Allison, Shaun and Panagiotopoulos, Aristotelis (2021) Dynamical obstructions to classification by (co)homology and other TSI-group invariants. Transactions of the American Mathematical Society, 374 (12). pp. 8793-8811. ISSN 0002-9947. doi:10.1090/tran/8475. https://resolver.caltech.edu/CaltechAUTHORS:20220309-966711000
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Abstract
In the spirit of Hjorth’s turbulence theory, we introduce “unbalancedness”: a new dynamical obstruction to classifying orbit equivalence relations by actions of Polish groups which admit a two-sided invariant metric (TSI). Since abelian groups are TSI, unbalancedness can be used for identifying which classification problems cannot be solved by classical homology and cohomology theories. In terms of applications, we show that Morita equivalence of continuous-trace C*-algebras, as well as isomorphism of Hermitian line bundles, are not classifiable by actions of TSI groups. In the process, we show that the Wreath product of any two non-compact subgroups of S_∞ admits an action whose orbit equivalence relation is generically ergodic against any action of a TSI group and we deduce that there is an orbit equivalence relation of a CLI group which is not classifiable by actions of TSI groups.
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| Alternate Title: | Dynamical obstructions for classification by actions of TSI groups | |||||||||
| Additional Information: | © Copyright 2021 by the authors. Received by editor(s): August 23, 2020. Received by editor(s) in revised form: December 3, 2020, and May 7, 2021. Published electronically: September 29, 2021. We want to acknowledge the hospitality and financial support of the California Institute of Technology during the visit of S.A. in the winter of 2020. We are grateful to A. Shani, M. Lupini, J. Bergfalk, and A.S. Kechris for all the useful and inspiring discussions, as well as to S. Coskey and J.D. Clemens for sharing an early draft of [CC] with us. We would also like to thank the anonymous referee for their valuable comments and for raising our attention to several subtle errors in an earlier version of this paper. | |||||||||
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| Subject Keywords: | Polish group, invariant metric, generically ergodic, turbulence, TSI, CLI, Borel reduction, continuous-trace -algebra, Morita equivalence, Hermitian line bundle | |||||||||
| Issue or Number: | 12 | |||||||||
| Classification Code: | MSC (2020): Primary 54H05, 37B02, 54H11; Secondary 46L35, 55R15 | |||||||||
| DOI: | 10.1090/tran/8475 | |||||||||
| Record Number: | CaltechAUTHORS:20220309-966711000 | |||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20220309-966711000 | |||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
| ID Code: | 113854 | |||||||||
| Collection: | CaltechAUTHORS | |||||||||
| Deposited By: | George Porter | |||||||||
| Deposited On: | 10 Mar 2022 20:37 | |||||||||
| Last Modified: | 10 Mar 2022 20:37 |
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