Z2Z2-cohomology and spectral properties of flat manifolds of diagonal type

We study the cohomology groups with Z2Z2-coefficients for compact flat Riemannian manifolds of diagonal type MΓ=Γ∖RnMΓ=Γ∖Rn by explicit computation of the differentials in the Lyndon–Hochschild–Serre spectral sequence. We obtain expressions for Hj(MΓ,Z2)Hj(MΓ,Z2), j=1,2j=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)H2(MΓ,Z2), and isospectral 4-manifolds M,M′M,M′ having the same Z2Z2-cohomology where w2(M)=0 and w2(M′)≠0. We compute the Z2Z2-cohomology of all generalized Hantzsche–Wendtnn-manifolds for n=3,4,5n=3,4,5 and we study H2H2 and w2 for a large nn-dimensional family, KnKn, with explicit computation for a subfamily of examples due to Lee and Szczarba.