Skip to main content
SpringerLink
Log in

Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 1

  • Published:
Journal of Homotopy and Related Structures Aims and scope Submit manuscript

Abstract

Let \(T= S^1\times D^2\) be the solid torus, \(\mathcal {F}\) the Morse–Bott foliation on T into 2-tori parallel to the boundary and one singular circle \(S^1\times 0\), which is the central circle of the torus T, and \(\mathcal {D}(\mathcal {F},\partial T)\) the group of diffeomorphisms of T fixed on \(\partial T\) and leaving each leaf of the foliation \(\mathcal {F}\) invariant. We prove that \(\mathcal {D}(\mathcal {F},\partial T)\) is contractible. Gluing two copies of T by some diffeomorphism between their boundaries, we will get a lens space \(L_{p,q}\) with a Morse–Bott foliation \(\mathcal {F}_{p,q}\) obtained from \(\mathcal {F}\) on each copy of T. We also compute the homotopy type of the group \(\mathcal {D}(\mathcal {F}_{p,q})\) of diffeomorphisms of \(L_{p,q}\) leaving invariant each leaf of \(\mathcal {F}_{p,q}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data availability

No data for the paper is available.

Notes

  1. Those notations are used in different parts of the paper, so the reader may skip this subsection and refer to it on necessity.

  2. In general, it is not a group, since \(U\) might not be not invariant under diffeomorphisms from \(\mathcal {D}_{}(M,B,{p})\)

References

  1. Hatcher, A.: Algebraic Topology, p. 544. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  2. Palais, R.S.: Homotopy theory of infinite dimensional manifolds. Topology 5, 1–16 (1966). https://doi.org/10.1016/0040-9383(66)90002-4

    Article  MathSciNet  MATH  Google Scholar 

  3. Smale, S.: Diffeomorphisms of the \(2\)-sphere. Proc. Am. Math. Soc. 10, 621–626 (1959). https://doi.org/10.1090/S0002-9939-1959-0112149-8

    Article  MathSciNet  MATH  Google Scholar 

  4. Earle, C.J., Eells, J.: The diffeomorphism group of a compact Riemann surface. Bull. Am. Math. Soc. 73, 557–559 (1967). https://doi.org/10.1090/S0002-9904-1967-11746-4

    Article  MathSciNet  MATH  Google Scholar 

  5. Earle, C.J., Eells, J.: A fibre bundle description of Teichmüller theory. J. Differ. Geom. 3, 19–43 (1969). https://doi.org/10.4310/jdg/1214428816

    Article  MATH  Google Scholar 

  6. Earle, C.J., Schatz, A.: Teichmüller theory for surfaces with boundary. J. Differ. Geom. 4, 169–185 (1970). https://doi.org/10.4310/jdg/1214429381

    Article  MATH  Google Scholar 

  7. Gramain, A.: Le type d’homotopie du groupe des difféomorphismes d’une surface compacte. Ann. Sci. École Norm. Sup. 4(6), 53–66 (1973). https://doi.org/10.24033/asens.1242

    Article  MATH  Google Scholar 

  8. Hatcher, A.E.: A proof of the Smale conjecture, \({\rm Diff}(S^{3})\simeq {\rm O}(4)\). Ann. Math. (2) 117(3), 553–607 (1983). https://doi.org/10.2307/2007035

    Article  MathSciNet  MATH  Google Scholar 

  9. Gabai, D.: The Smale conjecture for hyperbolic 3-manifolds: \({\rm Isom}(M^3)\simeq {\rm Diff}(M^3)\). J. Differ. Geom. 58(1), 113–149 (2001). https://doi.org/10.4310/jdg/1090348284

    Article  MathSciNet  Google Scholar 

  10. Hong, S., Kalliongis, J., McCullough, D., Rubinstein, J.H.: Diffeomorphisms of Elliptic 3-Manifolds. Lecture Notes in Mathematics, vol. 2055, p. 155. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31564-0

    Book  MATH  Google Scholar 

  11. Novikov, S.P.: Differentiable sphere bundles. Izv. Akad. Nauk SSSR Ser. Mat. 29, 71–96 (1965)

    MathSciNet  Google Scholar 

  12. Schultz, R.: Improved estimates for the degree of symmetry of certain homotopy spheres. Topology 10, 227–235 (1971). https://doi.org/10.1016/0040-9383(71)90007-3

    Article  MathSciNet  MATH  Google Scholar 

  13. Hajduk, B.: On the homotopy type of diffeomorphism groups of homotopy spheres. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26(12), 1003–10061979 (1978)

