Free actions of cyclic groups of order $2^{n}$ on $S^{1}\times S^{2}$
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- by Gerhard X. Ritter PDF
- Proc. Amer. Math. Soc. 46 (1974), 137-140 Request permission
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 ${M_4} = {{\mathbf {P}}^3}\# {{\mathbf {P}}^3}$. 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
- G. R. Livesay, Fixed point free involutions on the $3$-sphere, Ann. of Math. (2) 72 (1960), 603โ611. MR 116343, DOI 10.2307/1970232
- P. M. Rice, Free actions of $Z_{4}$ on $S^{3}$, Duke Math. J. 36 (1969), 749โ751. MR 248814
- Gerhard X. Ritter, Free $Z_{8}$ actions on $S^{3}$, Trans. Amer. Math. Soc. 181 (1973), 195โ212. MR 321078, DOI 10.1090/S0002-9947-1973-0321078-8
- Yoko Tao, On fixed point free involutions of $S^{1}\times S^{2}$, Osaka Math. J. 14 (1962), 145โ152. MR 140092
Additional Information
- © Copyright 1974 American Mathematical Society
- 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