Torsion homology and regulators of isospectral manifolds

TL;DR

This paper explores the relationship between torsion homology and regulators in isospectral manifolds constructed via Sunada's method, revealing new connections and examples of torsion differences.

Contribution

It establishes a link between torsion in integral homology and G-module structure, and provides examples where p-torsion homology differs for primes dividing the group order.

Findings

p-primary torsion homology is isomorphic for primes not dividing |G|

Examples of p-torsion homology differences for primes up to 71

Conjecture that such differences exist for all primes p

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.