the branch of geometry that studies continuity, described, for example, in terms of limits. The variety of manifestations of continuity in mathematics and the wide spectrum of approaches to its study have split topology into a number of branches, such as general topology and algebraic topology, which differ in content and methodology and are only weakly interconnected.
General topology. General topology is the branch of topology that is primarily concerned with the axiomatic study of continuity. Algebra and general topology are the bases of the modern set theoretical approach to mathematics.
There are many, usually nonequivalent, axiomatizations of continuity. The generally accepted axiomatization is based on the concept of an open set. A topology on a set A– is a set of its subsets, called open, such that (1) the empty set ø and X are open and (2) the union of an arbitrary number of open sets and the intersection of a finite number of open sets are open. A set with a specified topology is called a topological space. In a topological space X it is possible to define all the basic concepts of elementary analysis connected with continuity. For example, a neighborhood of a point x ∈ X is an open set containing x; a set A ⊂ X is closed if its complement X\\A is open; the closure of a set A is the smallest closed set containing A; A is dense in X if the closure of A is X.
Our definitions imply that the sets ∅ and X are both open and closed. A topological space X is connected if ∅ and X are the only sets in X that are both open and closed. In intuitive terms, a connected space consists of a single “piece” and a disconnected space of many.
A subset A of a topological space X has a natural topology consisting of the intersections of A with the open sets of X. The set A with its natural topology is a subset of the space X. A metric space becomes a topological space if we define an open set as a set each of whose points has an ∊-neighborhood—that is, a disk of radius ∊ centered on that point and contained in the set. In particular, every subset of n-dimensional Euclidean space IRn is a topological space. The study of these spaces and of metric spaces is by now a standard part of general topology.
The two major concerns of the study of subspaces of IRn are (1) the study of subsets of IRn of arbitrary complexity subject to restrictions of a general nature (here we might mention the theory of continua, that is, connected, bounded, and closed sets) and (2) the study of ways of embedding in IRn such simple topological spaces as spheres and disks. We note that spheres, for example, can be embedded in IRn in very complex ways.
An open cover of a topological space X is a family of its open sets whose union is X. The topological space X is said to be compact if every open cover of the space contains a finite subcover. The classical Heine-Borel theorem asserts that every closed and bounded subset of IRn is compact. It turns out that all the fundamental theorems of elementary analysis involving bounded open sets, for example, the theorem of Weierstrass to the effect that a continuous function on such a set attains its minimum and its maximum, hold in arbitrary compact topological spaces. This fact explains the important role of compact topological spaces in modern mathematics in general and in the context of existence theorems in particular. The singling out of the class of compact topological spaces was a major achievement of general topology that also proved to be of importance to all of mathematics.
An open cover {Vβ} is a refinement of a cover {Uα} if for every β there is an α such that Vβ ⊂ Uα. A cover {Vβ} is locally finite if every point x ∈ X has a neighborhood that intersects only a finite number of elements of that cover. A topological space is para-compact if each of that cover. A topological space is paracompact if each of its open covers has a locally finite refinement. The class of paracompact spaces is very large and contains, in particular, all metrizable topological spaces, that is, spaces X in which it is possible to introduce a metric ρ such that the topology generated in X by ρ coincides with the topology on A”. There are other classes of topological spaces obtained as a result of imposition of conditions related to compactness.
The multiplicity of an open cover is the largest number k such that there are k elements of the cover with nonempty intersection. The dimension, dim X, of a topological space X is the smallest number n such that each open cover of X has an open refinement of multiplicity ≤ n + 1. This terminology is justified by the fact that in elementary geometric situations dim X coincides with the relevant concept of dimension; for example, dim IRn = n. There are numerical functions defined on a topological space X that differ from dim X but agree with it in simple cases. Such functions are studied in dimension theory, the most geometric part of general topology. Dimension theory is the only theory providing a clear and sufficiently general definition of the concept of a geometric figure and, in particular, of a curve and a surface.
Separation axioms yield important classes of topological spaces. A relevant example is the T2, or Hausdorff, axiom, which requires any two points to have nonintersecting neighborhoods. A Hausdorff space is a topological space with the T2 property. At one time, the topological spaces commonly encountered in mathematics were almost exclusively Hausdorff spaces. Such, for example, are all metric spaces. Non-Hausdorff spaces play an increasingly important role in analysis and geometry.
Completely regular spaces are subspaces of compact Hausdorff spaces. These spaces can be characterized by a separation axiom that requires that for every x0 ∈ X and every closed set F ⊂ X not containing x0 there exist a continuous function g:X → [0,1], which is 0 at x0 and 1 at F.
Locally compact spaces are open subspaces of compact Hausdorff spaces. In the class of Hausdorff spaces they are characterized by a property that every point of such a space has a neighborhood with compact closure. Euclidean spaces are locally compact. A locally compact space can be compactified by the addition of a single point. Thus, adddition of a single point to the plane yields the sphere of complex numbers, and one point compactification of IRn yields the sphere Sn.
