Editing Characteristic class

{{Use American English|date=January 2019}}{{Short description|Association of cohomology classes to principal bundles
}}
In [[mathematics]], a '''characteristic class''' is a way of associating to each [[principal bundle]] of ''X'' a [[cohomology]] class of ''X''. The cohomology class measures the extent to which the bundle is "twisted" and whether it possesses [[section (fiber bundle)|section]]s. Characteristic classes are global [[topological invariant|invariant]]s that measure the deviation of a [[principle of locality|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.

==Definition==

Let ''G'' be a [[topological group]], and for a topological space <math>X</math>, write <math>b_G(X)</math> for the set of [[isomorphism class]]es of [[principal bundle|principal ''G''-bundles]] over <math>X</math>. This <math>b_G</math> is a [[contravariant functor]] from '''Top''' (the [[category (mathematics)|category]] of topological spaces and [[continuous function]]s) to '''Set''' (the category of [[set (mathematics)|set]]s and [[function (mathematics)|function]]s), sending a map <math>f\colon X\to Y</math> to the [[pullback bundle|pullback]] operation <math>f^*\colon b_G(Y)\to b_G(X)</math>.

A '''characteristic class''' ''c'' of principal ''G''-bundles is then a [[natural transformation]] from <math>b_G</math> to a cohomology functor <math>H^*</math>, regarded also as a functor to '''Set'''.

In other words, a characteristic class associates to each principal ''G''-bundle <math>P\to X</math> in <math>b_G(X)</math> 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==
{{redirect|Characteristic number}}

Characteristic classes are elements of cohomology groups;<ref>Informally, characteristic classes "live" in cohomology.</ref> one can obtain integers from characteristic classes, called '''characteristic numbers'''. Some important examples of characteristic numbers are [[Stiefel–Whitney class#Stiefel–Whitney numbers|Stiefel–Whitney numbers]], [[Chern class#Chern numbers|Chern numbers]], [[Pontryagin class#Pontryagin numbers|Pontryagin numbers]], and the [[Euler class#Relations to other invariants|Euler characteristic]].

Given an oriented manifold ''M'' of dimension ''n'' with [[fundamental class]] <math>[M] \in H_n(M)</math>, and a ''G''-bundle with characteristic classes <math>c_1,\dots,c_k</math>, 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 [[monomial]]s of degree ''n'' in the characteristic classes, or equivalently the partitions of ''n'' into <math>\mbox{deg}\,c_i</math>.

Formally, given <math>i_1,\dots,i_l</math> such that <math>\sum \mbox{deg}\,c_{i_j} = n</math>, the corresponding characteristic number is:
:<math>c_{i_1}\smile c_{i_2}\smile \dots \smile c_{i_l}([M])</math>

where <math>\smile</math> denotes the [[cup product]] of cohomology classes.
These are notated variously as either the product of characteristic classes, such as <math>c_1^2</math>, or by some alternative notation, such as <math>P_{1,1}</math> for the [[Pontryagin class#Pontryagin numbers|Pontryagin number]] corresponding to <math>p_1^2</math>, or <math>\chi</math> for the Euler characteristic.

From the point of view of [[de Rham cohomology]], one can take [[differential form]]s representing the characteristic classes,<ref>By [[Chern–Weil theory]], these are polynomials in the curvature; by [[Hodge theory]], one can take harmonic form.</ref> 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.

This also works for non-orientable manifolds, which have a <math>\mathbf{Z}/2\mathbf{Z}</math>-orientation, in which case one obtains <math>\mathbf{Z}/2\mathbf{Z}</math>-valued characteristic numbers, such as the Stiefel-Whitney numbers.

Characteristic numbers solve the oriented and unoriented [[Cobordism#Cobordism classes|bordism question]]s: two manifolds are (respectively oriented or unoriented) cobordant if and only if their characteristic numbers are equal.

==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]].

==Stability==
In the language of [[stable homotopy theory]], the [[Chern class]], [[Stiefel–Whitney 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: <math>c(V \oplus 1) = c(V)</math>. More abstractly, it means that the cohomology class in the [[classifying space]] for <math>BG(n)</math> pulls back from the cohomology class in <math>BG(n+1)</math> under the inclusion <math>BG(n) \to BG(n+1)</math> (which corresponds to the inclusion <math>\mathbf{R}^n \to \mathbf{R}^{n+1}</math> and similar). Equivalently, all finite characteristic classes pull back from a stable class in <math>BG</math>.

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 <math>H^k(X)</math> (hence pulls back from <math>H^k(BO(k))</math>, so it can't pull back from a class in <math>H^{k+1}</math>, as the dimensions differ.

==See also==
*[[Segre class]]
*[[Euler characteristic]]
*[[Chern class]]

== Notes ==
<references/>

== References ==

* {{cite book|first=Shiing-Shen|last=Chern|authorlink=Shiing-Shen Chern|title=Complex manifolds without potential theory|publisher=Springer-Verlag Press|year=1995|isbn=0-387-90422-0}} {{ISBN|3-540-90422-0}}.
*: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.
* {{citation|first=Allen|last=Hatcher|authorlink=Allen Hatcher|url=https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html|title= Vector bundles & K-theory}}
* {{cite book|first=Dale|last=Husemoller|authorlink=Dale Husemoller|title=Fibre bundles|publisher=McGraw Hill|year= 1966|edition= 3rd Edition, Springer 1993|isbn=0387940871}}
* {{cite book|first1=John W.|last1=Milnor|authorlink1=John Milnor| first2=Jim|last2=Stasheff| authorlink2=James D. Stasheff|title=Characteristic classes|series=Annals of Mathematics Studies|volume=76|publisher=[[Princeton University Press]], Princeton, NJ; [[University of Tokyo Press]], Tokyo|year= 1974|isbn=0-691-08122-0}}

{{DEFAULTSORT:Characteristic Class}}
[[Category:Characteristic classes| ]]