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Published December 2021 | Accepted Version + Published
Journal Article Open

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

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.

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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Published - S0002-9947-2021-08475-5.pdf

Accepted Version - 2004.07409.pdf

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Additional details

Created:
August 22, 2023
Modified:
October 23, 2023