    MathSciNet  MATH  Google Scholar 

  14. Dwyer, W.G., Szczarba, R.H.: On the homotopy type of diffeomorphism groups. Ill. J. Math. 27(4), 578–596 (1983)

    MathSciNet  MATH  Google Scholar 

  15. Kupers, A.: Some finiteness results for groups of automorphisms of manifolds. Geom. Topol. 23(5), 2277–2333 (2019). https://doi.org/10.2140/gt.2019.23.2277

    Article  MathSciNet  MATH  Google Scholar 

  16. Berglund, A., Madsen, I.: Rational homotopy theory of automorphisms of manifolds. Acta Math. 224(1), 67–185 (2020). https://doi.org/10.4310/acta.2020.v224.n1.a2

    Article  MathSciNet  MATH  Google Scholar 

  17. Smolentsev, N.K.: Diffeomorphism groups of compact manifolds. Sovrem. Mat. Prilozh. 37, Geometriya, 3–100 (2006). https://doi.org/10.1007/s10958-007-0471-0

    Article  Google Scholar 

  18. Milnor, J.: Differential topology forty-six years later. Not. Am. Math. Soc. 58(6), 804–809 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Fukui, K., Ushiki, S.: On the homotopy type of \(F{\rm Diff} (S^{3},\)\({\cal{F} }_{R})\). J. Math. Kyoto Univ. 15, 201–210 (1975). https://doi.org/10.1215/kjm/1250523125

    Article  MathSciNet  MATH  Google Scholar 

  20. Fukui, K.: On the homotopy type of some subgroups of \( {\rm Diff}(M^{3})\). Jpn. J. Math. (N.S.) 2(2), 249–267 (1976). https://doi.org/10.4099/math1924.2.249

    Article  MATH  Google Scholar 

  21. Herman, M.-R.: Simplicité du groupe des difféomorphismes de classe \(C^{\infty }\), isotopes à l’identité, du tore de dimension \(n\). C. R. Acad. Sci. Paris Sér. A-B 273, 232–234 (1971)

    MathSciNet  MATH  Google Scholar 

  22. Thurston, W.: Foliations and groups of diffeomorphisms. Bull. Am. Math. Soc. 80, 304–307 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mather, J.N.: The vanishing of the homology of certain groups of homeomorphisms. Topology 10, 297–298 (1971). https://doi.org/10.1016/0040-9383(71)90022-X

    Article  MathSciNet  MATH  Google Scholar 

  24. Mather, J.N.: Simplicity of certain groups of diffeomorphisms. Bull. Am. Math. Soc. 80, 271–273 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  25. Epstein, D.B.A.: The simplicity of certain groups of homeomorphisms. Compos. Math. 22, 165–173 (1970)

    MathSciNet  MATH  Google Scholar 

  26. Ling, W.: Factorizable groups of homeomorphisms. Compos. Math. 51(1), 41–50 (1984)

    MathSciNet  MATH  Google Scholar 

  27. Anderson, R.D.: The algebraic simplicity of certain groups of homeomorphisms. Am. J. Math. 80, 955–963 (1958). https://doi.org/10.2307/2372842

    Article  MathSciNet  MATH  Google Scholar 

  28. Rybicki, T.: The identity component of the leaf preserving diffeomorphism group is perfect. Monatsh. Math. 120(3–4), 289–305 (1995). https://doi.org/10.1007/BF01294862

    Article  MathSciNet  MATH  Google Scholar 

  29. Rybicki, T.: Isomorphisms between leaf preserving diffeomorphism groups. Soochow J. Math. 22(4), 525–542 (1996)

    MathSciNet  MATH  Google Scholar 

  30. Rybicki, T.: Homology of the group of leaf preserving homeomorphisms. Demonstr. Math. 29(2), 459–464 (1996)

    MathSciNet  MATH  Google Scholar 

  31. Rybicki, T.: The flux homomorphism in the foliated case. In: Differential Geometry and Applications (Brno, 1998), pp. 413–418. Masaryk Univ., Brno (1999)

  32. Rybicki, T.: On foliated, Poisson and Hamiltonian diffeomorphisms. Differ. Geom. Appl. 15(1), 33–46 (2001). https://doi.org/10.1016/S0926-2245(01)00042-0

    Article  MathSciNet  MATH  Google Scholar 

  33. Haller, S., Teichmann, J.: Smooth perfectness through decomposition of diffeomorphisms into fiber preserving ones. Ann. Glob. Anal. Geom. 23(1), 53–63 (2003). https://doi.org/10.1023/A:1021280213742

