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Relative pairing in cyclic cohomology and divisor flows

Published online by Cambridge University Press:  11 February 2008

Matthias Lesch ,
Henri Moscovici  and
Markus J. Pflaum
Matthias Lesch
Affiliation:
Mathematisches Institut, Universität Bonn, Beringstr. 1, 53115 Bonn, Germany, [email protected].
Henri Moscovici
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA, [email protected].
Markus J. Pflaum
Affiliation:
Department of Mathematics, University of Colorado UCB 395, Boulder, CO 80309, USA, [email protected].
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Abstract

We construct invariants of relative K-theory classes of multiparameter dependent pseudodifferential operators, which recover and generalize Melrose's divisor flow and its higher odd-dimensional versions of Lesch and Pflaum. These higher divisor flows are obtained by means of pairing the relative K-theory modulo the symbols with the cyclic cohomological characters of relative cycles constructed out of the regularized operator trace together with its symbolic boundary. Besides giving a clear and conceptual explanation to the essential features of the divisor flows, namely homotopy invariance, additivity and integrality, this construction allows to uncover the previously unknown even-dimensional counterparts. Furthermore, it confers to the totality of these invariants a purely topological interpretation, that of implementing the classical Bott periodicity isomorphisms in a manner compatible with the suspension isomorphisms in both K-theory and in cyclic cohomology. We also give a precise formulation, in terms of a natural Clifford algebraic suspension, for the relationship between the higher divisor flows and the spectral flow.

Keywords

noncommutative geometrypseudodifferential operatorscyclic cohomologyK-theorydivisor flow
Type
Research Article
Information
Journal of K-Theory , Volume 3 , Issue 2 , April 2009 , pp. 359 - 407
Copyright
Copyright © ISOPP 2009

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