$${\mathbb{Z}}_2^k$$ -Manifolds are isospectral on forms

Abstract

We obtain a simple formula for the multiplicity of eigenvalues of the HodgeLaplace operator, f , acting on sections of the full exterior bundle * (TM) = n p=0 p (TM)over an arbitrary compact flat Riemannian n-manifold M with holonomy group Z k 2 , 1 ≤ k ≤ n − 1. This formula implies that any two such manifolds having isospectral lattices of translations are isospectral with respect to f . As a consequence, we construct a large family of pairwise f -isospectral and nonhomeomorphic n-manifolds of cardinality greater than 2 (n−1)(n−2) 2 .

Z2-cohomology and spectral properties of flat manifolds of diagonal type

“…with B 2 i = Id, then we can assume that b j · e i = 0 or 1 2 for every 1 ≤ i ≤ n, 1 ≤ j ≤ k. Although this definition clearly imposes a restriction on Γ , this is still a very rich class of Bieberbach groups (see for instance [9]). …”

“…Note that a GHW group is of type HW if and only if it is orientable. We refer to [3,9,24] for more facts on these groups. For instance, in [9] it is shown that all GHW manifolds are isospectral on forms; that is, for the full Hodge-Laplacian acting on p-forms for all 0 ≤ p ≤ n. This also holds for the family K n .…”

“…We now consider a family K n (see [9] and also [3]) which is a subfamily of the generalized Hantzsche-Wendt manifolds. The GHW manifolds consist of those Bieberbach groups in dimension n having holonomy group Z n−1 2 .…”

“…(5) We obtain expressions for H 1 (M Γ , Z 2 ) and H 2 (M Γ , Z 2 ) for an infinite family, K n , studied in [9], and we compute it explicitly for the subfamily introduced by Lee and Szczarba in [10] (Theorems 4.4 and 4.5).…”