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Free actions of cyclic groups of order $ 2\sp{n}$ on $ S\sp{1}\times S\sp{2}$


Author: Gerhard X. Ritter
Journal: Proc. Amer. Math. Soc. 46 (1974), 137-140
MSC: Primary 57E25
DOI: https://doi.org/10.1090/S0002-9939-1974-0350768-2
MathSciNet review: 0350768
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Abstract: In [4] Y. Tao proved that if $ h$ is a fixed point free involution of $ {S^1} \times {S^2}$, then $ ({S^1} \times {S^2})/h$ must be homeomorphic to either $ {M_1} = {S^1} \times {S^2}$, or $ {M_2} = {{\mathbf{K}}^3}$, or $ {M_3} = {S^1} \times {{\mathbf{P}}^2}$ or $ {M_4} = {{\mathbf{P}}^3}\char93 {{\mathbf{P}}^3}$. In this paper we extend this result to free actions of $ {Z_{{2^n}}}$ on $ {S^1} \times {S^2}$, showing that, for $ n > 1,({S^1} \times {S^2})/{Z_{{2^n}}}$ must be homeomorphic to either $ {M_1}$ or $ {M_2}$.


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

  • [1] 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)
  • [2] P. M. Rice, Free actions of $ {Z_4}$ on $ {S^3}$, Duke Math. J. 36 (1969), 749-751. MR 40 #2064. MR 0248814 (40:2064)
  • [3] G. X. Ritter, Free actions of $ {Z_8}$ on $ {S^3}$, Trans. Amer. Math. Soc. 181 (1973), 195-212. MR 0321078 (47:9611)
  • [4] Y. Tao, On fixed point free involutions of $ {S^1} \times {S^2}$, Osaka Math. J. 14 (1962), 145-152. MR 25 #3515. MR 0140092 (25:3515)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0350768-2
Keywords: Free actions, piecewise linear, polyhedral, isotopic, manifold
Article copyright: © Copyright 1974 American Mathematical Society

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