Annals of Global Analysis and Geometry 18: 47–59, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. 47 Abelian Hypercomplex 8-Dimensional Nilmanifolds ISABEL G. DOTTI ⋆ FaMAF, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina. e-mail: idotti@mate.uncor.edu ANNA FINO ⋆⋆ Dipartimento di Matematica, Universitá di Torino, Via Carlo Alberto 10, 10123 Torino, Italy. e-mail: fino@dm.unito.it (Received: 31 July 1998; accepted: 22 March 1999) Abstract. We study invariant Abelian hypercomplex structures on 8-dimensional nilpotent Lie groups. We prove that a group N admitting such a structure is either Abelian or an Abelian extension of a group of type H . We determine the Poincaré polynomials of the associated nilmanifolds and study the existence of symplectic and quaternionic structures on such spaces. Mathematics Subject Classifications (1991): Primary 53C56, 53C30. Key words: hypercomplex, groups of type H , nilmanifold. 1. Introduction Throughout this paper we will concentrate ourselves on the case of a nilmanifold M = Ŵ\N , N a nilpotent Lie group of dimension 8, Ŵ a discrete subgroup, such that N is endowed with an Abelian invariant hypercomplex structure, that is an invariant hypercomplex structure {J i } i =1,2 such that for each {J i } i =1,2 any two (1, 0)-vector fields commute. Our purpose is two-fold. On the one hand, we prove that a nilpotent 8- dimensional nilpotent Lie group admitting an invariant Abelian hypercomplex structure is either Euclidean space or a trivial extension of a group of type H . Moreover, any invariant metric compatible with the whole sphere of complex structure is a central modification of the type H metric (see Theorem 4.1). On the other hand, we study some topological and geometrical properties of the associated compact nilmanifolds (see Sections 5 and 6). More precisely, using Nomizu’s theorem we compute the real cohomology and we study the existence of symplectic and quaternionic structures on such spaces. ⋆ Research partially supported by Conicet, Conicor, SecytU.N.C, Argentina and I.C.T.P., Trieste. ⋆⋆ Research partially supported by MURST and CNR of Italy.