Editing Characteristic class (section)

==Characteristic numbers==
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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.