A map f: X → Y of the topological space X into a topological space Y is continuous if for every open set V ⊂ Y the set f–1(V) is open in X. A continuous map f is a homeomorphism if it is one-to-one and its inverse f–1 is continuous. Such a map establishes a one-to-one correspondence between the open sets in X and Y that commutes with the operations of union and intersection. It follows that X and Y have the same topological properties, that is, properties formulated in terms of open sets. Therefore, from the point of view of topology, homeomorphic topological spaces (topological spaces X and Y for which there exist at least one homeomorphism X → Y) are regarded as being essentially the same (just as in Euclidean geometry we regard two figures that differ by a motion as being essentially the same). For example, a circle, the boundary of a square, and the boundary of a hexagon are homeomorphic, or “topologically identical.” More generally, any two simple closed curves, that is, closed curves without double points, are homeomorphic. On the other hand, a circle and a straight line are not homeomorphic, since removing a point from a circle does not disconnect the circle but does disconnect the line. Similarly, a line is not homeomorphic with a plane nor is a circle to a figure 8 curve. Removing two points rather than one shows that a circle is not homeomorphic with a plane.
Let {Xα} be a family of topological spaces. Let X be the Cartesian product of the Xα, that is, the set of all families {xα with xα ∈ Xα. For every α, the rule pα({xα}) = xα defines a map pα : X → Xα, called a projection. In general, there are many ways of topologizing X so that the maps pα are continuous. Among these topologies there is a least topology, that is, a topology contained in all the others. The set X with this topology is called the topological product of the Xα and is denoted by the symbol ∏Xα. If the number of spaces is finite, say n, then we write
The open sets of X can be described as unions of finite intersections of sets of the form pα–1(Uα), where Uα is open in Xα. The topological space X is characterized, up to a homeomorphism, by the following universal property: for every family of continuous maps fα : Y → Xα there exists a unique continuous map f: Y → X such that pα ⃘ f = fα for all α. The space IRn is the topological product of n copies of the number line. One of the most important theorems of general topology is the assertion that the topological product of compact topological spaces is compact.
Let X be a topological space, Y a set, and p : X → Y a map of X onto Y. To exemplify this situation, let Y be the quotient set of X corresponding to some equivalence relation on X and let p be the natural map that associates to each x ∈ X its equivalence class. It is natural to ask for a topology on Y in which the map p is continuous. Define V ⊂ Y to be open if f–1(V) ⊂ X is open in X. This topology on Y is of the required kind and is richest in open sets. Y with this topology is called the quotient space of X determined by p. The quotient space Y has the property that a map f: Y → Z is continuous if and only if the map f ⃘ p: X → Z is continuous. A continuous map p: X → Y is called a quotient map if the topological space Y is the quotient space of X determined by p. A continuous map p: X → Y is open if the image p(U) of an open set U ⊂ X is open in Y and closed if the image p(F) of a closed set F ⊂ X is closed in Y. Continuous open maps and continuous closed maps f: X → Y with f(X) = Y are quotient maps.
Let X be a topological space, A a subset of X, and f: X → Y a continuous map. We assume that the topological spaces X and Y are disjoint and topologize their union X ∪ Y by defining an open set in X ∪ Y to be the union of an open set in X and an open set in Y. Next we introduce in X ∪ Y the smallest equivalence relation with the property a ~ f(a) for all a ∈ A. The resulting quotient space is denoted by X ∪ fY and is said to be the result of gluing X to Y along A by means of the continuous map f. This simple and intuitive construction is very useful, since it enables us to make relatively complex spaces out of simple ones. If Y is a single point, then we denote X ∪ fY by X/A and say that it is obtained from X by contracting A to a point. For example, if X is a disk with boundary circle A, then X/A is homeomorphic with a sphere.
Uniform spaces. Topology of uniform spaces is the axiomatic study of uniform continuity. The usual definition of uniform continuity given in analysis carries over directly to mappings of metric spaces. It is therefore natural for metric spaces to provide the point of departure for axiomatizations of uniform continuity. Two axiomatic approaches to uniform continuity have been studied in detail. They are based on the concepts of a proximity and a uniformity, respectively.
Let A and B be subsets of a metric space X. We shall say that A is near B, and write AδB, if for every ∊ > 0 there are points a ∈ A and b ∈ B such that the distance between a and b is < ∊. If we take the basic properties of this relation as axioms, then we arrive at the following definition: A relation δ on the subsets of a set X is called a proximity for X if (1) ∅ δ̅X (δ̅ is the negation of δ), (2) Aδ̅B1 and Aδ̅B2 ⇔ Aδ̅(B1 ∪ B2), (3) {x}δ̅{y} ⇔ x ± y, and (4) if Aδ̅B, then there exists a set Cδ̅B such that Aδ̅(X\\C). A set together with a proximity relation is called a proximity space. A mapping of a proximity space X into a proximity space Y is equicontinuous if the images of near sets in X are near sets in Y. Two proximity spaces are equimorphic if there exists a one-to-one equicontinuous map X → Y whose inverse is equicontinuous. Such a map is called an equimorphism. In the class of proximity spaces, equimorphic spaces are essentially the same. A proximity space, like a metric space, can be made into a topological (Hausdorff) space by defining U ⊂ X to be open if {x}δ̅(X\\U) for all x ∈ U. With this definition, equicontinuous maps turn out to be continuous.
