Editing Todd class (section)

== Definition ==
To define the Todd class <math>\operatorname{td}(E)</math> where <math>E</math> is a complex vector bundle on a [[topological space]] <math>X</math>, it is usually possible to limit the definition to the case of a [[Whitney sum]] of [[line bundle]]s, by means of a general device of characteristic class theory, the use of [[Chern roots]] (aka, the [[splitting principle]]). For the definition, let 

::<math> Q(x) = \frac{x}{1 - e^{-x}}=\sum_{i=0}^\infty \frac{B_i}{i!}x^i = 1 +\dfrac{x}{2}+\dfrac{x^2}{12}-\dfrac{x^4}{720}+\cdots</math>
be the [[formal power series]] with the property that the coefficient of <math>x^n</math> in <math>Q(x)^{n+1}</math> is 1, where <math>B_i</math> denotes the <math>i</math>-th [[Bernoulli number]] (with <math>B_1 = +\frac{1}{2}</math>).  Consider the coefficient of <math>x^j</math> in the product 

:<math> \prod_{i=1}^m Q(\beta_i x) \ </math>

for any <math>m > j</math>.  This is symmetric in the <math>\beta_i</math>s and homogeneous of weight <math>j</math>: so can be expressed as a polynomial <math>\operatorname{td}_j(p_1,\ldots, p_j)</math> in the [[elementary symmetric function]]s <math>p</math> of the <math>\beta_i</math>s.  Then <math>\operatorname{td}_j</math> defines the '''Todd polynomials''': they form a [[multiplicative sequence]] with <math>Q</math> as characteristic [[power series]].
  
If <math>E</math> has the <math>\alpha_i</math> as its [[Chern roots]], then the '''Todd class'''

:<math>\operatorname{td}(E) = \prod Q(\alpha_i)</math>

which is to be computed in the [[cohomology ring]] of <math>X</math> (or in its completion if one wants to consider infinite-dimensional manifolds). 

The Todd class can be given explicitly as a formal power series in the Chern classes as follows:

:<math>\operatorname{td}(E) = 1 + \frac{c_1}{2} + \frac{c_1^2 +c_2}{12} + \frac{c_1c_2}{24} + \frac{-c_1^4 + 4 c_1^2 c_2 + c_1c_3 + 3c_2^2 - c_4}{720} + \cdots </math>

where the cohomology classes <math>c_i</math> are the Chern classes of <math>E</math>, and lie in the cohomology group <math>H^{2i}(X)</math>. If <math>X</math> is finite-dimensional then most terms vanish and <math>\operatorname{td}(E)</math> is a polynomial in the Chern classes.