Vector bundle automorphisms preserving Morse-Bott foliations

Abstract

Let $M$ be a smooth manifold and $\mathcal{F}$ a Morse-Bott foliation on $M$ with a compact critical manifold $\Sigma$. Denote by $\mathcal{D}(\mathcal{F})$ the group of diffeomorphisms of $M$ leaving invariant each leaf of $\mathcal{F}$. Under certain assumptions on $\mathcal{F}$ it is shown that the computation of the homotopy type of $\mathcal{D}(\mathcal{F})$ reduces to three rather independent groups: the group of diffeomorphisms of $\Sigma$, the group of vector bundle automorphisms of some regular neighborhood of $\Sigma$, and the subgroup of $\mathcal{D}(\mathcal{F})$ consisting of diffeomorphisms fixed near $\Sigma$. Examples of computations of homotopy types of groups $\mathcal{D}(\mathcal{F})$ for such foliations are also presented.