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The nonexistence of free $ S\sp{1}$ actions on some homotopy spheres


Author: Reinhard Schultz
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
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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.


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  • [1] D. W. Anderson, E. H. Brown, Jr. and F. P. Peterson, SU-cobordism, KO-characteristic numbers, and the Kervaire invariant, Ann. of Math. (2) 83 (1966), 54-67. MR 32 #6470. MR 0189043 (32:6470)
  • [2] G. E. Bredon, A $ {\Pi _\ast }$-module structure for $ {\theta _\ast }$ and applications to transformation groups, Ann. of Math. (2) 86 (1967), 434-448. MR 36 #4570. MR 0221518 (36:4570)
  • [3] W.-C. Hsiang, A note on free differentiable actions of $ {S^1}$ and $ {S^3}$ on homotopy spheres, Ann. of Math. (2) 83 (1966), 266-272. MR 33 #731. MR 0192506 (33:731)
  • [4] R. Lee, Non-existence of free differentiable actions of $ {S^1}$ and $ {Z_2}$ on homotopy spheres, Proc. Conference on Transformation Groups (New Orleans, 1967), Springer-Verlag, New York, 1968, pp. 208-209. MR 39 #6352. MR 0245040 (39:6352)
  • [5] M. Mahowald and M. Tangora, Some differentials in the Adams spectral sequence, Topology 6(1967), 349-369. MR 35 #4924. MR 0214072 (35:4924)
  • [6] R. Schultz, Smooth structures on $ {S^p} \times {S^q}$, Ann. of Math. (2) 90 (1969), 187-198. MR 0250321 (40:3560)
  • [7] -, Improved estimates for the degree of symmetry of certain homotopy spheres, Topology (to appear). MR 0283822 (44:1052)
  • [8] H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Studies, no. 49, Princeton Univ. Press, Princeton, N. J., 1962. MR 26 #777. MR 0143217 (26:777)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0271985-3
Keywords: Inertia group, exotic sphere, free differentiable action, PL and homotopy smoothings
Article copyright: © Copyright 1971 American Mathematical Society

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