What is Homology Theory?

In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.

Simple explanation

At the intuitive level homology is taken to be an equivalence relation, such that chains C and D are homologous on the space X if the chain C − D is a boundary of a chain of one dimension higher. The simplest case is in graph theory, with C and D vertices and homology with a meaning coming from the oriented edge E from P to Q having boundary Q — P. A collection of edges from D to C, each one joining up to the one before, is a homology. In general, a k-chain is thought of as a formal combination

where the are integers and the are k-dimensional simplices on X. The boundary concept here is that taken over from the boundary of a simplex; it allows a high-dimensional concept which for k = 1 is the kind of telescopic cancellation seen in the graph theory case. This explanation is in the style of 1900, and proved somewhat naive, technically speaking.

Example of a torus surface

For example if X is a 2-torus T, a one-dimensional cycle on T is in intuitive terms a linear combination of curves drawn on T, which closes up on itself (cycle condition, equivalent to having no net boundary). If C and D are cycles each wrapping once round T in the same way, we can find explicitly an oriented area on T with boundary C − D. Topologists can prove that the homology classes of 1-cycles with integer coefficients form a free abelian group with two generators, one generator for each of the two different ways round the 'doughnut'.

The nineteenth century

This level of understanding was common property in the mathematics of the nineteenth century, starting with the idea of Riemann surface. At the end of the century, the work of Poincaré had provided a much more general, though still intuitively-based, setting.

For example, it is considered that the general Stokes' theorem was first stated in 1899 by Poincaré: it involves necessarily both an integrand (we would now say, a differential form), and a region of integration (a p-chain), with two kinds of boundary operators, one of which in modern terms is the exterior derivative, and the other a geometric boundary operator on chains that includes orientation and can be used for homology theory. The two boundaries appear as adjoint operators, with respect to integration.

Twentieth century beginnings

Rather loose, geometric arguments with homology were only gradually replaced at the beginning of the twentieth century by rigorous techniques. To begin with, the style of the era was to use combinatorial topology (the fore-runner of today's algebraic topology). That assumes that the spaces treated are simplicial complexes, while the most interesting spaces are usually manifolds, so that artificial triangulations have to be introduced to apply the tools. Pioneers such as Solomon Lefschetz and Marston Morse still preferred a geometric approach. The combinatorial stance did allow Brouwer to prove foundational results such as the simplicial approximation theorem, at the base of the idea that homology is a functor (as it would later be put). Brouwer was able to prove the Jordan curve theorem, basic for complex analysis, and the invariance of domain, using the new tools; and remove the suspicion attached to topological arguments as handwaving.

Towards algebraic topology

The transition to algebraic topology is usually attributed to the influence of Emmy Noether, who insisted that homology classes lay in quotient groups — a point of view now so fundamental that it is taken as a definition. In fact Noether in the period from 1920 onwards was with her students elaborating the theory of modules for any ring, giving rise when the two ideas were combined to the concept of homology with coefficients in a ring. Before that, coefficients (that is, the sense in which chains are linear combinations of the basic geometric chains traced on the space) had usually been integers, real or complex numbers, or sometimes residue classes mod 2. In the new setting, there would be no reason not to take residues mod 3, for example: to be a cycle is then a more complex geometric condition, exemplified in graph theory terms by having the number of incoming edges at every vertex a multiple of 3. But in algebraic terms, the definitions present no new problem. The universal coefficient theorem explains that homology with integer coefficients determines all other homology theories, by use of the tensor product; it is not anodyne, in that (as we would now put it) the tensor product has derived functors that enter into a general formulation.

Cohomology, and singular homology

The 1930s were the decade of the development of cohomology theory, as several research directions grew together and the De Rham cohomology that was implicit in Poincaré's work cited earlier became the subject of definite theorems. Cohomology and homology are dual theories, in a sense that required detailed working out; at the same time it was realised that homology, that was, simplicial homology, was far from being at the end of its story. The definition of singular homology avoided the need for the apparatus of triangulations, at a cost of moving to infinitely-generated modules.

Axiomatics and extraordinary theories

The development of algebraic topology from 1940 to 1960 was very rapid, and the role of homology theory was often as 'baseline' theory, easy to compute and in terms of which topologists sought to calculate with other functors. The axiomatisation of homology theory by Eilenberg and Steenrod (the Eilenberg-Steenrod axioms) revealed that what various candidate homology theories had in common was, roughly speaking, some exact sequences and in particular the Mayer-Vietoris theorem, and the dimension axiom that calculated the homology of the point. The dimension axiom was relaxed to admit the (co)homology derived from topological K-theory, and cobordism theory, in a vast extension to the extraordinary (co)homology theories that became standard in homotopy theory. These can be easily characterised for the category of CW complexes.

• List of cohomology theories

Current state of homology theory

For more general (i.e. worse-behaved) spaces, recourse to ideas from sheaf theory brought some extension of homology theories, particularly the Borel-Moore homology for locally compact spaces.

The basic chain complex apparatus of homology theory has long since become a separate piece of technique in homological algebra, and has been applied independently, for example to group cohomology. Therefore there is no longer one homology theory, but many homology and cohomology theories in mathematics.