Torsion homology and regulators of isospectral manifolds

Bartel, A. and Page, A. (2016) Torsion homology and regulators of isospectral manifolds. Journal of Topology, 9(4), pp. 1237-1256. (doi:10.1112/jtopol/jtw023)

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Abstract

Given a finite group G, a G-covering of closed Riemannian manifolds, and a so-called G-relation, a construction of Sunada produces a pair of manifolds M_1 and M_2 that are strongly isospectral. Such manifolds have the same dimension and the same volume, and their rational homology groups are isomorphic. We investigate the relationship between their integral homology. The Cheeger-Mueller Theorem implies that a certain product of orders of torsion homology and of regulators for M_1 agrees with that for M_2. We exhibit a connection between the torsion in the integral homology of M_1 and M_2 on the one hand, and the G-module structure of integral homology of the covering manifold on the other, by interpreting the quotients Reg_i(M_1)/Reg_i(M_2) representation theoretically. Further, we prove that the p-primary torsion in the homology of M_1 is isomorphic to that of M_2 for all primes p not dividing #G. For p = 71, we give examples of pairs of isospectral hyperbolic 3-manifolds for which the p-torsion homology differs, and we conjecture such examples to exist for all primes p.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Bartel, Dr Alex
Authors: Bartel, A., and Page, A.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of Topology
Publisher:Wiley
ISSN:1753-8416
ISSN (Online):1753-8424
Published Online:03 November 2016
Copyright Holders:Copyright © 2016 London Mathematical Society
First Published:First published in Journal of Topology 9(4): 1237-1256
Publisher Policy:Reproduced in accordance with the publisher copyright policy

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