|
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. Homology
is taken to be an equivalence relation, such that cycles
C and D are homologous on the space X if the cycle C − D is a
boundary of a cycle of one dimension higher.
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
attaching 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, coeffients
(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 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.
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 theory 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.
|