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$ {\rm PL}$ involutions on the nonorientable $ 2$-sphere bundle over $ S\sp{1}$


Author: Paik Kee Kim
Journal: Proc. Amer. Math. Soc. 55 (1976), 449-452
DOI: https://doi.org/10.1090/S0002-9939-1976-0394717-1
MathSciNet review: 0394717
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Abstract | References | Additional Information

Abstract: We show that there are exactly nine distinct $ {\text{PL}}$ involutions on the nonorientable $ 2$-sphere bundle over $ {S^1}$, up to $ {\text{PL}}$ equivalences. This, together with results of [1], [3] and [8], classifies all $ {\text{PL}}$ involutions on the $ 2$-sphere bundles over $ {S^1}$.


References [Enhancements On Off] (What's this?)

  • [1] R. L. Fremon, Finite cyclic group actions on $ {S^1} \times {S^n}$, Thesis, Michigan State University, East Lansing, Mich., 1969.
  • [2] P. K. Kim and J. L. Tollefson, Splitting the $ PL$ involutions on nonprime $ 3$-manifolds (to appear).
  • [3] K. W. Kwun, Piecewise linear involutions of $ {S^1} \times {S^2}$, Michigan Math. J. 16 (1969), 93-96. MR 39 #3495. MR 0242161 (39:3495)
  • [4] G. R. Livesay, Fixed point free involutions on the $ 3$-sphere, Ann. of Math. (2) 72 (1960), 603-611. MR 22 #7131. MR 0116343 (22:7131)
  • [5] -, Involutions with two fixed points on the three-sphere, Ann. of Math. (2) 78 (1963), 582-593. MR 27 #5257. MR 0155323 (27:5257)
  • [6] P. A. Smith, Transformations of finite period. II, Ann. of Math. (2) 40 (1939), 690-711. MR 1, 30. MR 0000177 (1:30c)
  • [7] J. L. Tollefson, Free involutions on non-prime $ 3$-manifolds, Osaka J. Math. 7 (1970), 161-164. MR 42 #1092. MR 0266184 (42:1092)
  • [8] -, Involutions on $ {S^1} \times {S^2}$ and other $ 3$-manifolds, Trans. Amer. Math. Soc. 183 (1973), 139-152. MR 48 #5081. MR 0326738 (48:5081)
  • [9] F. Waldhausen, Über Involution der $ 3$-Sphäre, Topology 8 (1969), 81-91. MR 38 #5209. MR 0236916 (38:5209)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0394717-1
Keywords: $ 2$-sphere bundles over $ {S^1}$, $ {\text{PL}}$ involution, fixed-point set
Article copyright: © Copyright 1976 American Mathematical Society

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