The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

Abstract

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V k to a decreasing family of k foliations F i on a manifold M . We have shown that there exists a ( 1 , 1 ) tensor J of V k such that J k ≠ 0 , J k + 1 = 0 and we defined by L J ( V k ) the Lie Algebra of vector fields X on V k such that, for each vector field Y on V k , [ X , J Y ] = J [ X , Y ] . In this note, we study the first Chevalley–Eilenberg Cohomology Group, i.e. the quotient space of derivations of L J ( V k ) by the subspace of inner derivations, denoted by H 1 ( L J ( V k ) ) .