    Article  MathSciNet  MATH  Google Scholar 

  34. Abe, K., Fukui, K.: On the first homology of automorphism groups of manifolds with geometric structures. Cent. Eur. J. Math. 3(3), 516–528 (2005). https://doi.org/10.2478/BF02475921

    Article  MathSciNet  MATH  Google Scholar 

  35. Lech, J., Rybicki, T.: Groups of \(C^{r,s}\)-diffeomorphisms related to a foliation. In: Geometry and Topology of Manifolds, vol. 76, pp. 437–450. Banach Center Publ., Institute of Mathematics of the Polish Acad. Sci. Inst. Math, Warsaw (2007). https://doi.org/10.4064/bc76-0-21

  36. Fukui, K.: Homologies of the group \({\rm Diff}^{\infty }({ R}^{n},\,0)\) and its subgroups. J. Math. Kyoto Univ. 20(3), 475–487 (1980). https://doi.org/10.1215/kjm/1250522211

    Article  MathSciNet  MATH  Google Scholar 

  37. Rybicki, T.: On the group of diffeomorphisms preserving a submanifold. Demonstr. Math. 31(1), 103–110 (1998)

    MathSciNet  MATH  Google Scholar 

  38. Maksymenko, S.: Local inverses of shift maps along orbits of flows. Osaka J. Math. 48(2), 415–455 (2011). arXiv:0806.1502

    MathSciNet  MATH  Google Scholar 

  39. Lech, J., Michalik, I.: On the structure of the homeomorphism and diffeomorphism groups fixing a point. Publ. Math. Debr. 83(3), 435–447 (2013). https://doi.org/10.5486/PMD.2013.5551

    Article  MathSciNet  MATH  Google Scholar 

  40. Fomenko, A.T.: Symplectic topology of completely integrable Hamiltonian systems. Uspekhi Mat. Nauk 44(1(265)), 145–173248 (1989). https://doi.org/10.1070/RM1989v044n01ABEH002006

    Article  MathSciNet  MATH  Google Scholar 

  41. Vũ Ngoc, S.: On semi-global invariants for focus-focus singularities. Topology 42(2), 365–380 (2003). https://doi.org/10.1016/S0040-9383(01)00026-X

    Article  MathSciNet  MATH  Google Scholar 

  42. Scárdua, B., Seade, J.: Codimension one foliations with Bott–Morse singularities. I. J. Differ. Geom. 83(1), 189–212 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Mafra, A., Scárdua, B., Seade, J.: On smooth deformations of foliations with singularities. J. Singul. 9, 101–110 (2014)

    MathSciNet  MATH  Google Scholar 

  44. Wiesendorf, S.: Taut submanifolds and foliations. J. Differ. Geom. 96(3), 457–505 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Martínez-Alfaro, J., Meza-Sarmiento, I.S., Oliveira, R.D.S.: Singular levels and topological invariants of Morse Bott integrable systems on surfaces. J. Differ. Equ. 260(1), 688–707 (2016). https://doi.org/10.1016/j.jde.2015.09.008

    Article  MathSciNet  MATH  Google Scholar 

  46. Martínez-Alfaro, J., Meza-Sarmiento, I.S., Oliveira, R.D.S.: Singular levels and topological invariants of Morse–Bott foliations on non-orientable surfaces. Topol. Methods Nonlinear Anal. 51(1), 183–213 (2018)

    MathSciNet  MATH  Google Scholar 

  47. Evangelista-Alvarado, M., Suárez-Serrato, P., Torres Orozco, J., Vera, R.: On Bott–Morse foliations and their Poisson structures in dimension three. J. Singul. 19, 19–33 (2019). https://doi.org/10.5427/jsing.2019.19b

    Article  MathSciNet  MATH  Google Scholar 

  48. Maksymenko, S.: Smooth shifts along trajectories of flows. Topol. Appl. 130(2), 183–204 (2003). https://doi.org/10.1016/S0166-8641(02)00363-2

    Article  MathSciNet  MATH  Google Scholar 

  49. Maksymenko, S.: Stabilizers and orbits of smooth functions. Bull. Sci. Math. 130(4), 279–311 (2006). https://doi.org/10.1016/j.bulsci.2005.11.001

    Article  MathSciNet  MATH  Google Scholar 

  50. Maksymenko, S.: Homotopy types of stabilizers and orbits of Morse functions on surfaces. Ann. Glob. Anal. Geom. 29(3), 241–285 (2006). https://doi.org/10.1007/s10455-005-9012-6

