Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 2

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 singular circles and parallel 2-tori. In the previous paper Khokliuk and Maksymenko (J Homotopy Relat Struct 18:313–356. https://doi.org/10.1007/s40062-023-00328-z, 2024) 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.