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

• Published in: arXiv.org 2024-04
• Main Author: Maksymenko, Sergiy
• Format: Article
• Language: English
• Subjects: Gluing; Isomorphism; Lenses; Topology; Toruses
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2301.12447

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.