Editing Chern class (section)

===The Chern character===
Chern classes can be used to construct a homomorphism of rings from the [[topological K-theory]] of a space to (the completion of) its rational cohomology. For a line bundle ''L'', the Chern character ch is defined by

<math display="block">\operatorname{ch}(L) = \exp(c_1(L)) := \sum_{m=0}^\infty \frac{c_1(L)^m}{m!}.</math>

More generally, if <math>V = L_1 \oplus \cdots \oplus L_n</math> is a direct sum of line bundles, with first Chern classes <math>x_i = c_1(L_i),</math> the Chern character is defined additively
<math display="block"> \operatorname{ch}(V)  = e^{x_1} + \cdots + e^{x_n} :=\sum_{m=0}^\infty \frac{1}{m!}(x_1^m + \cdots + x_n^m). </math>

This can be rewritten as:<ref>(See also {{slink||Chern polynomial}}.) Observe that when ''V'' is a sum of line bundles, the Chern classes of ''V'' can be expressed as [[elementary symmetric polynomials]] in the <math>x_i</math>, <math>c_i(V) = e_i(x_1,\ldots,x_n).</math>
In particular, on the one hand
<math display="block">c(V) := \sum_{i=0}^n c_i(V),</math>
while on the other hand
<math display="block">\begin{align}
c(V) &= c(L_1 \oplus \cdots \oplus L_n) \\
&= \prod_{i=1}^n c(L_i) \\
&= \prod_{i=1}^n (1+x_i) \\
&= \sum_{i=0}^n e_i(x_1,\ldots,x_n)
\end{align}</math>
Consequently, [[Newton's identities#Expressing power sums in terms of elementary symmetric polynomials|Newton's identities]] may be used to re-express the power sums in ch(''V'') above solely in terms of the Chern classes of ''V'', giving the claimed formula.</ref>

<math display="block"> \operatorname{ch}(V) = \operatorname{rk}(V) + c_1(V) + \frac{1}{2}(c_1(V)^2 - 2c_2(V)) + \frac{1}{6} (c_1(V)^3 - 3c_1(V)c_2(V) + 3c_3(V)) + \cdots.</math>

This last expression, justified by invoking the [[splitting principle]], is taken as the definition ''ch(V)'' for arbitrary vector bundles ''V''.

If a connection is used to define the Chern classes when the base is a manifold (i.e., the [[Chern–Weil theory]]), then the explicit form of the Chern character is
<math display="block">\operatorname{ch}(V)=\left[\operatorname{tr}\left(\exp\left(\frac{i\Omega}{2\pi}\right)\right)\right]</math>
where {{math|Ω}} is the [[curvature form|curvature]] of the connection.

The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product.  Specifically, it obeys the following identities:

<math display="block">\operatorname{ch}(V \oplus W) = \operatorname{ch}(V) + \operatorname{ch}(W)</math>
<math display="block">\operatorname{ch}(V \otimes W) = \operatorname{ch}(V) \operatorname{ch}(W).</math>

As stated above, using the Grothendieck additivity axiom for Chern classes, the first of these identities can be generalized to state that ''ch'' is a [[homomorphism]] of [[abelian group]]s from the [[K-theory]] ''K''(''X'') into the rational cohomology of ''X''.  The second identity establishes the fact that this homomorphism also respects products in ''K''(''X''), and so ''ch'' is a homomorphism of rings.

The Chern character is used in the [[Hirzebruch–Riemann–Roch theorem]].