Todd class

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. 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. 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. To define the Todd class td ⁡ ( E ) {displaystyle operatorname {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 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-th Bernoulli number. Consider the coefficient of x j {displaystyle x^{j}} in the product for any m > j {displaystyle m>j} . This is symmetric in the β i {displaystyle eta _{i}} s and homogeneous of weight 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 of the β i {displaystyle eta _{i}} s. Then td j {displaystyle operatorname {td} _{j}} defines the Todd polynomials: they form a multiplicative sequence with Q as characteristic power series. If E has the αi as its Chern roots, then the Todd class which is to be computed in the cohomology ring of X (or in its completion if one wants to consider infinite-dimensional manifolds).