Editing Chern class (section)

=== 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. -->