On the First Homology of the Groups of Foliation Preserving Diffeomorphisms for Foliations with Singularities of Morse Type

Abstract

Let R n be an n-dimensional Euclidean space and F ϕ be the foliation defined by levels of a Morse function ϕ : R n → R. We determine the first homology of the identity component of the foliation preserving diffeomorphism group of (R n , F ϕ ). Then we can apply it to the calculation of the first homology of the foliation preserving diffeomorphism groups for codimension one compact foliations with singularities of Morse type.