Free $S^{1}$ actions and the group of diffeomorphisms
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- by Kai Wang PDF
- Trans. Amer. Math. Soc. 191 (1974), 113-127 Request permission
Abstract:
Let ${S^1}$ act linearly on ${S^{2p - 1}} \times {D^{2q}}$ and ${D^{2p}} \times {S^{2q - 1}}$ and let $f:{S^{2p - 1}} \times {S^{2q - 1}} \to {S^{2p - 1}} \times {S^{2q - 1}}$ be an equivariant diffeomorphism. Then there is a well-defined ${S^1}$ action on ${S^{2p - 1}} \times {D^{2q}}{ \cup _f}{D^{2p}} \times {S^{2q - 1}}$. An ${S^1}$ action on a homotopy sphere is decomposable if it can be obtained in this way. In this paper, we will apply surgery theory to study in detail the set of decomposable actions on homotopy spheres.References
-
W. Browder, Surgery on simply connected manifolds, Mimeographed notes, Princeton University, Princeton, N. J., 1969.
- William Browder, Surgery and the theory of differentiable transformation groups, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 1–46. MR 0261629
- William Browder and Ted Petrie, Semi-free and quasi-free $S^{1}$ actions on homotopy spheres, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 136–146. MR 0263111
- Wu-chung 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 192506, DOI 10.2307/1970431
- Hsu-tung Ku and Mei-chin Ku, Characteristic spheres of free differentiable actions of $S^{1}$ and $S^{3}$ on homotopy spheres, Trans. Amer. Math. Soc. 156 (1971), 493–504. MR 276992, DOI 10.1090/S0002-9947-1971-0276992-7
- G. R. Livesay and C. B. Thomas, Free $Z_{2}$ and $Z_{3}$ actions on homotopy spheres, Topology 7 (1968), 11–14. MR 220277, DOI 10.1016/0040-9383(86)90011-X D. Sullivan, Triangulating homotopy equivalences, Thesis, Princeton University, Princeton, N. J., 1965.
- C. T. C. Wall, Surgery on compact manifolds, London Mathematical Society Monographs, No. 1, Academic Press, London-New York, 1970. MR 0431216
- Kai Wang, Free and semi-free smooth actions of $S^{1}$ and $S^{3}$ on homotopy spheres, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 298, Springer, Berlin, 1972, pp. 119–130. MR 0362384 —, Free ${S^1}$ actions and the group of diffeomorphisms, Notices Amer. Math. Soc. 136 (1972), 346. Abstract #72T-G48. —, Stable tangent bundles of the gluing spaces and its application to transformation groups (preprint). —, Differentiable ${S^1}$ actions on homotopy spheres (preprint).
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 191 (1974), 113-127
- MSC: Primary 57E15; Secondary 57D10
- DOI: https://doi.org/10.1090/S0002-9947-1974-0356106-8
- MathSciNet review: 0356106