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. Euclid, detail from The School of Athens by Raphael. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

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 G-bundles. 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^*. An isomorphism class is a collection of mathematical objects isomorphic with a certain mathematical object. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... This article is about sets in mathematics. ... Partial plot of a function f. ... In mathematics, a pullback bundle or induced bundle is a common construction in the theory of fiber bundles. ...

A characteristic class c of principal G-bundles is then a natural transformation from b_G to a cohomology functor H^*, regarded also as a functor to Set. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

In other words, a characteristic class associates 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.

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 Gauss-Bonnet theorem. In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ... Section can be: A cross section (in the common sense or the physics sense) In mathematics: A conic section A section of a fiber bundle or sheaf A Caesarean section In UK law, Section 28 In the fictional Star Trek universe, Section 31 A military unit A section (land) is... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) 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). ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In category theory, see covariant functor. ... Obstruction theory is a mathematical theory concerned with when a topological manifold has a piecewise linear structure and when a piecewise linear manifold has a differentiable structure. ... Curvature is the amount by which a geometric object deviates from being flat. ... The 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). ...

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 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. Stiefel-Whitney classes arise in mathematics as a type of characteristic class associated to real vector bundles . ... This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, the Pontryagin classes are certain characteristic classes. ... In the theory of Lie groups in mathematics, especially those that are compact, a special role is played by the torus groups. ... In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). ... In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. ... In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more (flourishing roughly 1885-1935) done internationally in birational geometry, particularly on algebraic surfaces. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...

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. In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... In mathematics, a classifying space in homotopy theory of a discrete group G is, roughly speaking, a path connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

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. In mathematics, K-theory is, firstly, an extraordinary cohomology theory which consists of topological K-theory. ... In mathematics, cobordism is a relation between manifolds, based on the idea of boundary. ...

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 mathematics, informally speaking, a foliation is a kind of clothing worn on a manifold, cut from a stripy fabric. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

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. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... Simon Kirwan Donaldson, born in Cambridge in 1957, is a mathematician famous for his work on exotic four-dimensional spaces in differential geometry using instantons, and the discovery of new differential invariants. ... This article or section is in need of attention from an expert on the subject. ... Chen Xingshen Shiing-Shen Chern (陳省身; pinyin: Chén Xǐngshēn; October 26, 1911 – December 3, 2004) was a Chinese-American mathematician, one of the leading differential geometers of the twentieth century. ... In mathematics, the Chern-Simons forms are certain secondary characteristic classes. ...

See also

• Stiefel-Whitney class
• Chern class
• Pontryagin class
• Euler class

Stiefel-Whitney classes arise in mathematics as a type of characteristic class associated to real vector bundles . ... This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, the Pontryagin classes are certain characteristic classes. ... In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. ...

References

• 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 0-691-08122-0].
• Shiing-Shen Chern, Complex Manifolds Without Potential Theory (Springer-Verlag Press, 1995) [ISBN 0-387-90422-0, ISBN 3-540-904220-0]. 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.