The class of topological spaces obtained in the above manner from proximity spaces coincides with the class of completely regular topological spaces. If X is a completely regular space, then there is a one-to-one correspondence between the proximities on X and the so-called compactifications (bicompact extensions) bX, that is, compact Hausdorff topological spaces containing X as an everywhere-dense subspace. The proximity δ corresponding to a bX is characterized by the fact that AδB if and only if the closures of A and B intersect in bX. In particular, for every compact Hausdorff space X there is a unique proximity on X that generates its topology.
The second approach is based on the fact that uniform continuity in a metric space X can be defined in terms of the relation “the points x and y are at most ∊ apart.” The general concept of a relation is that of a subset U of the Cartesian product X × X. The identity relation is the diagonal Δ ⊂ X × X, that is, the set of pairs (x,x), x ∈ X. The inverse of a relation U is the relation U–1 = {(x, y)} with (y, x) ∈ U. The product U ⃘ V of relations U and V consists of all (x, y) for which there is a z ∈ X such that (x, z) ∈ U, (z,y) ∈ V. A uniformity for a set X is a family {U} of relations U called neighborhoods of the diagonal such that (1) the intersection of two neighborhoods of the diagonal contains a neighborhood of the diagonal, (2) every neighborhood of the diagonal contains Δ and the intersection of all the neighborhoods of the diagonal coincides with the diagonal, (3) if U is a neighborhood of the diagonal, then so is U–1, and (4) for every neighborhood U of the diagonal there exists a neighborhood W of the diagonal such that W ⃘ W ⊂ U. The pair (X,{U}) is a uniform space. A map f: X → Y of a uniform space X into a uniform space Y is uniformly continuous if the map f × f: X × X → Y × Y has the property that the preimage of every neighborhood V ⊂ Y × Y of the diagonal contains a neighborhood of the diagonal in X × X Two uniform spaces X and Y are uniformly isomorphic if there exists a one-to-one uniformly continuous map X → Y with a uniformly continuous inverse. The map is a uniform isomorphism. In the class of uniform spaces, uniformly isomorphic spaces are essentially the same.
Every uniformity for X determines a proximity via the following definition: AδB if and only if (A × B) ∪ U + ∅ for every neighborhood U ⊂ X × X, with Δ ⊂ U. The uniformly continuous maps turn out to be equicontinuous.
Algebraic topology. Suppose that to each topological space X in a certain class there is associated an algebraic object—for example, a group or ring—h(X), and to each continuous map f: X → Y, a homomorphism h(f):h(X) → h(Y), which is the identity homomorphism if f is the identity map. If h(f1, ⃘ f2) = h(f1,) ⃘ h(f2), then we say that h is a functor. Most problems of algebraic topology are connected in some way with the following extension problem: given a continuous map f: A → Y of a subspace A ⊂ X into a topological space Y, find a continuous map g: X → Y that coincides with f on A; in other words, f = g · i, where i is the inclusion map; i: A → X, i(a) = a for all a ∈ A. If such a continuous map g exists, then for every functor there exists a homomorphism φ : h(X) → h(Y) such that h(f) = φ ⃘ h(i); in fact, φ = h(g). It follows that if the homomorphism φ fails to exist even for one functor, then the mapping g fails to exist.
Almost all methods of algebraic topology can, in essence, be reduced to this simple principle. For example, there exists a functor h whose value on the disk En is a trivial group, and on the boundary Sn–1 of the disk, a nontrivial group. This fact alone implies the nonexistence of a continuous map p : En → Sn–1 (called a retraction) such that p is fixed on Sn–1, that is, such that the map p·i (where i:Sn–1 → En is the inclusion map) is the identity map. [If such a p existed, then the identity map on the group h(Sn–1) would be the composition of the maps h(i):h(Sn–1) → h(En) and h(p):h(En) → h{Sn–1), which is impossible in view of the trivial nature of the group h(En).] So far, this essentially elementary geometric fact, whose intuitively obvious physical significance (for n = 2) is that of the possibility of stretching a drum on a circular hoop, has not been proved without the use of methods of algebraic topology. Its immediate consequence is the assertion that every continuous map f: En → En has at least one fixed point; that is, the equation f(x) = x has at least one solution in En [if f(x) ± x for all x ∈ En, we could obtain a retraction p: En → Sn–1 by taking p(x) to be the point of Sn–1 collinear with x and f(x) and such that the segment with endpoints f(x) and p(x) contains x]. This fixed point theorem was one of the earliest theorems of algebraic geometry and became the source of various theorems concerned with the existence of solutions of equations.
In general, the more complex the algebraic structure of the objects h(X), the easier it is to establish the nonexistence of the homomorphism φ. That is why algebraic topology considers algebraic objects of extremely involved nature, and the needs of algebraic topology have provided a strong stimulus for the development of abstract algebra.
