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A counterexample to the HK-conjecture that is principal

Part of: Topological and differentiable algebraic systems Selfadjoint operator algebras

Published online by Cambridge University Press:  02 May 2022

ROBIN J. DEELEY
ROBIN J. DEELEY*
Affiliation:
Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, CO 80309-0395, USA
*
e-mail: robin.deeley@colorado.edu
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Abstract

Scarparo has constructed counterexamples to Matui’s HK-conjecture. These counterexamples and other known counterexamples are essentially principal but not principal. In the present paper, a counterexample to the HK-conjecture that is principal is given. Like Scarparo’s original counterexample, our counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free. The examples discussed here do satisfy the rational version of the HK-conjecture.

Keywords

the HK-conjecturegroupoidsK-theoryhomology

MSC classification

Primary: 46L80: $K$-theory and operator algebras (including cyclic theory) 22A22: Topological groupoids (including differentiable and Lie groupoids)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 6 , June 2023 , pp. 1829 - 1846
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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