of mathematics, the pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain. ...


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 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). This is a special case of the Hurewicz theorem of algebraic topology. In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... 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. ... 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. ... In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory. ...


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). The term connection has various uses, including: An act of connecting two or more physical entities in a physical sense or connecting concepts in memory or imagination, see below Telecommunications circuit switching That which connects, relates or joins: An electrical connection A telecommunication circuit such as a fiber-optic connection... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ...


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. In mathematics, one method of defining a group is by a presentation. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... 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 one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ...


Related concepts

The fundamental group measures the 1-dimensional hole structure of a space. For studying "higher-dimensional holes", the homotopy groups are used. The elements of the n-th homotopy group of X are homotopy classes of (basepoint-preserving) maps from Sn to X. In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. ...


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 loop space of a topological space X is the topological space of continuous maps from the circle S1 to X with the compact-open topology. ...


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 groupoid, the fundamental groupoid of the space. In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. ...


 
 

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