A topological space X is called a CW-complex if it is the union of an increasing sequence of subspaces X0 ⊂ ··· Xn–1 ⊂ Xn–1 ⊂ Xn ··· and the following conditions hold: (1) a set U ⊂ X is open in X if and only if for all n the set U ⊂ Xn is open in Xn, (2) Xn, the n-skeleton of X, is obtained from Xn–1 by gluing a class of n-dimensional disks to Xn–1 along their respective (n – 1)-dimensional boundary spheres (using arbitrary continuous maps from the spheres into Xn–1), and (3) X0 consists of isolated points. We see that, roughly speaking, a CW-complex is representable as the union of sets, called cells, homeomorphic with open disks. The spaces studied in algebraic topology are almost exclusively CW-complexes, since they provide a setting that accommodates the specific nature of the problems of algebraic topology. Actually, the CW-complexes of interest in algebraic topology are essentially the polytopes. In general, however, such restriction of the class of CW-complexes complicates matters, since the class of polytopes is not closed under a number of useful operations on CW-complexes.
Two continuous maps f,g:X → Y are homotopic if one can be continuously deformed into the other, that is, if there exists a family of continuous maps f1: X → Y that depends continuously on the parameter t ∈ [0,1] and f0 = f, f1 = g. This means that the equation F(x,t) = f(x), x ∈ X, t ∈ [0,1], defines a continuous map F: X × [0,1] → Y. The map F and the family {f1} are each referred to as a homotopy from f to g. The relation of homotopy on the set of continuous maps X → Y is an equivalence relation, and the set of equivalence classes (homotopy classes) is denoted by [X, Y]. Homotopy theory is the study of the relation of homotopy and, in particular, the study of the sets [X, Y]. For most topological spaces of interest, the sets [X, Y] are finite or countably infinite and can be explicitly and effectively computed. Two topological spaces X and Y are of the same homotopy type if there exist continuous maps f: X → Y, g: Y → X such that g·f: X → X and f·g: Y → Y are homotopic to the corresponding identity maps. In homotopy theory such spaces are considered to be essentially the same, since they have the same homotopic invariants.
In many cases, particularly in the case of CW-complexes, the solvability of the extension problem depends on the homotopy class of the continuous map f: A → Y in the sense that if for fthere exists an extension g: X → Y, then for any homotopy f1: A → Y, with f0 = f, there exists an extension g1: X → Y such that g0 = g. Hence, instead of f we need only consider [f] and therefore only homotopically invariant functors (cofunctors) h, that is, functors such that h(f0) = h(f1) if f0 and f1 are homotopic. This links algebraic topology and homotopy theory to such an extent that they may be regarded as a single discipline.
For any topological space Y the equations h(X) = [X,Y] and h(f) = [φ ⃘ f], where f: X1 → X2 and φ: X2 → Y, define a homotopically invariant cofunctor h, which is said to represent the topological space Y. This is a standard, and essentially unique, way of constructing homotopically invariant cofunctors. For h(X) to be, say, a group, it is necessary to choose Y in a suitable manner, for example, to require that Y is a topological group (in general, this is not quite so: it is necessary to choose a point x0 in X and to consider only the continuous maps and homotopies that carry x0 to the group identity; in what follows we ignore this technical complication). Actually, it suffices that Y is a topological group in the “homotopic sense, ” that is, that the axioms of associativity and the existence of an inverse element, which are assertions of equality of certain maps, hold “up to a homotopy.” Such topological spaces are called H-spaces. In summary, every H-space Y determines a homotopically invariant cofunctor h(X) = [X,Y] whose values are groups.
Dually, for any topological space Y the equations h(X) = [Y,X] and h{f) = [f ⃘ φ], where f: X1 → X2 and φ: Y → X1, define a functor h. For h{X) to be a group, Y must have an algebraic structure dual, in a certain specific sense, to the structure of an H-space. A topological space with such a structure is a co-H-space. An example of a co-H-space is an n-dimensional sphere Sn for n ≥ 1. Thus for every topological space the map πnX = [Sn,X] defines a group πnX, n ≥ 1, called the nth homotopy group of X. For n = 1, this group coincides with the fundamental group. For n > 1, the group πnX is commutative. If π1X = {1}, then X is said to be simply connected.
A CW-complex X is a K(G,n) space if πiX = 0 for i ± n and πnX = G. Such a CW-complex exists for n ≥ 1 and for every group G (supposed commutative for n > 1) and is unique up to homotopic equivalence. For n > 1, as well as for n = 1 if G is commutative, the space K{G,n) is an H-space and therefore represents a certain group Hn(X;G) = [X;K(G,n)]. This group is the n-dimensional cohomology group of the topological space X with coefficient group G. It is a typical representative of a series of important cofunctors, including the K-functor KO(X) = [X,BO], representable by the infinite dimensional Grassmanian BO and the group ΩnX of oriented cobordisms.
If G is a ring, then the direct sum H*(X;G) of the groups Hn(X;G) is an algebra over G. In addition, G has a very complex algebraic structure, which, for G = Zp, the cyclic group of order p, includes the action on H*(X;G) of a noncommutative algebra Up known as the Steenrod algebra. The complexity of this structure makes it possible to work out effective, but far from simple, methods for computing the groups Hn(X;G) and to establish connections between these groups and other homotopically invariant functors, such as πnX. The latter functors can frequently be explicitly computed with the aid of these connections.
