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Torsion homology and regulators of isospectral manifolds

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 strongly isospectral arithmetic hyperbolic 3-manifolds for which the p-torsion homology differs.
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https://hal.inria.fr/hal-01671812
Contributor : Aurel Page <>
Submitted on : Friday, December 22, 2017 - 4:19:50 PM
Last modification on : Monday, May 6, 2019 - 5:34:01 PM

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Alex Bartel, Aurel Page. Torsion homology and regulators of isospectral manifolds. Journal of topology, Oxford University Press, 2016, 9 (4), pp.1237 - 1256. ⟨10.1112/jtopol/jtw023⟩. ⟨hal-01671812⟩

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