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Relative pairing in cyclic cohomology and divisor flows

Matthias Lesch·Henri Moscovici·Markus Pflaum
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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 all the essential features of the divisor flows, 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.

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Matthias Lesch, Henri Moscovici, Markus Pflaum. 2006-12-22. Relative pairing in cyclic cohomology and divisor flows. https://doi.org/10.1017/is008001021jkt051

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