    Article  MathSciNet  MATH  Google Scholar 

  51. Maksymenko, S., Feshchenko, B.: Orbits of smooth functions on \(2\)-torus and their homotopy types. Matematychni Studii 44(1), 67–84 (2015). arXiv:1409.0502

    MathSciNet  MATH  Google Scholar 

  52. Maksymenko, S.: Deformations of functions on surfaces by isotopic to the identity diffeomorphisms. Topol. Appl. 282, 107312–48 (2020). https://doi.org/10.1016/j.topol.2020.107312

    Article  MathSciNet  MATH  Google Scholar 

  53. Kuznietsova, I., Maksymenko, S.: Reversing orientation homeomorphisms of surfaces. Proc. Int. Geom. Cent. 13(4), 129–159 (2020). https://doi.org/10.15673/tmgc.v13i4.1953

    Article  MathSciNet  MATH  Google Scholar 

  54. Kuznietsova, I., Soroka, Y.: First Betti numbers of orbits of Morse functions on surfaces. Ukraïn. Mat. Zh. 73(2), 179–200 (2021). https://doi.org/10.37863/umzh.v73i2.2383

    Article  MathSciNet  MATH  Google Scholar 

  55. Kudryavtseva, E.A.: Realization of smooth functions on surfaces as height functions. Mat. Sb. 190(3), 29–88 (1999). https://doi.org/10.1070/SM1999v190n03ABEH000392

    Article  MathSciNet  MATH  Google Scholar 

  56. Kudryavtseva, E.A.: Special framed Morse functions on surfaces. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4, 14–20 (2012). https://doi.org/10.3103/S0027132212040031

    Article  MathSciNet  MATH  Google Scholar 

  57. Kudryavtseva, E.A.: The topology of spaces of Morse functions on surfaces. Math. Notes 92(1–2), 219–236 (2012). https://doi.org/10.1134/S0001434612070243. (translation of Mat. Zametki 92 (2012), no. 2, 241–261)

    Article  MathSciNet  MATH  Google Scholar 

  58. Kudryavtseva, E.A.: On the homotopy type of spaces of Morse functions on surfaces. Mat. Sb. 204(1), 79–118 (2013). https://doi.org/10.1070/SM2013v204n01ABEH004292

    Article  MathSciNet  Google Scholar 

  59. Kudryavtseva, E.A.: Topology of spaces of functions with prescribed singularities on the surfaces. Dokl. Akad. Nauk 468(2), 139–142 (2016). https://doi.org/10.1134/s1064562416030066

    Article  MathSciNet  Google Scholar 

  60. Khokhliuk, O., Maksymenko, S.: Diffeomorphisms preserving Morse–Bott functions. Indag. Math. (N.S.) 31(2), 185–203 (2020). https://doi.org/10.1016/j.indag.2019.12.004

    Article  MathSciNet  MATH  Google Scholar 

  61. Khokhliuk, O., Maksymenko, S.: Smooth approximations and their applications to homotopy types. Proc. Int. Geom. Cent. 13(2), 68–108 (2020). https://doi.org/10.15673/tmgc.v13i2.1781

    Article  MathSciNet  Google Scholar 

  62. Khokhliuk, O., Maksymenko, S.: Foliated and compactly supported isotopies of regular neighborhoods (2022) arXiv:2208.05876

  63. Ivanov, N.V.: Groups of diffeomorphisms of Waldhausen manifolds. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 66, 172–176209 (1976). https://doi.org/10.1007/BF01098421. (studies in topology, II)

    Article  MathSciNet  MATH  Google Scholar 

  64. Lima, E.L.: On the local triviality of the restriction map for embeddings. Comment. Math. Helv. 38, 163–164 (1964). https://doi.org/10.1007/BF02566913

    Article  MathSciNet  MATH  Google Scholar 

  65. Reidemeister, K.: Homotopieringe und Linsenräume. Abh. Math. Sem. Univ. Hamburg 11(1), 102–109 (1935). https://doi.org/10.1007/BF02940717

    Article  MathSciNet  MATH  Google Scholar 

  66. Bonahon, F.: Difféotopies des espaces lenticulaires. Topology 22(3), 305–314 (1983). https://doi.org/10.1016/0040-9383(83)90016-2

    Article  MathSciNet  MATH  Google Scholar 

  67. Kalliongis, J., Miller, A.: Geometric group actions on lens spaces. Kyungpook Math. J. 42(2), 313–344 (2002)

    MathSciNet  MATH  Google Scholar 

  68. Cerf, J.: Topologie de certains espaces de plongements. Bull. Soc. Math. France 89, 227–380 (1961). https://doi.org/10.24033/bsmf.1567

