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Let G be a group, and for a topological space X, write bG(X) for the set of isomorphism classes of principal G-bundles. This is a functor from Top to Set, sending a map f to the pullback operation f*. A characteristic class c of principal G-bundles is then a natural transformation from bG to a cohomology functor H*, regarded also as a functor to Set.
In other words, we want to associate to any principal G-bundle P → X an element c(P) in H*(X) such that, if f : Y → X is a continuous map, then c(f *P) = f *c(P). On the left is the class of the pullback of P to Y; on the right is the image of the class of P under the induced map in cohomology.
Characteristic classes are in an essential way phenomena of cohomology theory — they are contravariantContravariant is a mathematical term with a precise definition in tensor analysis. It specifies precisely the method (direction of projection) used to derive the components by projecting the magnitude of the tensor quantity onto the coordinate system bein constructions, in the way that a section is a kind of function on a space, and to lead to a contradiction from the existence of a section we do need that variance. In fact cohomology theory grew up after homologyIn mathematics (especially algebraic topology and abstract algebra), homology is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). See homology theory and homotopy theory, which are both covariantIn category theory, see covariant functor. In tensor analysis, a covariant coordinate system is reciprocal to a corresponding contravariant coordinate system. Roughly speaking, a covariant tensor is a vector field that defines the topology of a space; it 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 curvatureCurvature is the amount by which an geometric object deviates from being flat''. The word flat might have very different meaning depending on the object considered (for curves it is a straight line and for surfaces it is a Euclidean plane). In this articl invariants was a particular reason to make a theory, to prove a general Gauss-Bonnet theoremThe Gauss- Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). Suppose is a compact two-dimensional orient.
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 classStiefel-Whitney classes arise in mathematics as a type of characteristic class associated to real vector bundles. They are denoted , taking values in , the cohomology groups with mod coefficients. Naturally enough, we say that is the th Stiefel-Whitney cl, the Chern classHomology theory Algebraic topology Differential geometry Differential topology In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are a particular type of characteristic class associated to complex vector bund, and the Pontryagin classes) 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 Grassmannians, 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 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 groups and unitary groups as G. Once the cohomology H*(BG) was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in H*(X) 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 'extra-ordinary' 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 foliations of manifolds; 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 Chern have also proved important: see Chern-Simons theory.