The cohomology groups were introduced after the homology groups Hn(X;G). The groups Hn(X;G) are the homotopy groups Hn(X;G) of a certain CW-complex M(X;G) that is constructed out of given CW-complex X and a group G and is uniquely determined by X and G. The homology and cohomology groups are, in a definite sense, dual to each other and their theories are essentially equivalent. The algebraic structure associated with homology groups is somewhat unusual—for example, they form a coalgebra rather than an algebra—and that is why cohomology groups are commonly used in computations. Nevertheless, there are problems in which it is more convenient to work with homology groups, which is why these groups are also studied. Homology theory is the branch of algebraic topology that investigates and uses homology and cohomology groups of arbitrary topological spaces and their applications. It turns out that, in general, different ways of constructing these groups for spaces other than compact CW-complexes lead to different results, so that for topological spaces that are not CW-complexes we obtain a whole series of different homology and cohomology groups. General homology theory is mainly applied in dimension theory and in duality theory, which studies the connection between the topological properties of pairs of complementary subsets of a topological space, and its development has been largely stimulated by the needs of these theories.
Piecewise linear topology. A subset P ⊂ IRn is a cone with vertex a and base B if each point of P belongs to a unique segment of the form ab with b ∈ B. A subset X ∈ IRn is a polytope if each of its points has a neighborhood in X whose closure is a cone with compact base. A continuous map f: X → Y on polytopes is piecewise linear if it is linear on the rays of every conical neighborhood of every point x ∈ X. A one-to-one piecewise linear map with piecewise linear inverse is a piecewise linear isomorphism. Piecewise linear topology investigates polytopes and their piecewise linear maps. In piecewise linear topology, two polytopes connected by a piecewise linear isomorphism are regarded as being essentially the same.
A subset X ⊂ IRn is a (compact) polytope if and only if it can be represented as the union of (a finite number of) convex polyhedra. Every polytope can be represented as a union of simplexes any two of which do not meet at all, or if they do meet, then their intersection is a face of each of them. Such a representation is a triangulation of the polytope. Each triangulation is determined by its simplicial complex, that is, its set of vertices subdivided into subsets that are sets of vertices of simplexes. In this way, the study of polytopes reduces to the study of the simplicial complexes of their triangulations. For example, from the simplicial complex it is possible to compute the homology and cohomology groups of a polytope. This is done as follows.
(1) A simplex with ordered vertices is an ordered simplex of the given triangulation (or simplicial complex) K. Formal linear combinations of ordered simplexes of given dimension n with coefficients from a given group G are n-dimensional chains. There is a natural way of making them into a group, which we denote by the symbol Cn(K;G).
(2) By removing from an ordered n-dimensional simplex σ the ith vertex, 0 ≤ i ≤ n, we obtain an (n – 1)-dimensional simplex σ(i). The chain σa = σ(0) – σ(1) + ··· (–1)nσ(n) is the boundary of σ. The map ∂ is extended by linearity to a homomorphism ∂: Cn(K;G).
(3) Chains c with ∂c = 0 are cycles and form the group of cycles Zn(K;G).
(4) Chains of the form ∂c are boundaries and form the group Bn(K;G) of boundaries.
(5) It can be shown that Bn(K;G) ⊂ Zn(K;G), that is, boundaries are cycles. This enables us to form the quotient group Hn(K;G) = Zn(K;G)/Bn(K;G). It turns out that the group Hn(K;G) is isomorphic with the homology group Hn(X;G) of the polytope X, whose triangulation is K. A similar construction using cochains in place of chains (cochains are arbitrary functions defined on the ordered simplexes with values in the group G) yields the cohomology groups.
Apart from minor modifications, the above construction marked, in a very real sense, the beginning of algebraic topology. The original construction made use of oriented simplexes, that is, classes of ordered simplexes that differ by even permutations of the vertices. This construction was extended and generalized in a great number of ways. In particular, its algebraic content gave rise to homological algebra.
The most general definition of a simplicial complex is that of a set with distinguished subsets called simplexes such that every subset of a simplex is a simplex. Such a simplicial complex is the simplicial complex of a triangulation of a polytope if and only if the number of elements of an arbitrary distinguished subset does not exceed a certain fixed number. It is possible to generalize the concept of polytope to include so-called infinite-dimensional polytopes. Then any simplicial complex is the simplicial complex of a triangulation of a polytope (the geometric realization of the complex).
To every open cover {Uα} of a topological space X we can associate a simplicial complex, whose vertices are the elements Uα of the cover and whose distinguished subsets are just those subsets whose elements have a nonempty intersection. This simplicial complex is the nerve of the cover. There is a definite sense in which the nerves of all covers of X approximate X. By a suitable limiting process, it is possible to obtain from the homology and cohomology groups of the nerves the homology and cohomology groups of X. This idea is the basis of almost all constructions of general homology theory. The approximation of a topological space by the nerves of its open covers plays an important role in general topology.
