Editing Characteristic class (section)

==Motivation==

Characteristic classes are phenomena of [[cohomology theory]] in an essential way &mdash; they are [[covariance and contravariance of functors|contravariant]] constructions, in the way that a [[Section (category theory)|section]] is a kind of function ''on'' a space, and to lead to a contradiction from the existence of a section one does need that variance. In fact cohomology theory grew up after [[Homology (mathematics)|homology]] and [[homotopy theory]], which are both [[Covariance|covariant]] theories based on mapping ''into'' a space; and characteristic class theory in its infancy in the 1930s (as part of [[obstruction theory]]) was one major reason why a 'dual' theory to homology was sought. The characteristic class approach to [[curvature]] invariants was a particular reason to make a theory, to prove a general [[Gauss–Bonnet theorem]].

When the theory was put on an organised basis around 1950 (with the definitions reduced to homotopy theory) it became clear that the most fundamental characteristic classes known at that time (the [[Stiefel–Whitney class]], the [[Chern class]], and the [[Pontryagin class]]es) were reflections of the classical linear groups and their [[maximal torus]] structure. What is more, the Chern class itself was not so new, having been reflected in the [[Schubert calculus]] on [[Grassmannian]]s, and the work of the [[Italian school of algebraic geometry]]. On the other hand there was now a framework which produced families of classes, whenever there was a [[vector bundle]] involved.

The prime mechanism then appeared to be this: Given a space ''X'' carrying a vector bundle, that implied in the [[CW complex|homotopy category]] a mapping from ''X'' to a [[classifying space]] ''BG'', for the relevant linear group ''G''. For the homotopy theory the relevant information is carried by compact subgroups such as the [[orthogonal group]]s and [[unitary group]]s of ''G''. Once the cohomology <math>H^*(BG)</math> was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in <math>H^*(X)</math> in the same dimensions. For example the [[Chern class]] is really one class with graded components in each even dimension.

This is still the classic explanation, though in a given geometric theory it is profitable to take extra structure into account. When cohomology became 'extraordinary' with the arrival of [[K-theory]] and [[cobordism theory]] from 1955 onwards, it was really only necessary to change the letter ''H'' everywhere to say what the characteristic classes were.

Characteristic classes were later found for [[foliation]]s of [[manifold]]s; they have (in a modified sense, for foliations with some allowed singularities) a classifying space theory in [[homotopy]] theory.

In later work after the ''rapprochement'' of mathematics and [[physics]], new characteristic classes were found by [[Simon Donaldson]] and [[Dieter Kotschick]] in the [[instanton]] theory. The work and point of view of [[Shiing-Shen Chern|Chern]] have also proved important: see [[Chern–Simons|Chern–Simons theory]].