Homotopy types of diffeomorphisms groups of simplest Morse-Bott foliations on lens spaces, 2

TL;DR

This paper investigates the homotopy types of groups of leaf-preserving and foliated diffeomorphisms of Morse-Bott foliations on lens spaces, showing their homotopy equivalence under certain conditions.

Contribution

It proves that the inclusion of leaf-preserving diffeomorphism groups into foliated diffeomorphism groups is a homotopy equivalence for these lens space foliations.

Findings

Weak homotopy types of leaf-preserving diffeomorphism groups computed previously.

Inclusion into foliated diffeomorphism groups is a homotopy equivalence.

Results extend understanding of diffeomorphism groups of Morse-Bott foliations on lens spaces.

Abstract

Let $\mathcal{F}$ be a Morse-Bott foliation on the solid torus $T=S^1\times D^2$ into $2$-tori parallel to the boundary and one singular central circle. Gluing two copies of $T$ by some diffeomorphism between their boundaries, one gets a lens space $L_{p,q}$ with a Morse-Bott foliation $\mathcal{F}_{p,q}$ obtained from $\mathcal{F}$ on each copy of $T$ and thus consisting of two singluar circles and parallel $2$-tori. In the previous paper [O. Khokliuk, S. Maksymenko, Journ. Homot. Rel. Struct., 2024, 18, 313-356] there were computed weak homotopy types of the groups $\mathcal{D}^{lp}(\mathcal{F}_{p,q})$ of leaf preserving (i.e. leaving invariant each leaf) diffeomorphisms of such foliations. In the present paper it is shown that the inclusion of these groups into the corresponding group $\mathcal{D}_{+}^{fol}(\mathcal{F}_{p,q})$ of foliated (i.e. sending leaves to leaves) diffeomorphisms which do not interchange singular circles are homotopy equivalences.