Definition
To define the Todd class td(E) where E is a complex vector bundle on a topological space X, it is usually possible to limit the definition to the case of a Whitney sum of line bundles, by means of a general device of characteristic class theory, the use of Chern roots (aka, the splitting principle). For the definition, let
be the formal power series with the property that the coefficient of xn in Q(x)n+1 is 1 (where the Bi are Bernoulli numbers).
If E has the αi as its Chern roots, then
which is to be computed in the cohomology ring of X (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:
- td(E) = 1 + c1/2 + (c12+c2)/12 + c1c2/24 + (−c14 + 4c12c2 + c1c3 + 3c22 − c4)/720 + ...
where the cohomology classes ci are the Chern classes of E, and lie in the cohomology group H2i(X). If X is finite dimensional then most terms vanish and td(E) is a polynomial in the Chern classes.
Read more about this topic: Todd Class
Famous quotes containing the word definition:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)