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

Abstract

We study the cohomology groups with Z2-coefficients for compact flat Riemannian manifolds of diagonal type MΓ = Γ {set minus} Rn by explicit computation of the differentials in the Lyndon-Hochschild-Serre spectral sequence. We obtain expressions for Hj (MΓ, Z2), j = 1, 2 and give an effective criterion for the non-vanishing of the second Stiefel-Whitney class w2 (MΓ). We apply the results to exhibit isospectral pairs with special cohomological properties; for instance, we give isospectral 5-manifolds with different H2 (MΓ, Z2), and isospectral 4-manifolds M, M′ having the same Z2-cohomology where w2 (M) = 0 and w2 (M′) ≠ 0. We compute the Z2-cohomology of all generalized Hantzsche-Wendtn-manifolds for n = 3, 4, 5 and we study H2 and w2 for a large n-dimensional family, Kn, with explicit computation for a subfamily of examples due to Lee and Szczarba.