Topology of manifolds. A paracompact Hausdorff space is an n-dimensional manifold if it is locally Euclidean, that is, if each of its points has a neighborhood, known as a coordinate patch, that is homeomorphic to IRn. In this way, each point of a patch is assigned n numbers x1,....xn, its local coordinates. In the intersection of two patches the local coordinates of one patch are functions of the local coordinates of the other patch. These overlap functions determine overlap homeomorphisms of appropriate open sets in IRn.
A homeomorphism of open sets in IRn is a t-homeomorphism. A homeomorphism that is a piecewise linear isomorphism is a p-homeomorphism, and a homeomorphism given by smooth, that is, infinitely differentiable, functions is an s-homeomorphism.
Let α = t, p, or s. A manifold X is an α-manifold if we choose a cover of X such that the overlap homeomorphisms are α-homeomorphisms. Such a cover determines an α-topology on X. Thus, a p-manifold is simply a manifold and a p-manifold is a piecewise linear manifold. Every piecewise linear manifold is a polytope. In the class of polytopes, the n-dimensional piecewise linear manifolds are characterized by the fact that each point of such a manifold has a neighborhood that can be mapped onto an n-dimensional cube by a piecewise linear isomorphism, s-manifolds are smooth, or differentiable, manifolds. An α-map is continuous, piecewise linear, or smooth (that is, given in terms of local coordinates by smooth functions) according to whether α = t, p, or s. A one-to-one α-map is an α-homeomorphism (a diffeomorphism if α = s). α-manifolds X and Y are α-homeomorphic (for α = s, diffeomorphic) if there exists an α-homeomorphism X → Y. The theory of α-manifolds is the study of α-manifolds and their α-maps; α-homeomorphic α-manifolds are regarded as being essentially the same. The theory of p-manifolds is a branch of piecewise linear topology. The theory of s-manifolds is also known as smooth topology.
The fundamental method of modern manifold theory is to reduce its problems to those of algebraic topology for suitable topological spaces. This close connection between manifold theory and algebraic topology made possible the solution of many difficult geometric problems and stimulated the development of algebraic topology.
Examples of smooth manifolds are n-dimensional surfaces in IRn that have no singular points. A basic embedding theorem asserts that every smooth manifold is diffeomorphic to such a surface (for N ≥ 2n + 1). Analogous results hold for α = t and α = P.
Every p-manifold is a t-manifold. It turns out that on every s-manifold it is possible to introduce in a natural way a p-topology, usually referred to as a Whitehead triangulation. Every α-manifold, where α is p or s, is an α′-manifold, where α′ is t or p. It is natural to ask on what α′-manifolds it is possible to define an α-topology, and if so, what is the number of such topologies (if α′ = p, then this α′-manifold is said to be smoothed, and if α′ = t, then we call it triangulable). The answer to this question depends on the dimension n.
There exist just two one-dimensional manifolds, the circle S1 and the line IR. S1 is compact and IR is not. For every α = p,s there is a unique α-topology on the t-manifolds S1 and IR.
Similarly, on every two-dimensional manifold (surface) there exists a unique α-topology, and it is easy to describe all compact connected surfaces (the description of noncompact connected surfaces, while possible, is more involved). For surfaces to be homeomorphic, it suffices that they are homotopically equivalent. The homotopy type of a surface is uniquely determined by its homology groups. A surface is orientable or nonorientable. Examples of orientable surfaces are the sphere S2 and the torus T2. Let X and Y be two connected n-dimensional α-manifolds. We remove a disk in each manifold and glue the manifolds together along the spherical boundaries of the “holes.” If we take elementary precautions, then the result is again an α-manifold, the connected sum X # Y of X and Y. A simple example of a connected sum is T2 # T2; it looks like a pretzel. The sphere Sn is the neutral element of this construction in the sense that Sn # X = X for all X. In particular, S2 # T2 = T2. It turns out that every orientable surface is homeomorphic to a connected sum of the form S2 # T2 # ··· # T2. The number p of summands T2 is the genus of the surface and has the value 0 for S2 and 1 for T2. A model of a surface of genus p is a sphere with p handles. Every nonorientable surface is homeomorphic with the connected sum IRP2 # ··· # IRP2 of projective planes IRP2. Such a surface may be thought of as a sphere with a certain number of Möbius strips.
On every three-dimensional topological manifold for every α = p,s there exists a unique α-topology. Also, we can describe all homotopy types of three-dimensional manifolds, except that in this case we cannot rely on the homology groups alone. At this writing (1976), even for the case of compact connected manifolds, we have no complete description of all three-dimensional manifolds of specified homotopy type. Nor has this problem been solved for simply connected manifolds; such manifolds are homotopically equivalent to the sphere S3. The Poincaré conjecture states that such a manifold is homeomorphic to S3.
The question of existence and uniqueness of α-topologies (α = p,s) for four-dimensional (compact and connected) manifolds remains unsolved, and their homotopy types have been described only under the additional assumption of simple connectivity. We do not know the status of the Poincaré conjecture for these manifolds.
