Examples of foliations with infinite dimensional special cohomology

Abstract

We present two examples of foliations with infinite dimensional basic symplectic and complex cohomologies, along with a general sufficient condition for such phenomena. This puts restrictions on possible generalisations of several finiteness results from Riemannian foliations to any broader class. The examples are also noteworthy for the unusual behaviour of their basic de Rham cohomology.

On the Symplectic Structures in Frame Bundles and the Finite Dimension of Basic Symplectic Cohomologies

“…We refer the interested reader to [21,24], and [2,6] for a more in-depth treatment of the topic in the non-foliated and foliated cases, respectively.…”

“…The foliated analogues (cf. [2,6,11,16]) help to better understand the limitations of the theories. For example, there is always a compatible metric for a symplectic structure on a manifold, whereas there need not be one for a transverse symplectic structure of a foliation, and in the present paper we highlight some consequences it has for the symplectic cohomologies defined in [21].…”

“…[24] and [21,Section 3.2.1]). However, in [6] we recently gave an example of a very simple non-Riemannian transversally symplectic foliation, which shows that there is no possibility to extend Theorem 1.1 to all transversally symplectic foliations, or indeed to any other interesting broad class. We include it in the last section, in Example 5.1.…”