Todd class

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. The Todd class plays a fundamental role in generalising the classical Riemann-Roch theorem to higher dimensions, in the Hirzebruch-Riemann-Roch theorem and 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 Hirzebruch.

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. For the definition, let

:"Q"("x") = "x"/(1 − "e"−"x")

considered as a formal power series; the expansion can be made explicit in terms of Bernoulli numbers. If "E" has the α"i" as its Chern roots, then

:"td"("E") = Π "Q"(α"i"),

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 + "c"1/2 + ("c"12+"c"2)/12 + "c"1"c"2/24 + ...where the cohomology classes "c""i" are the Chern classes of "E", and lie in the cohomology group H2"i"("X"). If "X" is finite dimensional then most terms vanish and "td"("E") is a polynomial in the Chern classes.


*J. Todd, "The arithmetical theory of algebraic loci", Proc. London Math. Soc., 43 (1937) pp. 190–225
*F. Hirzebruch, "Topological methods in algebraic geometry", Springer (1978)
*springer|id=T/t092930|title=Todd class|author=M.I. Voitsekhovskii

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Todd Pacific Shipyards — Corporation was founded in 1916 as the William H. Todd Corporation through the merger of Robins Dry Dock Repair Company of Erie Basin, Brooklyn, New York, the Tietjen Long Dry Dock Company of Hoboken, New Jersey, and the Seattle Construction Dry… …   Wikipedia

  • Todd Field — (2007) Todd Field (* 24. Februar 1964 in Pomona, Kalifornien; eigentlich William Todd Field) ist ein US amerikanischer Schauspieler, Filmregisseur, Drehbuchautor, Filmproduzent und Filmkomponist …   Deutsch Wikipedia

  • Todd Boyd — Born Detroit, Michigan Occupation Professor, Author, Media Commentator, Producer Influenced by Malcolm X, Miles Davis, Gordon Parks, Quincy Jones, Howard Cosell …   Wikipedia

  • Todd Strasser — (born May, 5th, 1950[1] in New York City, New York)[2] is an American author[3] of more than 130 young adult and middle grade novels and novelizations, some written under the pen names Morton Rhue[4] and T.S. Rue …   Wikipedia

  • Todd Hammel — Todd Layne Hammel (born December 7, 1966 in Durant, Oklahoma) is an Arena Football League quarterback playing most recently for the Colorado Crush. He has played for 10 teams since his AFL career began in 1992. He is currently a free agent. He… …   Wikipedia

  • Todd Solondz — lors du Festival international du film de Guadalajara en mars 2009 Données clés Nom de naissance …   Wikipédia en Français

  • Todd J. Greenwald — es un productor, escritor, guionista, director y actor estadounidense. Es mejor conocido por crear la serie original de Disney Channel, Los Hechiceros de Waverly Place. Contenido 1 Filmografía 1.1 Productor Ejecutivo 1.2 …   Wikipedia Español

  • Class Action (band) — Class Action Phreek Origin New York, New York, U.S. Genres Garage house, Post disco, disco Years active 1984, 1978 Labels …   Wikipedia

  • Todd Blackburn — Todd Daniel Blackburn (born October 25, 1974) is a former U.S. Army Ranger deployed with B Company, 3rd Battalion, to Mogadishu, Somalia during Operation Gothic Serpent in 1993. He was wounded in action on October 3 1993, aged 20, during an… …   Wikipedia

  • Todd P. Kennett — Todd Kennett (b. April 14, 1970, Middlebury, Connecticut) is a coach of the Division I Collegiate lightweight rowing program at Cornell University. In 2006, his varsity boat captured both the Eastern Sprints Regatta and the Intercollegiate Rowing …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”