Remarks on a Künneth formula for foliated de Rham cohomology

Mélanie Bertelson

Vol. 252 (2011), No. 2, 257–274

DOI: 10.2140/pjm.2011.252.257

Abstract

The validity of the Künneth formula 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. We present versions of the Künneth formula when both factors have Hausdorff foliated cohomology and when one factor has finite-dimensional foliated cohomology and a compact underlying manifold. We also give a counterexample to an alternative version of the Künneth formula. The proof of the second result involves a right inverse for the foliated de Rham differential.

Keywords

foliation, tangential de Rham cohomology, Künneth, tensor product, right inverse

Mathematical Subject Classification 2000

Primary: 46A32, 46E10, 53C12, 53D17

Secondary: 46A63, 46M18, 46A04

Milestones

Received: 10 April 2008

Revised: 9 March 2011

Accepted: 11 July 2011

Published: 29 October 2011