The existence of a social choice model on a preference space P is a topological, even homotopical problem. It has been solved 50 years ago, under different terminology, by the author and, a little later, jointly with T. Ganea and P.J. Hilton. P must be an H-space and either contractible or homotopy equivalent to a product of Eilenberg-MacLane spaces over the rationals.

In an introductory lecture to the "Borel-Seminar" on `2-homology in Bern, summer 2002, I described among various others a new class of finitely presented groups where the first `2-Betti number vanishes. Namely, all infi- nite lattices in arbitrary connected Lie groups apart from special exceptions. These exceptions are the lattices commensurable with those in PSL2(R), the isometry group of the hyperbolic plane. In the general case it follows that the deficiency of the lattice is � 1. The lattices with vanishing first `2-Betti number and deficiency equal to 1 are of special interest. They are examined in Part Two of this paper. After writing down the present detailed survey of these aspects of my lecture I learned that the class was NOT that new: It had appeared, not long before, in a paper by John Lott (L) on the deficiency of lattices. In spite of a considerable overlap between the two papers there are some dif- ferences in methods and motivation. As for the methods, for example, my treatment of harmonic L2-forms is reduced to the cocompact case (thanks to Gaboriau's proportionality principle) and thus does not use the singular `2-theory of Cheeger-Gromov nor their interesting but complicated approach to the non-cocompact case in (CH-G2). Or, in Part Two, the additivity of (virtual) cohomology dimensions for group extensions is a simplifying tool. My motivation for Part Two was the discussion of knot groups, since they have exactly the respective properties; this is not in (L), where the motivation for the special lattices is to show that, apart from few exceptions, lattices have deficiency � 0 (which contains various special results by Lubotzky). I thank Marc Burger for good remarks and suggestions.

. The complex group algebra CG of a countable group G can be imbedded in the von Neumann algebra NG of G . If G is torsion-free, and if P is a finitely generated projective module over CG it is proved that the central-valued trace of NG⊗ _ CGP , i.e. of an idempotent CG -matrix A defining P is equal to the canonical trace κ (P) times identity I . It follows that κ (P) characterizes the isomorphism type of NG⊗ _ CGP .¶If κ (P) is an integer, e.g., if the weak Bass conjecture holds for G then NG⊗ _ C GP is free. It is also shown that for certain classes of groups geometric arguments can be used to prove the Bass conjecture.

These are notes from a mini-course at the ETH Zurich addressed to faculty and advanced students.Its purpose was to provide a first acquaintance of the Hilbert space methods in algebraic topology which were initated by Atiyah in 1976 and have become a quite general and important tool during more recent years.Prerequisites are basic algebraic topology of cell-complexes and basic concepts Of Hilbert spaces.The definitions (Hilbert-G-module, yon Neumann dimension, reduced (co)homology, ~2-Betti numbers of finite complexes) are given, as well as complete proofs of main properties such as homotopy invariance, Poincar~ duality, etc. Applications which cannot, or not easily, be done without g2-Betti numbers concern (partial) Euler characteristic, finitely presented groups, and 4manifolds; the Cheeger-Gromov lemma on amenable groups is stated and proved.The integrality conjecture known as "Atiyah conjecture" is formulated in a most general way and discussed.A word about our systematic use of the group of harmonic chains, isomorphic to both homology and cohomology groups.To prepare the ground this is illustrated, in a preliminary chapter, by the elementary case of (co-)homology with real coefficients of a finite cell-complex X.The chain groups Ci(X) are finite dimensional vector spaces with a natural scalar product where the cells form an orthonormal basis.Boundary d and coboundary 5 are adjoint maps; Ci decomposes into three mutually orthogonal subspaces: dCi+l, 5Ci-1, and the kernel