Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Finite, Fiber-preserving Group Actions on Elliptic 3-manifolds

  • Peet, Benjamin (Department of Mathematics, St. Martin's University Lacey)
  • Received : 2020.05.16
  • Accepted : 2021.12.08
  • Published : 2022.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

In two previous papers the author presented a general construction of finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds. In this paper we restrict our attention to elliptic 3-manifolds. For illustration of our methods a constructive proof is given that orientation-reversing and fiber-preserving diffeomorphisms of Seifert manifolds do not exist for nonzero Euler class, in particular elliptic 3-manifolds. Each type of elliptic 3-manifold is then considered and the possible group actions that fit the given construction. This is shown to be all but a few cases that have been considered elsewhere. Finally, a presentation for the quotient space under such an action is constructed and a specific example is generated.

Keywords

References

  1. F. Bonahon, Geometric structures on 3-manifolds, Handbook of geometric topology, 93-164, North-Holland, Amsterdam(2002).
  2. G. Davidoff C. D. Olds, A. Lax and G. P. Davidoff, The geometry of numbers, volume 41, Cambridge University Press(2000).
  3. J. Kalliongis and R. Ohashi, Finite actions on the 2-sphere, the projective plane and i-bundles over the projective plane, Ars Math. Contemp., 15(2)(2018), 297-321 https://doi.org/10.26493/1855-3974.806.c9d
  4. J. Kalliongis and A. Miller, Geometric group actions on lens spaces, Kyungpook Math. J., 42(2)(2002), 313-313.
  5. J. Kalliongis and A. Miller, The symmetries of genus one handlebodies, Canad. J. Math, 43(2)(1991), 371-404. https://doi.org/10.4153/CJM-1991-022-1
  6. J. M. Lee, Smooth manifolds, In Introduction to Smooth Manifolds, Springer(2003).
  7. W. D. Neumann and F. Raymond, Seifert manifolds, plumbing, µ-invariant and orientation reversing maps, Lecture Notes in Math., Springer(1978).
  8. B. Peet, Finite, fiber- and orientation-preserving group actions on totally orientable Seifert manifolds, Ann. Math. Sil., 33(1)(2019), 235--265. https://doi.org/10.2478/amsil-2019-0007
  9. B. Peet, Finite, fiber- and orientation-preserving actions on orientable seifert manifolds with non-orientable base space, JP Journal of Geometry and Topology, 23(1)(2019), 45-68. https://doi.org/10.17654/gt023010045
  10. P. Scott, The geometries of 3-manifolds, Bull. Lond. Math. Soc., 15(5)(1983), 401-487. https://doi.org/10.1112/blms/15.5.401
  11. W. P. Thurston, The geometry and topology of 3-manifolds, Lecture Notes(1979).