Some remarks on a K\"unneth formula for foliated de Rham cohomology

Abstract

The K\"unneth formula is one of the basic tools for computing cohomology. Its validity for foliated cohomology, that is, for the tangential de Rham cohomology of a foliated manifold, is investigated. The main difficulty encountered is the non-Hausdorff nature of the foliated cohomology spaces, forbidding the completion of the tensor product. The results presented here are a K\"unneth formula when both factors have Hausdorff foliated cohomology, a K\"unneth formula when one factor has Hausdorff finite-dimensional foliated cohomology and a counterexample to an alternative version of the K\"unneth formula. The proof of the second result involves a right inverse for the foliated de Rham differential.