Z_2^k-manifolds are isospectral on forms

Abstract

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