It is remarkable that for compact connected manifolds of dimension n ≥ 5 the situation is radically different. For such manifolds, all fundamental problems are solved in principle in the sense that they can be reduced to problems of algebraic topology. Every smooth manifold can be embedded as a smooth (n-dimensional) surface in IRN, and the tangent vectors to X form a new smooth manifold TX, the tangent fiber bundle of the smooth manifold X. Quite generally, a vector fiber bundle over a topological space X is a topological space E and a continuous map π: E → X such that (1) for every x ∈ X the preimage (fiber) π–1(X) is a vector space, (2) there exists an open cover {Uα} of X such that for every α the preimage π–1(Uα) is homeomorphic to the product Uα × IRn, and (3) there exists a homeomorphism π–1(Uα) → Uα × IRn that maps every fiber π–1(x), → Uα, linearly onto the vector space {x} × IRn. For E = TX, the continuous map α associates to every tangent vector its point of tangency, so that the fiber α–1(s) is the space tangent to X at x. It turns out that every vector fiber bundle over a compact space X determines an element of the group KO(X). Thus, in particular, for every compact connected manifold there is a definite element in the group KO(X) corresponding to the tangent fiber bundle over X. This group element is the tangential invariant of the smooth manifold X. There is an analogous construction for every α. For α = p and α = t the role of the group KO(X) is played by groups denoted by KPL(X) and KTop(X), respectively. Every α-manifold X determines in an appropriate group [KO(X), KPL(X), or KTop(X)] an element—its α-tangential invariant. There are natural homomorphisms KO(X) → KPL(X) → KTop(X), and one can show that if X is an n-dimensional (n ≥ 5) compact connected α′-manifold, α′ = t,p, then it is possible to introduce in X an α-topology (α = p if α′ = t and α = s if α′ = p) if and only if the α′-tangential invariant of X lies in the image of the appropriate group [KPL(X) for α′ = t and KO(X) for α′ = p]. The number of such topologies is finite and equals the number of elements of a certain quotient set of the set [X,Yα], where Yα is a certain topological space constructed in a special manner (for α = s, the space Yα is denoted by PL/O, and for α = p, by the symbol Top/PL). In this way the question of the existence and uniqueness of an α-topology is reduced to a problem of homotopy theory. The homotopy type of the topological space PL/O is rather involved and up to now (1976) has not been fully computed. We do know that πi(PL/O) = 0 for i ≤ 6, which implies that every piecewise linear manifold of dimension n ≤ 7 can be smoothed, and for n ≤ 6 this can be done in just one way. The homotopy type of the topological space Top/PL turns out to be remarkably simple, namely, Top/PL is homotopically equivalent to K(ℤ2,3). It follows that the number of piecewise linear topologies on a manifold does not exceed the number of elements of the group H3(X, ℤ2). Such topologies are known to exist if H4(X, ℤ2) = 0 but may not exist if H4(X, ℤ2) ± 0.
In particular, on the sphere Sn there exists just one piecewise linear topology. There can be many smooth topologies on Sn, for example, on S7 there are 28 different smooth topologies. On a torus Tn, the topological products of n copies of a circle S1, for n ≥ 5 there are many different piecewise smooth topologies, all of which admit a smooth topology. Thus, beginning with dimension 5, there exist homeomorphic but nondifferomorphic smooth manifolds. Spheres with such properties exist beginning with dimension?.
It is natural to solve the problem of describing (up to an α-homeomorphism) all n-dimensional (n ≥ 5) compact connected manifolds in two stages: to look first for conditions of homotopic equivalence of α-manifolds and then for conditions under which homotopically equivalent α-manifolds are α-homeomorphic. The first problem is a problem in homotopy theory and may be viewed as completely solved in that theory. Basically, the second problem is also completely solved, at least for simply connected α-manifolds. The main tool for its solution is the technique of “decomposition into handles” adapted to higher dimensions. Using this technique, we can prove, for example, the Poincaré conjecture for n-dimensional (n ≥ 5) manifolds (a compact connected manifold homotopically equivalent to a sphere is homeomorphic to it).
In addition to α-manifolds, we can consider α-manifolds with boundary (bordered α-manifolds). The boundary of such manifolds consists of points that have neighborhoods homeomorphic to the half-space xn ≥ 0 of IRn. The boundary is an (n – 1)-dimensional, generally not connected, α-manifold. Two n-dimensional compact α-manifolds are (co)bordant if there exists an (n + 1)-dimensional compact manifold with boundary W such that W is the union of disjoint smooth manifolds α-homeomorphic to X and Y, respectively. If the embeddings X → W and Y → W are homotopically equivalent, then the smooth manifolds are h-cobordant. Using decompositions into handles we can show that for n ≥ 5 simply connected α-manifolds are α-homeomorphic if they are h-cobordant. This theorem on h-cobordism supplies us with a better method for deciding when α-manifolds are α-homeomorphic (in particular, it implies the Poincaré conjecture). A similar but more involved result is true for α-manifolds that are not simply connected.
