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

TL;DR

This paper investigates the applicability of the K"unneth formula to foliated de Rham cohomology, addressing challenges posed by non-Hausdorff spaces and providing conditions under which the formula holds or fails.

Contribution

It establishes new K"unneth formulas for foliated cohomology under specific Hausdorff conditions and presents a counterexample to an alternative formulation.

Findings

K"unneth formula holds when both factors have Hausdorff cohomology

K"unneth formula holds when one factor has finite-dimensional Hausdorff cohomology

Counterexample shows limitations of alternative K"unneth formulations

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.