Z_2-cohomology and spectral properties of flat manifolds of diagonal type

S. CONSOLE; R.J. MIATELLO; J.P. ROSSETTI

JOURNAL OF GEOMETRY AND PHYSICS

ELSEVIER SCIENCE BV

Año: 2010 vol. 60 p. 760 - 781

0393-0440

We study the cohomology groups with Z_2-coefficients for compact flat Riemannian manifolds of diagonal type M_G = G R^n by explicit computation of the differentials in the Lyndon–Hochschild–Serre spectral sequence. We obtain expressions for H^j(M_G,Z_2),j = 1, 2 and give an effective criterion for the non-vanishing of the second Stiefel–Whitney class w_2(M_G). We apply the results to exhibit isospectral pairs with special cohomological properties; for instance, we give isospectral 5-manifolds with different H^2(M_G,Z_2), and isospectral 4-manifolds M, M´ having the same Z_2-cohomology where w_2(M)=0 and w_2(M´) not 0. We compute the Z_2-cohomology of all generalized Hantzsche–Wendt n-manifolds for n = 3, 4, 5 and we study H^2 and w_2 for a large n-dimensional family, K_n , with explicit computation for a subfamily of examples due to Lee and Szczarba.