A Combinatorial Formula for the Pontrjagin Classes
نویسندگان
چکیده
A combinatorial formula for the Pontrjagin classes of a triangulated manifold is given. The main ingredients are oriented matroid theory and a modified formulation of Chern-Weil theory.
منابع مشابه
Local formulae for combinatorial Pontrjagin classes
By p(|K|) denote the characteristic class of a combinatorial manifold K given by the polynomial p in Pontrjagin classes of K. We prove that for any polynomial p there exists a function taking each combinatorial manifold K to a rational simplicial cycle z(K) such that: (1) the Poincaré dual of z(K) represents the cohomology class p(|K|); (2) a coefficient of each simplex ∆ in the cycle z(K) is d...
متن کاملThe Existence of Combinatorial Formulae for Characteristic Lasses
Given a characteristic class on a locally ordered combinatorial manifold M there exists a cocycle which represents the class on M and is locally defined, i.e. its value on o E M depends only on the ordered star st(or, M). For rational classes the dependence on order disappears. There is also a locally defined cycle which carries the dual homology class. For some time it has been known that ther...
متن کاملHodge’s harmonic p-sets and Pontrjagin classes
This paper shows how Hodge’s theory of harmonic p-sets (a discrete version of his theory of harmonic forms) allows a new approach to be taken to the problem of providing a combinatorial definition of the Pontrjagin classes of a compact manifold. This approach is then related to the author’s definition of flag vectors for hypergraphs, and other objects constructed out of vertices and cells. It i...
متن کاملVaradarajan's book began as a set of lecture notes in a course on Lie groups and is intended to "introduce the aspiring graduate student as well as the nonspecialist mathematician to the fundamental themes
bundle are defined using the Chern classes of the complexification of the vector bundle. These are integral characteristic classes in dimensions divisible by 4. With this, all of the characteristic classes to be considered in the book have been defined. The rest of the body of the text is devoted to applications of the characteristic classes. First is an application to partial calculation of th...
متن کاملAbsolute Torsion and Eta-invariant
In a recent joint work with V. Turaev [6], we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion, it has the advantage of having a well-determined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector bundles, and not only for unimodular ones, as is classical Reidemeister torsion. In this p...
متن کامل