PSEUDODIFFERENTIAL EXTENSION AND TODD CLASS

PSEUDODIFFERENTIAL EXTENSION AND TODD CLASS Denis PERROT Universite de Lyon, Universite Lyon 1, CNRS, UMR 5208 Institut Camille Jordan, 43, bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France December 8, 2011 Abstract Let M be a closed manifold. Wodzicki shows that, in the stable range, the cyclic cohomology of the associative algebra of pseudodifferential sym- bols 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 Poincare dual of the Todd class. As an immediate corollary we obtain a purely algebraic proof of the Atiyah-Singer index theorem for elliptic pseudodif- ferential operators on closed manifolds. Keywords: Pseudodifferential operators, K-theory, cyclic cohomology. MSC 2000: 19D55, 19K56, 58J42. 1 Introduction Let M be a closed, not necessarily orientable, smooth manifold and denote by CL(M) the algebra of classical, one-step polyhomogeneous pseudodifferential operators on M . The space of smoothing operators L?∞(M) is a two-sided ideal in CL(M), and we call the quotient CS(M) = CL(M)/L?∞(M) the algebra of formal symbols on M .

• ?m endowed


• index theorem


• over


• over all


• manifold


• pseudodifferential operators


• residue cocycle


• symbol map


• associative algebra


• acting