    Article  MathSciNet  MATH  Google Scholar 

  69. Palais, R.S.: Local triviality of the restriction map for embeddings. Comment. Math. Helv. 34, 305–312 (1960). https://doi.org/10.1007/BF02565942

    Article  MathSciNet  MATH  Google Scholar 

  70. Wajnryb, B.A.: Mapping class group of a handlebody. Fund. Math. 158(3), 195–228 (1998). https://doi.org/10.4064/fm-158-3-195-228

    Article  MathSciNet  MATH  Google Scholar 

  71. Brody, E.J.: The topological classification of the lens spaces. Ann. Math. 2(71), 163–184 (1960). https://doi.org/10.2307/1969884

    Article  MathSciNet  MATH  Google Scholar 

  72. Gadgil, S.: Cobordisms and Reidemeister torsions of homotopy lens spaces. Geom. Topol. 5, 109–125 (2001). https://doi.org/10.2140/gt.2001.5.109

    Article  MathSciNet  MATH  Google Scholar 

  73. Hatcher, A.: Homeomorphisms of sufficiently large \(P^{2}\)-irreducible \(3\)-manifolds. Topology 15(4), 343–347 (1976). https://doi.org/10.1016/0040-9383(76)90027-6

    Article  MathSciNet  MATH  Google Scholar 

  74. Ivanov, N.V.: Homotopies of automorphism spaces of some three-dimensional manifolds. Dokl. Akad. Nauk SSSR 244(2), 274–277 (1979)

    MathSciNet  Google Scholar 

  75. Ivanov, N.V.: Corrections: “Homotopies of automorphism spaces of some three-dimensional manifolds’’. Dokl. Akad. Nauk SSSR 244, 274–277 (1979). (MR 80k:57015. Dokl. Akad. Nauk SSSR (no. 6), 1288 (1979))

    MathSciNet  Google Scholar 

  76. Rubinstein, J.H.: On \(3\)-manifolds that have finite fundamental group and contain Klein bottles. Trans. Am. Math. Soc. 251, 129–137 (1979). https://doi.org/10.2307/1998686

    Article  MathSciNet  MATH  Google Scholar 

  77. Hatcher, A.: On the diffeomorphism group of \(S^{1}\times S^{2}\). Proc. Am. Math. Soc. 83(2), 427–430 (1981). https://doi.org/10.2307/2043543

    Article  MATH  Google Scholar 

  78. Ivanov, N.V.: Homotopy of spaces of diffeomorphisms of some three-dimensional manifolds. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 72–103164165. Studies in topology, IV (1982)

  79. Balmer, R., Kleiner, B.: Ricci flow and contractibility of spaces of metrics. (2019). arXiv:1909.08710

  80. Kalliongis, J., Miller, A.: Actions on lens spaces which respect a Heegaard decomposition. Topol. Appl. 130(1), 19–55 (2003). https://doi.org/10.1016/S0166-8641(02)00196-7

    Article  MathSciNet  MATH  Google Scholar 

  81. Putman, A.: Answer on MathOverflow to the question: homotopy type of \({\rm Diff} (\mathbb{R}{P}^3)\). https://mathoverflow.net/q/431229. Accessed 26 Sept 2022

Download references

Acknowledgements

The authors are grateful to Andy Putman for the information on MathOverflow ( [81]) about the homotopy type of \(\mathcal {D}(\mathbb {R}{P}^3)\) and reference to the paper [79], to Ryan Budney and Neil Strickland for comments, and to Ivan Smith for reference to the paper [20] by Fukui. The authors also thanks the anonymous Referee for careful reading the paper, useful comments, and suggestions.

Author information

Authors and Affiliations

  1. Liceum “Educator”, Yelyzavety Chavdar St, 11a, Kyiv, 02140, Ukraine

    Oleksandra Khokhliuk

  2. Algebra and Topology Department, Institute of Mathematics NAS of Ukraine, Tereshchenkivska str., 3, Kyiv, 01024, Ukraine

    Sergiy Maksymenko

Authors
  1. Oleksandra Khokhliuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergiy Maksymenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergiy Maksymenko.

Additional information

Communicated by Benson Farb

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The contributions of each of the authors to this paper are equal.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khokhliuk, O., Maksymenko, S. Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 1. J. Homotopy Relat. Struct. 18, 313–356 (2023). https://doi.org/10.1007/s40062-023-00328-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40062-023-00328-z

Keywords

Mathematics Subject Classification

Search

Navigation