Dynamical obstructions to classification by (co)homology and other TSI-group invariants
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- by Shaun Allison and Aristotelis Panagiotopoulos PDF
- Trans. Amer. Math. Soc. 374 (2021), 8793-8811
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 $S_{\infty }$ 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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Additional Information
- Shaun Allison
- Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
- ORCID: 0000-0003-0239-1730
- Email: shaunpallison@gmail.com
- Aristotelis Panagiotopoulos
- Affiliation: Department of Mathematics, Caltech, 1200 E. California Blvd, Pasadena, California 91125
- MR Author ID: 1132435
- ORCID: 0000-0002-7695-4842
- Email: aristotelis.panagiotopoulos@gmail.com
- 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
- Additional Notes: 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
- © Copyright 2021 by the authors
- Journal: Trans. Amer. Math. Soc. 374 (2021), 8793-8811
- MSC (2020): Primary 54H05, 37B02, 54H11; Secondary 46L35, 55R15
- DOI: https://doi.org/10.1090/tran/8475
- MathSciNet review: 4337929