Vol. 252, No. 2, 2011

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Remarks on a Künneth formula for foliated de Rham cohomology

Mélanie Bertelson

Vol. 252 (2011), No. 2, 257–274

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.

foliation, tangential de Rham cohomology, Künneth, tensor product, right inverse
Mathematical Subject Classification 2000
Primary: 46A32, 46E10, 53C12, 53D17
Secondary: 46A63, 46M18, 46A04
Received: 10 April 2008
Revised: 9 March 2011
Accepted: 11 July 2011
Published: 29 October 2011
Mélanie Bertelson
Département de Mathématiques
Université Libre de Bruxelles
ULB, CP 218
Boulevard du Triomphe
B-1050 Bruxelles