The nonexistence of free $S^{1}$ actions on some homotopy spheres
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- by Reinhard Schultz
- Proc. Amer. Math. Soc. 27 (1971), 595-597
- DOI: https://doi.org/10.1090/S0002-9939-1971-0271985-3
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Abstract:
In this paper a necessary condition is given for the existence of a free differentiable action of the circle group ${S^1}$ on a $(4k + 1)$-dimensional homotopy sphere. This includes a previously known criterion due to R. Lee and yields additional examples of exotic spheres for which no such actions exist.References
- D. W. Anderson, E. H. Brown Jr., and F. P. Peterson, $\textrm {SU}$-cobordism, $\textrm {KO}$-characteristic numbers, and the Kervaire invariant, Ann. of Math. (2) 83 (1966), 54β67. MR 189043, DOI 10.2307/1970470
- Glen E. Bredon, A $\Pi _\ast$-module structure for $\Theta _\ast$ and applications to transformation groups, Ann. of Math. (2) 86 (1967), 434β448. MR 221518, DOI 10.2307/1970609
- Wu-chung Hsiang, A note on free differentiable actions of $S^{1}$ and $S^{3}$ on homotopy spheres, Ann. of Math. (2) 83 (1966), 266β272. MR 192506, DOI 10.2307/1970431
- Ronnie Lee, Non-existence of free differentiable actions of $S^{1}$ and Z$_{2}$ on homotopy spheres, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp.Β 208β209. MR 0245040
- Mark Mahowald and Martin Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967), 349β369. MR 214072, DOI 10.1016/0040-9383(67)90023-7
- Reinhard E. Schultz, Smooth structures on $S^{p}\times S^{q}$, Ann. of Math. (2) 90 (1969), 187β198. MR 250321, DOI 10.2307/1970687
- Reinhard Schultz, Improved estimates for the degree of symmetry of certain homotopy spheres, Topology 10 (1971), 227β235. MR 283822, DOI 10.1016/0040-9383(71)90007-3
- Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 27 (1971), 595-597
- MSC: Primary 57.47
- DOI: https://doi.org/10.1090/S0002-9939-1971-0271985-3
- MathSciNet review: 0271985