In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encountered where Chern classes exist — most notably in differential topology, the theory of complex manifolds and algebraic geometry. In rough terms, a Todd class acts like a reciprocal of a Chern class, or stands in relation to it as a conormal bundle does to a normal bundle. The Todd class plays a fundamental role in generalising the classical Riemann–Roch theorem to higher dimensions, in the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Hirzebruch–Riemann–Roch theorem.
History It is named for J. A. Todd, who introduced a special case of the concept in algebraic geometry in 1937, before the Chern classes were defined. The geometric idea involved is sometimes called the Todd-Eger class. The general definition in higher dimensions is due to Friedrich Hirzebruch.
Definition To define the Todd class td ( E ) {\displaystyle \operatorname {td} (E)} where E {\displaystyle E} is a complex vector bundle on a topological space X {\displaystyle 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
Q ( x ) = x 1 − e − x = ∑ i = 0 ∞ B i i ! x i = 1 + x 2 + x 2 12 − x 4 720 + ⋯ {\displaystyle 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 }
be the formal power series with the property that the coefficient of x n {\displaystyle x^{n}} in Q ( x ) n + 1 {\displaystyle Q(x)^{n+1}} is 1, where B i {\displaystyle B_{i}} denotes the i {\displaystyle i} -th Bernoulli number (with B 1 = + 1 2 {\displaystyle B_{1}=+{\frac {1}{2}}} ). Consider the coefficient of x j {\displaystyle x^{j}} in the product
∏ i = 1 m Q ( β i x ) {\displaystyle \prod _{i=1}^{m}Q(\beta _{i}x)\ }
for any m > j {\displaystyle m>j} . This is symmetric in the β i {\displaystyle \beta _{i}} s and homogeneous of weight j {\displaystyle j} : so can be expressed as a polynomial td j ( p 1 , … , p j ) {\displaystyle \operatorname {td} _{j}(p_{1},\ldots ,p_{j})} in the elementary symmetric functions p {\displaystyle p} of the β i {\displaystyle \beta _{i}} s. Then td j {\displaystyle \operatorname {td} _{j}} defines the Todd polynomials: they form a multiplicative sequence with Q {\displaystyle Q} as characteristic power series. If E {\displaystyle E} has the α i {\displaystyle \alpha _{i}} as its Chern roots, then the Todd class
td ( E ) = ∏ Q ( α i ) {\displaystyle \operatorname {td} (E)=\prod Q(\alpha _{i})}
which is to be computed in the cohomology ring of X {\displaystyle 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:
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