{"title":"Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 2","authors":"Sergiy Maksymenko","doi":"10.1007/s40062-024-00346-5","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\({\\mathcal {F}}\\)</span> be a Morse–Bott foliation on the solid torus <span>\\(T=S^1\\times D^2\\)</span> into 2-tori parallel to the boundary and one singular central circle. Gluing two copies of <i>T</i> by some diffeomorphism between their boundaries, one gets a lens space <span>\\(L_{p,q}\\)</span> with a Morse–Bott foliation <span>\\({\\mathcal {F}}_{p,q}\\)</span> obtained from <span>\\({\\mathcal {F}}\\)</span> on each copy of <i>T</i> 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 <span>\\({\\mathcal {D}}^{lp}({\\mathcal {F}}_{p,q})\\)</span> 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 <span>\\({\\mathcal {D}}^{fol}_{+}({\\mathcal {F}}_{p,q})\\)</span> of foliated (i.e. sending leaves to leaves) diffeomorphisms which do not interchange singular circles are homotopy equivalences.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 2","pages":"239 - 273"},"PeriodicalIF":0.7000,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Homotopy and Related Structures","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40062-024-00346-5","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍:
Journal of Homotopy and Related Structures (JHRS) is a fully refereed international journal dealing with homotopy and related structures of mathematical and physical sciences.
Journal of Homotopy and Related Structures is intended to publish papers on
Homotopy in the broad sense and its related areas like Homological and homotopical algebra, K-theory, topology of manifolds, geometric and categorical structures, homology theories, topological groups and algebras, stable homotopy theory, group actions, algebraic varieties, category theory, cobordism theory, controlled topology, noncommutative geometry, motivic cohomology, differential topology, algebraic geometry.