Remarks on a Künneth formula for foliated de Rham cohomology (Q657341)

The foliated cohomology of a foliated manifold plays an important role in various aspects of the theory of foliations, see, e.g., [\textit{N. M. Dos Santos}, Ergodic Theory Dyn. Syst. 27, No. 6, 1719--1735 (2007; Zbl 1127.37026); \textit{M. Bertelson}, Commun. Contemp. Math. 3, No. 3, 441--456 (2001; Zbl 1002.53056)]. One of the difficulties encountered while studying these cohomologies is the fact that they could be infinite dimensional non-Hausdorff. The author uses the theory of nuclear topological vector spaces to prove the Künneth formula for product foliations in some important cases: (a) both foliations have Hausdorff foliated cohomology; (b) one of the foliations is of a compact manifold and has finite dimensional foliated cohomology. The paper is divided into four sections, the first one dedicated to preliminaries, among them some relevant examples. The second contains the proof of the Künneth formula in case (a), the third in case (b). The last one is devoted to the calculation of the foliated cohomology of an example of the product foliation when both factors are infinite dimensional and one of them non-Hausdorff. The calculations show that, for this particular product foliation, the Künneth formula does not hold.