(Z₂ᵏ)-manifolds are isospectral on forms

الأصل · EN

We obtain a simple formula for the multiplicity of eigenvalues of the Hodge-Laplace operator, Δf, acting on sections of the full exterior bundle over an arbitrary compact flat Riemannian n-manifold M with holonomy group Z₂ᵏ, with 0<k<n. This formula implies that any two compact flat manifolds with holonomy group Z₂ᵏ having isospectral lattices of translations are isospectral on forms, that is, with respect to Δf. As a consequence, we construct a large family of pairwise Δf-isospectral and nonhomeomorphic n-manifolds of cardinality greater than 2⁽ⁿ⁻¹⁾⁽ⁿ⁻²⁾/².