Under the operation of connected sum the ℜn classes of cobordant compact α-manifolds form a commutative group. The identity of this group is the class of α-manifolds that are borders and thus cobordant to zero. It turns out that for α = s this group is isomorphic to the homotopy group π2n+1MO(n + 1) of a certain topological space MO{n + 1) known as a Thom space. A similar result holds for α = p,t. It follows that the methods of algebraic topology enable us, in principle, to compute the group ℜαn. In particular, it turns out that the group ℜsn is the direct sum of copies of xs2, whose number is equal to the number of partitions of n into summands that are not of the form 2m – 1. For example, ℜs3 = 0, since every three-dimensional compact smooth manifold is a boundary. On the other hand, ℜs2 = 2, since there exist cobordant surfaces that are not cobordant to zero; an example of such a surface is the projective plane IRP2.
M. M. POSTNIKOV
Fundamental stages of the development of topology. Isolated results of a topological nature were obtained in the 18th and 19th centuries. Examples of such results are Euler’s theorem on polyhedra, the classification of surfaces, and Jordan’s theorem, which asserts that a simple plane curve separates the plane into two parts. In the beginning of the 20th century, the general concept of a space was introduced into topology (M. Frechet of France defined a metric space, and F. Hausdorff of Germany, a topological space), the initial ideas of measure theory arose, proofs of the simplest properties of continuous maps were given (H. Lebesgue, France; L. Brouwer, the Netherlands), and polytopes were introduced (H. Poincaré, France) and their Betti numbers defined.
At the end of the first quarter of the 20th century, general topology flourished and the Moscow school of topology was founded. Foundations were laid for a general dimension theory (P. S. Uryson [Urysohn], USSR), the modern axiomatization of topological spaces was created (P. S. Aleksandrov), the theory of compact spaces was developed (Aleksandrov, Uryson), and the theorem on the product of such spaces was proved (A. N. Tikhonov). In addition, necessary and sufficient conditions for the metrizability of a space were given (Aleksandrov, Uryson), the concept of a locally finite cover was introduced (Aleksandrov), which was used by J. Diendonné (France) in 1944 to define para-cocompact spaces, and completely regular spaces were introduced (Tikhonov). The concept of a nerve was also defined, and thus general homology theory was founded (Aleksandrov). Under the influence of E. Noether (Germany), Betti numbers were realized as the ranks of homology groups, which is why these groups are also called Betti groups. L. S. Pontriagin (USSR), on the basis of his theory of characters, proved duality laws for closed sets.
In the second quarter of the 20th century, general topology continued to develop. A. Stone (USA) and E. Čech (Czechoslovakia) developed Tikhonov’s ideas and introduced the Stone-Čech, or maximal, (bi)compact extension of a completely regular space. Homology groups of arbitrary spaces were defined (Čech), multiplication was defined for cohomology groups (J. Alexander, USA; A. N. Kolmogorov, USSR), and the cohomology ring was constructed. At that time, combinatorial methods, based on the study of simplicial complexes, were preeminent; this explains why to this day, algebraic topology is also called combinatorial topology. Proximity and uniform spaces were introduced. There was an intensive development of homotopy theory (H. Hopf, Switzerland; Pontriagin), homotopy groups were defined (W. Hurewicz, USA), and the maps of smooth topology were used to compute them (Pontriagin). Homology and cohomology groups were axiomatized (N. Steenrod and S. Eilenberg, USA). The theory of fiber bundles was developed (H. Whitney, USA; Pontriagin), and CW-complexes were introduced (J. Whitehead, Great Britain).
In the second half of the 20th century, the modern school of general topology and homology theory developed in the USSR; work was done in dimension theory, on the metrization problem, in the theory of (bi)compact extensions, in the general theory of continuous maps (quotient, open, and closed), particularly the theory of absolutes, and in the theory of cardinal valued invariants (A. V. Arkhangel’skii, B. A. Pasynkov, V. I. Ponomarev, E. G. Skliarenko, Iu. M. Smirnov).
The efforts of a number of mathematicians resulted in the completion of homotopy theory (J. P. Serre and A. Cartan, France; M. M. Postnikov, USSR; Whitehead). At that time, major centers for the study of algebraic topology developed in the USA and Great Britain; interest in geometric topology revived. The theory of vector fiber bundles and K-functors was developed (M. Atiyah, Great Britain; F. Hirzebruch, Federal Republic of Germany), algebraic topology was used extensively in smooth topology (R. Thom, France) and algebraic geometry (Hirzebruch), (co)bordism theory was developed (B. A. Rokhlin, USSR; Thom; S. P. Novikov, USSR), as well as the theory of smoothing and triangulability (J. Milnor, USA).
Topology continues to develop in all directions and so does its range of applications.
A. A. MAL’TSEV
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Spanier, E. Algebraicheskaia topologiia. Moscow, 1971. (Translated from English.)A. A. MAL’TSEV
topology - topographic study of a given place (especially the history of the place as indicated by its topography); "Greenland's topology has been shaped by the glaciers of the ice age"
topology - the study of anatomy based on regions or divisions of the body and emphasizing the relations between various structures (muscles and nerves and arteries etc.) in that region