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

S. CÓNSOLE, R.J. MIATELLO, J.P. ROSSETTI

JOURNAL OF GEOMETRY AND PHYSICS

ELSEVIER SCIENCE BV

Año: 2010 vol. 60 p. 760 - 781

0393-0440

a b s t r a c t We study the cohomology groups with Z2-coefficients for compact flat Riemannian manifolds of diagonal type M by explicit computation of the differentials in the Lyndon-Hochschild-Serre spectral sequence. We obtain expressions for Hj (M ; Z2), j = 1; 2, and give an effective criterion for the non-vanishing of the second Stiefel-Whitney class w2(M). We apply the results to exhibit isospectral pairs with special cohomological properties; for instance, we give isospectral 5-manifolds with different H2(M; Z2), and isospectral 4-manifolds M, M´ having the same Z2-cohomology where w2 (M) is not 0 and  w2 (M´) = 0. We compute the Z2-cohomology of all generalized Hantzsche-Wendt n-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.Z2-coefficients for compact flat Riemannian manifolds of diagonal type M by explicit computation of the differentials in the Lyndon-Hochschild-Serre spectral sequence. We obtain expressions for Hj (M ; Z2), j = 1; 2, and give an effective criterion for the non-vanishing of the second Stiefel-Whitney class w2(M). We apply the results to exhibit isospectral pairs with special cohomological properties; for instance, we give isospectral 5-manifolds with different H2(M; Z2), and isospectral 4-manifolds M, M´ having the same Z2-cohomology where w2 (M) is not 0 and  w2 (M´) = 0. We compute the Z2-cohomology of all generalized Hantzsche-Wendt n-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. ´ 2010 Elsevier B.V. All rights reserved.2010 Elsevier B.V. All rights reserved. 0.