Characteristic class

In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not. In other words, characteristic classes are global invariants which measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry and algebraic geometry.
Contents
Definition
Let G be a topological group, and for a topological space X, write b_{G}(X) for the set of isomorphism classes of principal Gbundles. This b_{G} is a contravariant functor from Top (the category of topological spaces and continuous functions) to Set (the category of sets and functions), sending a map f to the pullback operation f*.
A characteristic class c of principal Gbundles is then a natural transformation from b_{G} to a cohomology functor H*, regarded also as a functor to Set.
In other words, a characteristic class associates to any principal Gbundle 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 numbers
Characteristic classes are elements of cohomology groups;^{[1]} one can obtain integers from characteristic classes, called characteristic numbers. Respectively: StiefelWhitney numbers, Chern numbers, Pontryagin numbers, and the Euler characteristic.
Given an oriented manifold M of dimension n with fundamental class , and a Gbundle with characteristic classes , one can pair a product of characteristic classes of total degree n with the fundamental class. The number of distinct characteristic numbers is the number of monomials of degree n in the characteristic classes, or equivalently the partitions of n into .
Formally, given such that , the corresponding characteristic number is:
These are notated various as either the product of characteristic classes, such as or by some alternative notation, such as P_{1,1} for the Pontryagin number corresponding to , or χ for the Euler characteristic.
From the point of view of de Rham cohomology, one can take differential forms representing the characteristic classes,^{[2]} take a wedge product so that one obtains a top dimensional form, then integrates over the manifold; this is analogous to taking the product in cohomology and pairing with the fundamental class.
This also works for nonorientable manifolds, which have a orientation, in which case one obtains valued characteristic numbers, such as the StiefelWhitney numbers.
Characteristic numbers solve the oriented and unoriented bordism questions: two manifolds are (respectively oriented or unoriented) cobordant if and only if their characteristic numbers are equal.
Motivation
Characteristic classes are in an essential way phenomena of cohomology theory — they are contravariant 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 homology and homotopy theory, which are both 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 GaussBonnet 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 StiefelWhitney class, the Chern class, 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 of 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 'extraordinary' with the arrival of Ktheory 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 reapprochement 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 ChernSimons theory.
Stability
In the language of stable homotopy theory, the Chern class, StiefelWhitney class, and Pontryagin class are stable, while the Euler class is unstable.
Concretely, a stable class is one that does not change when one adds a trivial bundle: . More abstractly, it means that the cohomology class in the classifying space for BG(n) pulls back from the cohomology class in BG(n + 1) under the inclusion (which corresponds to the inclusion and similar). Equivalently, all finite characteristic classes pull back from a stable class in BG.
This is not the case for the Euler class, as detailed there, not least because the Euler class of a kdimensional bundle lives in H^{k}(X) (hence pulls back from H^{k}(BO(k)), so it can’t pull back from a class in H^{k + 1}, as the dimensions differ.
From the perspective of the splitting principle, this corresponds to the stability of symmetric polynomials, and the instability of alternating polynomials, specifically the Vandermonde polynomial, which represents the Euler class.
See also
 Chern class
 Segre class
 StiefelWhitney class
 Pontryagin class
 Euler class
Notes
 ^ Informally, characteristic classes "live" in cohomology.
 ^ By ChernWeil theory, these are polynomials in the curvature; by Hodge theory, one can take harmonic form.
References
 Allen Hatcher, Vector Bundles & KTheory
 Milnor, John W.; Stasheff, James D. Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. vii+331 pp. ISBN 0691081220.
 ShiingShen Chern, Complex Manifolds Without Potential Theory (SpringerVerlag Press, 1995) ISBN 0387904220, ISBN 3540904220.
 The appendix of this book: "Geometry of Characteristic Classes" is a very neat and profound introduction to the development of the ideas of characteristic classes.
Categories: Algebraic topology
 Characteristic classes
Wikimedia Foundation. 2010.
Look at other dictionaries:
characteristic — adj Characteristic, individual, peculiar, distinctive are comparable when they mean indicating or revealing the special quality or qualities of a particular person or thing or of a particular group of persons or things. Characteristic stresses… … New Dictionary of Synonyms
class — n Class, category, genus, species, denomination, genre are compared here only in their general, nonspecial ized use, and the following comments may be inapplicable to such technical fields as philosophy and the sciences. Class is a very general… … New Dictionary of Synonyms
Class stratification — is a form of social stratification in which a society tends to divide into separate classes whose members have differential access to resources and power. An economic and cultural rift usually exists between different classes. People are usually… … Wikipedia
class — n: a group of persons or things having characteristics in common: as a: a group of persons who have some common relationship to a person making a will and are designated to receive a gift under the will but whose identities will not be determined … Law dictionary
characteristic — [kar΄ək tər is′tik, kar΄iktər is′tik] adj. [Gr charaktēristikos: see CHARACTER] of or constituting the special character; typical; distinctive [the characteristic odor of cabbage] n. 1. a distinguishing trait, feature, or quality; peculiarity 2.… … English World dictionary
class — ► NOUN 1) a set or category of things having a common characteristic and differentiated from others by kind or quality. 2) a system that divides members of a society into sets based on social or economic status. 3) a set in a society ordered by… … English terms dictionary
Class formation — In mathematics, a class formation is a structure used to organize the various Galois groups and modules that appear in class field theory. They were invented by Emil Artin and John Tate. Contents 1 Definitions 2 Examples of class formations 3 The … Wikipedia
characteristic — I. noun Date: 1664 1. a distinguishing trait, quality, or property 2. the integral part of a common logarithm 3. the smallest positive integer n which for an operation in a ring or field yields 0 when any element is used n times with the… … New Collegiate Dictionary
Characteristic set — The mathematical concept of a characteristic set was discovered in the late forties by J.F. Ritt. Besides Gröbner basis method, it provides an alternative algorithmic way for solving multivariate polynomial equations or differential equations.In… … Wikipedia
Characteristic based product configurator — A characteristic based product configurator is a product configurator extension which uses a set of discrete variables, called characteristics (or features), to define all the possible product variations. The characteristics There are two… … Wikipedia