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Dynamical obstructions to classification by (co)homology and other TSI-group invariants

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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1090/tran/8475DOIArticle
https://arxiv.org/abs/2004.07409arXivDiscussion Paper
ORCID:
AuthorORCID
Allison, Shaun0000-0003-0239-1730
Panagiotopoulos, Aristotelis0000-0002-7695-4842
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.
Funders:
Funding AgencyGrant Number
CaltechUNSPECIFIED
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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