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

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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.

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Notes

  1. In fact, it will become a right action if we define it by \(\mu (h,f) = f\circ h\). However, it will be convenient to use the terms “left-right” and “right” to refer the sides at which we apply the corresponding diffeomorphisms to \(f\).

  2. Usually an atlas of a manifold \(M\) is a collection of open embeddings \({\mathbb {R}}^{n} \supset U_i \xrightarrow {\psi _i}M\), \(i\in \Lambda \), from open subsets of \({\mathbb {R}}^{n}\) such that \(M= \cup _{i\in \Lambda } \psi _i(U_i)\). However, all the theory of manifolds will not be changed if one extends the notion of an atlas allowing each \(U_i\) to be an open subset of some n-manifold such that the corresponding transition functions are smooth maps.

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Acknowledgements

The author is sincerely grateful to anonymous Referee for very useful remarks and suggestions which allowed to improve the contents of the paper and exposition.

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  1. Algebra and Topology Department, Institute of Mathematics, NAS of Ukraine, Tereshchenkivska Str., 3, Kyiv, 01024, Ukraine

    Sergiy Maksymenko

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Maksymenko, S. Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 2. J. Homotopy Relat. Struct. 19, 239–273 (2024). https://doi.org/10.1007/s40062-024-00346-5

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