The cohomology class measures the extent to which the bundle is "twisted" and whether it possesses sections.
Characteristic classes are global invariants that 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.
The notion of characteristic class arose in 1935 in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds.
for the set of isomorphism classes of principal G-bundles over
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
A characteristic class c of principal G-bundles is then a natural transformation from
In other words, a characteristic class associates to each principal G-bundle
Given an oriented manifold M of dimension n with fundamental class
denotes the cup product of cohomology classes.
These are notated variously as either the product of characteristic classes, such as
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 integrate over the manifold; this is analogous to taking the product in cohomology and pairing with the fundamental class.
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.
Characteristic classes are phenomena of cohomology theory in an essential way — 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 one does need that variance.
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 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
was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in
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 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.
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
pulls back from the cohomology class in
This is not the case for the Euler class, as detailed there, not least because the Euler class of a k-dimensional bundle lives in
, so it can't pull back from a class in