Pseudodifferential extension and Todd class

2011/12/08 by Denis Perrot, Perrot, Denis · 1 citation
Mathematics · #FOS: Mathematics #K-Theory and Homology (math.KT) #math.KT

paper · pdf · doi:10.48550/arxiv.1112.1850

40 pages

arxiv created 2011/12/08 · arxiv updated 2011/12/09

Abstract

Let M be a closed manifold. Wodzicki shows that, in the stable range, the cyclic cohomology of the associative algebra of pseudodifferential symbols of order ≤ 0 is isomorphic to the homology of the cosphere bundle of M. In this article we develop a formalism which allows to calculate that, under this isomorphism, the Radul cocycle corresponds to the Poincaré dual of the Todd class. As an immediate corollary we obtain a purely algebraic proof of the Atiyah-Singer index theorem for elliptic pseudodifferential operators on closed manifolds.

Cited by

• Explicit Families of Spinor Representations

Related

• Homology of pseudodifferential operators I. Manifolds with boundary
• A cohomological formula for the Atiyah-Patodi-Singer index on manifolds with boundary
• Indices of pseudodifferential operators on open manifolds
• Families Index for Pseudodifferential Operators on Manifolds with Boundary
• The Radul cocycle, the Chern-Connes character, and manifolds with conical singularities