Algebraic topology

The branch of mathematics in which one studies such properties of geometrical figures (in a wider sense, of all objects for which one can speak of continuity), and their mappings into each other, which remain unchanged under continuous deformations (homotopies). In principle, the objective of algebraic topology is a complete listing of such properties. The very name of algebraic topology originates from the decisive role of algebraic notions and algebraic methods in solving problems in this field. The most important classes of objects whose properties are studied in algebraic topology include complexes (polyhedra, cf. Complex), which may be simplicial, cellular, etc.; manifolds (cf. Manifold), which may be open, closed or with boundary, and may in turn be divided into smooth (differentiable), analytic, complex-analytic, piecewise-linear or topological; and fibre bundles (fibrations, cf. Fibration) and their sections. The principal types of mappings considered in algebraic topology are arbitrary continuous, piecewise-linear and smooth mappings, and their most important subclasses are: homeomorphisms, in particular, continuous, piecewise-linear or smooth (diffeomorphisms); imbeddings of one object into another and also immersions (local imbeddings). (cf. Homeomorphism; Diffeomorphism.)

A very important notion in algebraic topology is that of a deformation. The principal types of deformations include: a homotopy, i.e. an arbitrary continuous (smooth, piecewise-linear) deformation of continuous mappings; an isotopy (continuous, smooth, piecewise-linear), i.e. a deformation of homeomorphisms, imbeddings or immersions where in the process of deformation the mapping is a homomorphism, an imbedding or an immersion at any moment of time.

The principal internal problems of algebraic topology include the problem of the classification of manifolds by homeomorphisms (continuous, smooth, piecewise-linear), the classification of imbeddings (or immersions) with respect to isotopies (regular homotopies), and the classification of general continuous mappings up to homotopy. An important intermediate role in the solution of these problems is played by the problem of the classification of complexes or manifolds by the so-called homotopy equivalence or homotopy type.

The following, somewhat special, problems played an important role in the development of algebraic topology.

1) The problem of imbedding is usually considered not in its general form, but rather as imbedding into a Euclidean space. A very important special case is knot theory (and the theory of links) in three-dimensional space, which was one of the origins of algebraic topology; braid theory is a related case.

2) An important role in the history of algebraic topology was played by the theory of homology invariants of the position of various sets in a Euclidean space and the duality laws (cf. Duality in algebraic topology), connecting the homology of a set and its complement.

3) A number of fundamental results concerned the computation of the algebraic number of fixed points of a mapping of a manifold into itself. Many fundamental facts were discovered for the fixed points of compact smooth groups of transformations, including cyclic groups of finite order.

4) Of major technical importance in the development of algebraic topology were methods developed to solve the problems on so-called cobordism: Does a manifold with boundary (cobordism) with a given closed manifold as boundary exist? Problems of this kind originally arose in the context of the computation of the homotopy groups of spheres (cf. Homotopy group). Important cases of the cobordism problem are solved in the language of characteristic classes (cf. Characteristic class).

5) Many facts are now available concerning the homology invariants of the singularities of vector fields, frame fields and tensor fields, as well as of the singularities of smooth mappings of manifolds, in particular into Euclidean space. The solution of this problem leads, in particular, to characteristic classes. An especially important case is that of stationary points of smooth functions on manifolds or of various functionals on path spaces (extremals); their connection with homology theory is important in the clarification of the geometrical structure of manifolds and in obtaining lower bounds for the number of extremals.

6) The study of the algebraic-topological properties of an important kind of special spaces — Lie groups — is closely connected with their algebraic structure, their representations and the calculus of variations on Lie groups (cf. Lie group). The results obtained for the topological structure of Lie groups form the base of numerous methods and facts of algebraic topology which apply to arbitrary manifolds. Algebraic topology of homogeneous manifolds is closely related to methods of Lie groups.

7) A major role in algebraic topology is played by special invariants connected with various algebraic structures over the fundamental group. The simplest invariants of this type appeared in knot theory and in the theory of three-dimensional manifolds; the development of their algebraic theory subsequently made substantial progress, and eventually formed a separate discipline — stable algebra or algebraic $K$-theory.

8) The analysis of the geometrical structure of a very large number of examples of the simplest and most frequently encountered manifolds (e.g. Lie groups, homogeneous spaces, manifolds of line elements and manifolds with discrete groups of motions), in conjunction with the fundamental principles of Riemannian geometry, led to the notion of a fibre bundle, which consists of a space (the total or fibre space), a base space, a projection of the total space onto the base space, a fibre (homeomorphic to a "fibre" of the projection), and the structure group of fibre transformations. In this context, one of the central problems in algebraic topology is the problem of the classification with respect to homotopy of fibre bundles and their sections. Principal fibre bundles and vector bundles are especially important (cf. Principal fibre bundle; Vector bundle).

A universal method for the solution of all principal problems in algebraic topology involves the construction of algebraic invariants which are effectively calculable in actual examples and which assume some discrete set of values; the value of the invariant may not vary during deformations in the corresponding class of mappings for the study of which the invariant has been constructed. A large number of essential invariants, the abundance of algebraic links between them and the difficulty in their calculation have determined the features of modern algebraic topology.

