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 Fundam. Prikl. Mat., 2016, Volume 21, Issue 5, Pages 61–77 (Mi fpm1758)

On finiteness conditions in twisted $K$-theory

M. A. Gerasimova

Lomonosov Moscow State University

Abstract: The aim of this (mostly expository) article is to show a connection between the finiteness conditions arising in twisted $K$-theory. There are two different conditions arising naturally in two main approaches to the problem of computing the index of the appropriate family of elliptic operators (the approach of Nistor and Troitsky and the approach of Mathai, Melrose, and Singer). These conditions are formulated absolutely differently, but in some sense they should be close to each other. In this paper, we find this connection and prove the corresponding formal statement. Thereby it is shown that these conditions map to each other. This opens a possibility to synthesize these approaches. It is also shown that the finiteness condition arising in the paper of Nistor and Troitsky is a special case of the finiteness condition that appears in the paper of Emerson and Meyer, where the theorem of Nistor and Troitsky is proved not only for the case of a bundle of Lie groups, but also for the case of a general groupoid.

 Funding Agency Grant Number Russian Science Foundation 16-11-10018

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Citation: M. A. Gerasimova, “On finiteness conditions in twisted $K$-theory”, Fundam. Prikl. Mat., 21:5 (2016), 61–77

Citation in format AMSBIB
\Bibitem{Ger16} \by M.~A.~Gerasimova \paper On finiteness conditions in twisted $K$-theory \jour Fundam. Prikl. Mat. \yr 2016 \vol 21 \issue 5 \pages 61--77 \mathnet{http://mi.mathnet.ru/fpm1758}