(Z_2^k)-manifolds are isospectral on forms

Abstract

We obtain a simple formula for the multiplicity of eigenvalues of the Hodge-Laplace operator, $Δ_f$, acting on sections of the full exterior bundle over an arbitrary compact flat Riemannian n-manifold M with holonomy group Z_2^k, with 0<k<n. This formula implies that any two compact flat manifolds with holonomy group Z_2^k having isospectral lattices of translations are isospectral on forms, that is, with respect to $Δ_f$. As a consequence, we construct a large family of pairwise $Δ_f$-isospectral and nonhomeomorphic n-manifolds of cardinality greater than $2^{(n-1)(n-2)/2}$.