Editing Chern class (section)

== Constructions ==

=== Via the Chern–Weil theory ===
{{main|Chern–Weil theory}}

Given a complex [[Hermitian metric|hermitian]] [[vector bundle]] ''V'' of [[vector bundle|complex rank]] ''n'' over a [[smooth manifold]] ''M'', representatives of each Chern class (also called a '''Chern form''') <math>c_k(V)</math> of ''V'' are given as the coefficients of the [[characteristic polynomial]] of the [[curvature form]] <math>\Omega</math> of ''V''.

<math display="block">\det \left(\frac {it\Omega}{2\pi} +I\right) = \sum_k c_k(V) t^k</math>

The determinant is over the ring of <math>n \times n</math> matrices whose entries are polynomials in ''t'' with coefficients in the commutative algebra of even complex differential forms on ''M''.  The [[curvature form]] <math>\Omega</math> of ''V'' is defined as
<math display="block">\Omega = d\omega+\frac{1}{2}[\omega,\omega]</math>
with ω the [[connection form]]  and ''d'' the [[exterior derivative]], or via the same expression in which ω is a [[gauge field]] for the [[gauge group]] of ''V''.  The scalar ''t'' is used here only as an [[indeterminate (variable)|indeterminate]] to [[generating function|generate]] the sum from the determinant, and ''I'' denotes the ''n'' × ''n'' [[identity matrix]].

To say that the expression given is a ''representative'' of the Chern class indicates that 'class' here means [[up to]] addition of an [[exact differential form]]. That is, Chern classes are [[cohomology class]]es in the sense of [[de Rham cohomology]].  It can be shown that the cohomology classes of the Chern forms do not depend on the choice of connection in ''V''.

If follows from the  matrix identity <math>\mathrm{tr}(\ln(X))=\ln(\det(X))</math> that <math> \det(X) =\exp(\mathrm{tr}(\ln(X)))</math>. Now applying the  [[Taylor series|Maclaurin series]] for <math>\ln(X+I)</math>, we get the following expression for the Chern forms:

<math display="block">\sum_k c_k(V) t^k = \left[ 1 
       + i \frac{\mathrm{tr}(\Omega)}{2\pi} t 
       +   \frac{\mathrm{tr}(\Omega^2)-\mathrm{tr}(\Omega)^2}{8\pi^2} t^2
       + i \frac{-2\mathrm{tr}(\Omega^3)+3\mathrm{tr}(\Omega^2)\mathrm{tr}(\Omega)-\mathrm{tr}(\Omega)^3}{48\pi^3} t^3
       + \cdots
       \right].</math>

=== Via an Euler class ===
One can define a Chern class in terms of an Euler class. This is the approach in the book by Milnor and Stasheff, and emphasizes the role of an [[orientation of a vector bundle]].

The basic observation is that a [[complex vector bundle]] comes with a canonical orientation, ultimately because <math>\operatorname{GL}_n(\Complex)</math> is connected. Hence, one simply defines the top Chern class of the bundle to be its Euler class (the Euler class of the underlying real vector bundle) and handles lower Chern classes in an inductive fashion.

The precise construction is as follows. The idea is to do base change to get a bundle of one-less rank. Let <math>\pi\colon E \to B</math> be a complex vector bundle over a [[paracompact space]] ''B''. Thinking of ''B'' as being embedded in ''E'' as the zero section, let <math>B' = E \setminus B</math> and define the new vector bundle:
<math display="block">E' \to B'</math>
such that each fiber is the quotient of a fiber ''F'' of ''E'' by the line spanned by a nonzero vector ''v'' in ''F'' (a point of ''B′'' is specified by a fiber ''F'' of ''E'' and a nonzero vector on ''F''.)<ref>Editorial note: Our notation differs from Milnor−Stasheff, but seems more natural.</ref> Then <math>E'</math> has rank one less than that of ''E''. From the [[Gysin sequence]] for the fiber bundle <math>\pi|_{B'}\colon B' \to B</math>:
<math display="block">\cdots \to \operatorname{H}^k(B; \Z) \overset{\pi|_{B'}^*} \to \operatorname{H}^k(B'; \Z) \to \cdots,</math>
we see that <math>\pi|_{B'}^*</math> is an isomorphism for <math>k < 2n-1</math>. Let
<math display="block">c_k(E) = \begin{cases}
{\pi|_{B'}^*}^{-1} c_k(E') & k < n\\
e(E_{\R}) & k = n \\
0 & k > n
\end{cases}</math>

It then takes some work to check the axioms of Chern classes are satisfied for this definition.

See also: [[Thom space#The Thom isomorphism|The Thom isomorphism]].<!--
== Via an elementary symmetric polynomial ==
This is the approach taken by topologists such as May or Hatcher. This approach leads very directly to related notions such as Chern characters. See the "Chern polynomial" section. -->