a b s t r a c tIn this paper we consider 2-step nilpotent Lie algebras, Lie groups and nilmanifolds associated with graphs. We present a combinatorial construction of the second cohomology group for these Lie algebras. This enables us to characterize those graphs giving rise to symplectic or contact nilmanifolds.In this paper we consider finite-dimensional real 2-step nilpotent Lie algebras and nilmanifolds associated with finite graphs. This is a small class of Lie algebras with a nice combinatorial structure. In fact they are only finitely many in a given dimension, while there are infinitely many (non-isomorphic) 2-step nilpotent Lie algebras already in dimension 9 [1]. However five out of the seven six-dimensional symplectic 2-step Lie algebras [2] are associated with graphs, and it is not clear to us how the symplectic 2-step Lie algebras are related in general to the ones considered here.Our interest in this class of Lie algebras and the corresponding nilmanifolds is twofold.On the one hand we are interested in the explicit construction of cohomology classes for nilpotent Lie algebras, which is in turn related to the Toral Rank Conjecture for nilpotent Lie algebras. We look for general constructions such as the one in [3]. The combinatorial construction we present in this paper can be seen as the first step in a more general approach to the construction of cohomology classes for certain classes of 2-step nilpotent Lie algebras. At this stage it constructs all of the (first and) second cohomology groups for 2-step algebras associated with graphs. Note that the 2-step free ones are included.On the other hand, we are interested in the geometry of nilmanifolds and in particular in the construction of geometric structures such as symplectic and contact forms.We show that the first and second cohomology groups of the Lie algebras considered here can be easily read off the graph directly. This gives us an easy criterion to decide, just by inspection, whether the Lie algebra associated with a given graph admits a symplectic or contact form. It turns out that the three-dimensional Heisenberg Lie algebra is the only one of this class with a contact form while those admitting a symplectic form are exactly those corresponding to a graph where each connected component has at least as many vertices as arrows. Moreover, the proof of the criterion yields an algorithm to construct such a form.

We construct homology classes for split metabelian Lie algebras. The Toral Rank Conjecture follows for this family. We exhibit the first nontrivial examples, of nongraded algebras in any dimension 10, for which the conjecture holds.  2005 Elsevier Inc. All rights reserved.For finite-dimensional complex Lie algebras, the construction of homology classes is a general problem of interest. The class of nilpotent Lie algebras deserves particular attention, since what is known suggests that their homology is "big" in contrast to the homology of, for instance, solvable Lie algebras. In this direction the Toral Rank Conjecture (TRC)[6] asserts that dim H * (n) 2 dim z for any finite-dimensional complex nilpotent Lie algebra n with center z.A motivation for our construction of homology classes is finding reasons for the validity of the TRC. This conjecture is known to hold for algebras of small dimension or with a center of small (co)dimension, for 2-step nilpotent Lie algebras and for some graded algebras (see [1][2][3]5,7,8]). In most cases, the proof is based on combinatorial arguments which do not construct homology classes.

This article describes the cohomology spaces of any Lie algebra containing a Lie algebra of Heisenberg type (whose cohomology was studied by Santharoubane) as an ideal of codimension 1. For instance, the twisted standard filiform Lie algebras are of this kind. We give an explicit formula for the Betti numbers of this Lie algebra, and use this to describe new families of algebras whose Betti numbers do not behave unimodally.

We prove that N( f ) = |L( f )| for any continuous map f of a given infranilmanifold with Abelian holonomy group of odd order. This theorem is the analogue of a theorem of Anosov for continuous maps on nilmanifolds. We will also show that although their fundamental groups are solvable, the infranilmanifolds we consider are in general not solvmanifolds, and hence they cannot be treated using the techniques developed for solvmanifolds.

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