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Torsion homology and regulators of isospectral manifolds
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Bartel, Alex and Page, Aurel (2016) Torsion homology and regulators of isospectral manifolds. Journal of Topology, 9 (4). pp. 1237-1256.
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Official URL: http://dx.doi.org/10.1112/jtopol/jtw023
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: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Torsion, Homology theory, Manifolds (Mathematics) | ||||||||
| Journal or Publication Title: | Journal of Topology | ||||||||
| Publisher: | Oxford University Press | ||||||||
| ISSN: | 1753-8416 | ||||||||
| Official Date: | December 2016 | ||||||||
| Dates: |
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| Volume: | 9 | ||||||||
| Number: | 4 | ||||||||
| Number of Pages: | 21 | ||||||||
| Page Range: | pp. 1237-1256 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Funder: | Engineering and Physical Sciences Research Council (EPSRC) | ||||||||
| Grant number: | EP/K034383/1 (EPSRC) | ||||||||
| Open Access Version: |
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