The calculation of algebraic-topological invariants of the simplest and most frequently encountered manifolds is not always a simple matter. Thus, the calculation of homology invariants of Lie groups and of many homogeneous spaces required much effort and involved the use of complicated methods. The calculation of homotypy groups is even more difficult. A number of the most important homotypy groups for Lie groups was calculated, against expectation, using the variational theory of geodesics; the knowledge of the table of these homotypy groups for Lie groups made the classification of certain vector bundles possible.

Most algebraic-topological invariants are a so-called functor on the category of topological spaces of the type studied. This means, roughly speaking, that the values of the invariant undergo natural transformations when the spaces are mapped into each other. For example, the fundamental groups of arbitrary spaces or their homology (cohomology) groups (rings) are connected by homomorphisms induced by continuous mappings; the characteristic classes (distinguished elements in (co)homology groups) are mapped into each other by the homomorphism corresponding to a morphism of manifolds; the result of cohomology operations performed on an element of the homology (cohomology) is converted, after a continuous mapping of the space, into the result of this operation performed on the image of this element, etc.

One of the principal properties, on which the study and the application of almost-all algebraic-topological invariants are based, is the fact that their effective construction is usually connected with an essential complementary geometrical structure. For instance, the construction of all the principal invariants up to homeomorphism for complexes involves a subdivision into simplices or cells, whereas the result of such a construction has to be invariant with respect to all continuous homeomorphisms, and even with respect to homotopy equivalences; examples are the fundamental group, the Euler characteristic, the homology group (Betti group), the cohomology ring and the cohomology operations (cf. Cohomology operation). In a similar manner, the structure of all the basic homeomorphism invariants for smooth manifolds involves their preliminary triangulation, i.e. reduction to complexes, or else to a significant extent the use of analytical tools, such as the construction of a homology ring by way of differential forms (skew-symmetric tensors) and differential operations on them, or the construction of characteristic classes using the singularities of vector fields, frame fields or tensor fields. Moreover, it may also be necessary to use tools of Riemannian geometry in certain cases, e.g. the definitions of the characteristic classes of a manifold or of a fibre bundle in terms of the Riemannian curvature play an important role, even though the result is invariant wih respect to all continuous homeomorphisms. In this context, the appearance of basic, effectively calculable invariants in the history of topology involved the difficult problem of proving the invariance of these quantities. This is another type of problem dealt with by algebraic topology. Thus, rational characteristic classes (or integrals of classes over cycles) proved to be topologically invariant and homotopically non-invariant. The totality of characteristic classes over the integers proved to be non-invariant with respect to continuous or even piecewise-linear homeomorphisms.

On the contrary, if an invariant is so constructed that it is invariant (by definition) with respect to continuous homeomorphisms (or some wider class of transformations — homotopy equivalences), then the calculation of such an invariant is usually difficult. The most important invariants of this type are the homotopy groups of classes of homotopic mappings of a sphere into the space under study. The calculation of even the simplest examples of homotopy groups involves major difficulties. Thus, the homotopy groups of the spheres themselves are only incompletely known, despite the large number of methods which were proposed for such calculations in the course of development of algebraic topology (cf. Homotopy group).

The method of classification of classes of homotopic mappings is based on homology theory in conjunction with the theory of fibre bundles with its apparatus of spectral sequences, as well as cohomology operations and their generalizations. The algebraic discipline which arose on the basis of the complicated computational tools of algebraic topology is known as homological algebra.

All the basic primary constructions of homology theory for complexes and smooth manifolds by way of triangulation or differential forms are effectively combinatorial — algebraic or analytic. However, it was noted in the course of development of algebraic topology that homology theory can be completely defined by a small number of formal properties (axioms) on which its computational tools are based:

1) Homotopic invariance of homology groups.

2) The homology groups are homomorphically mapped into each other under continuous mappings of spaces, and products of mappings correspond to the products of these homomorphisms (functoriality).

3) A certain form of interconnection is postulated between the cohomology of a complex, a subcomplex and a quotient complex (exactness axiom).

4) The so-called excision axiom.

5) Normalization: Only the homology groups for a point need be known, and they must be known to vanish in non-zero dimensions.

It was subsequently noted that objects of an altogether different geometrical nature may display all of the above properties, except for normalization. Such objects were called generalized or extra-ordinary homology theories. They were used to improve the calculation methods of algebraic topology. The most important example is $K$-theory, which is based on vector bundles over the space studied instead of their cycles or differential forms which served for the construction of the ordinary homology groups. Another important example is cobordism (bordism) theory, in which only the mappings of closed manifolds into the space under study are taken instead of arbitrary cycles, and only mappings of manifolds with boundary of some classes are taken instead of arbitrary boundaries.

The combination of the methods of algebraic topology with a small number of pure geometric facts applied to the structure of manifolds and their homeomorphism groups eventually resulted in the solution (which, from the modern point of view, may or may not be considered complete) of classification problems of manifolds with respect to all kinds of homeomorphisms (smooth, piecewise-linear, continuous), and in the solution of fundamental problems in the classification of imbeddings and immersions. Strangely enough, problems connected with three-dimensional and four-dimensional manifolds, where the fundamental problems are still (1977) unsolved, represent an exception.

Since most of the fundamental internal problems have now been solved, modern efforts in algebraic topology tend to concentrate on the application of these ideas in other fields.

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The axioms 1)–5) for a (co)homology theory are often known as the Eilenberg–Steenrod axioms (cf. Steenrod–Eilenberg axioms), [a3]. Some additional good references are [a1], [a2], [a4]–[a6].

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