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. Here, we investigate the relationship between their integral homology. The Cheeger–Müller 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 $${\rm Reg}_i(M_1)/{\rm Reg}_i(M_2)$$ representation theoretically. Further, we prove that the $$p^\infty $$-torsion in the homology of $$M_1$$ is isomorphic to that of $$M_2$$ for all primes $$p\nmid \#G$$. For $$p\leq 71$$, we give examples of pairs of strongly isospectral hyperbolic 3-manifolds for which the $$p$$-torsion homology differs, and we conjecture such examples to exist for all primes $$p$$.

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