Leafwise morse-novikov cohomological invariants of foliations

Abstract

The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has been utilized by many researchers to study important properties and invariants of a manifold. Morse-Novikov cohomology is defined using the differential d_?=d+?? , where ? is a closed 1-form. We study Morse-Novikov cohomology in the context of singular distributions given by the kernel of differential forms, and foliations of manifold. The kernel of a d_? closed form is involutive and hence gives a foliation of a manifold. A transversely oriented foliation of a Riemannian manifold uniquely determines leafwise Morse-Novikov cohomology groups, which are independent of the choice of metric in the sense that different metrics correspond to isomorphic groups. The relevant 1-form ?, which is always leafwise closed, can be chosen to be the mean curvature 1-form of the transverse distribution of the foliation. In the case of Riemannian foliations, we prove that the reduced leafwise Morse-Novikov cohomology groups satisfy the Hodge theorem and Poincar¿e duality. We also show that for general singular foliations, the isomorphism classes of the induced leafwise Morse-Novikov cohomology groups are foliated homotopy invariants.