Homotopy invariance of foliation Betti numbers

Abstract

Foliation Betti numbers, introduced by Connes [C I], bear a striking formal similarity to the Betti numbers of a Galois covering space [A]. Both appear in a index theorem for DeRham complexes, and both can be interpreted as the trace on a v o n Neumann algebra applied to projection operators. Dodziuk proved that the covering space Betti numbers are invariants of homotopy type [D]. The purpose of this paper is to prove the corresponding result for foliations: the leafwise Betti numbers of a foliation of a compact manifold which admits an invariant transverse measure are preserved by a measure preserving leafwise homotopy equivilence. Given two compact foliated manifolds with invariant transverse measures and a leaf-wise homotopy equivalence, we construct a v o n Neumann algebra containing the yon Neumann algebras of the individual foliations and we show that there is an involution in this algebra which intertwines the corresponding projections which give rise to the Betti numbers of the two foliations. The assumption that the homotopy equivalence is measure preserving implies that there is a trace on this new algebra which restricts to the traces on the von Neumann algebras of the two foliations. Intertwining implies that the projections have the same traces, and thus that the corresponding Betti numbers are the same. The main technical problems are the construction of a transversely measureable leaf-wise triangulation whose simplicies have volume and diameter bounded away from zero, and the construction of simplicial approximations to the leafwise homotopy equivilence and its inverse. The techniques of I-D] and [H-L I] are then applicable.