Differentiable group actions on homotopy spheres. III. Invariant subspheres and smooth suspensions

Schultz, Reinhard

doi:10.1090/S0002-9947-1983-0682728-6

Differentiable group actions on homotopy spheres. III. Invariant subspheres and smooth suspensions
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Trans. Amer. Math. Soc. 275 (1983), 729-750 Request permission

Abstract:

A linear action of an abelian group on a sphere generally contains a large family of invariant linear subspheres. In this paper the problem of finding invariant subspheres for more general smooth actions on homotopy spheres is considered. Classification schemes for actions with invariant subspheres are obtained; these are formally parallel to the classifications discussed in the preceding paper of this series. The realizability of a given smooth action as an invariant codimension two subsphere is shown to depend only on the ambient differential structure and an isotopy invariant. Applications of these results to specific cases are given; for example, it is shown that every exotic $10$-sphere admits a smooth circle action.

References

Glen E. Bredon, A $\Pi _\ast$-module structure for $\Theta _\ast$ and applications to transformation groups, Ann. of Math. (2) 86 (1967), 434–448. MR 221518, DOI 10.2307/1970609
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
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, Embedding smooth manifolds, Proc. Internat. Congr. Math. (Moscow, 1966) Izdat. “Mir”, Moscow, 1968, pp. 712–719. MR 0238335
William Browder, Diffeomorphisms of $1$-connected manifolds, Trans. Amer. Math. Soc. 128 (1967), 155–163. MR 212816, DOI 10.1090/S0002-9947-1967-0212816-0
William Browder, Free $Z_{p}$-actions on homotopy spheres, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp. 217–226. MR 0276982
William Browder, Surgery on simply-connected manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65, Springer-Verlag, New York-Heidelberg, 1972. MR 0358813, DOI 10.1007/978-3-642-50020-6
W. Browder and G. R. Livesay, Fixed point free involutions on homotopy spheres, Tohoku Math. J. (2) 25 (1973), 69–87. MR 321077, DOI 10.2748/tmj/1178241416
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
William Browder and Ted Petrie, Diffeomorphisms of manifolds and semifree actions on homotopy spheres, Bull. Amer. Math. Soc. 77 (1971), 160–163. MR 273636, DOI 10.1090/S0002-9904-1971-12646-0
Sylvain E. Cappell and Julius L. Shaneson, Topological knots and knot cobordism, Topology 12 (1973), 33–40. MR 321099, DOI 10.1016/0040-9383(73)90020-7
Sylvain E. Cappell and Julius L. Shaneson, The codimension two placement problem and homology equivalent manifolds, Ann. of Math. (2) 99 (1974), 277–348. MR 339216, DOI 10.2307/1970901
Sylvain E. Cappell and Julius L. Shaneson, Piecewise linear embeddings and their singularities, Ann. of Math. (2) 103 (1976), no. 1, 163–228. MR 407859, DOI 10.2307/1971003
Jean Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 5–173 (French). MR 292089
Morris W. Hirsch, Smooth regular neighborhoods, Ann. of Math. (2) 76 (1962), 524–530. MR 149492, DOI 10.2307/1970372
Morris W. Hirsch and John Milnor, Some curious involutions of spheres, Bull. Amer. Math. Soc. 70 (1964), 372–377. MR 176479, DOI 10.1090/S0002-9904-1964-11103-4
William D. Homer, Equivariant PL embeddings of spheres, Topology 19 (1980), no. 1, 51–63. MR 559476, DOI 10.1016/0040-9383(80)90031-2
William D. Homer, Singularities in lens spaces, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 155–158. MR 520531
Sören Illman, Smooth equivariant triangulations of $G$-manifolds for $G$ a finite group, Math. Ann. 233 (1978), no. 3, 199–220. MR 500993, DOI 10.1007/BF01405351
Hsü Tung Ku and Mei Chin Ku, Characteristic invariants of free differentiable 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. 19–40. MR 0377955
Serge Lang, Differential manifolds, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0431240
R. Lashof and M. Rothenberg, $G$-smoothing theory, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 211–266. MR 520506
J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR 180981, DOI 10.2307/1970561
J. Levine, Unknotting spheres in codimension two, Topology 4 (1965), 9–16. MR 179803, DOI 10.1016/0040-9383(65)90045-5
Chao Chu Liang, Browder-Livesay index invariant and equivariant knots, Michigan Math. J. 23 (1976), no. 4, 321–323 (1977). MR 461527
Chao Chu Liang, Involutions fixing codimension two knots, Pacific J. Math. 73 (1977), no. 1, 125–129. MR 482790, DOI 10.2140/pjm.1977.73.125
Chao Chu Liang, Knots fixed by $Z_{p}$-actions, and periodic links, Math. Ann. 233 (1978), no. 1, 49–54. MR 494131, DOI 10.1007/BF01351496
Chao Chu Liang, $\textbf {Z}_{(2)}$-knot cobordism in codimension two, and involutions on homotopy spheres, Trans. Amer. Math. Soc. 256 (1979), 89–97. MR 546908, DOI 10.1090/S0002-9947-1979-0546908-1
S. López de Medrano, Involutions on manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 59, Springer-Verlag, New York-Heidelberg, 1971. MR 0298698, DOI 10.1007/978-3-642-65012-3
Deane Montgomery and C. T. Yang, Free differentiable actions on homotopy spheres, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 175–192. MR 0245042
Ted Petrie, Pseudoequivalences of $G$-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 169–210. MR 520505
Melvin Rothenberg, Differentiable group actions on spheres, Proceedings of the Advanced Study Institute on Algebraic Topology (Univ. Aarhus, Aarhus, 1970) Mat. Inst., Aarhus Univ., Aarhus, 1970, pp. 455–475. MR 0301764
Mel Rothenberg, Torsion invariants and finite transformation groups, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 267–311. MR 520507
M. Rothenberg and J. Sondow, Nonlinear smooth representations of compact Lie groups, Pacific J. Math. 84 (1979), no. 2, 427–444. MR 568660, DOI 10.2140/pjm.1979.84.427
C. P. Rourke and B. J. Sanderson, Block bundles. III. Homotopy theory, Ann. of Math. (2) 87 (1968), 431–483. MR 232404, DOI 10.2307/1970714
Reinhard E. Schultz, Smooth structures on $S^{p}\times S^{q}$, Ann. of Math. (2) 90 (1969), 187–198. MR 250321, DOI 10.2307/1970687
Reinhard Schultz, Improved estimates for the degree of symmetry of certain homotopy spheres, Topology 10 (1971), 227–235. MR 283822, DOI 10.1016/0040-9383(71)90007-3
Reinhard Schultz, The nonexistence of free $S^{1}$ actions on some homotopy spheres, Proc. Amer. Math. Soc. 27 (1971), 595–597. MR 271985, DOI 10.1090/S0002-9939-1971-0271985-3
Reinhard Schultz, Semifree circle actions and the degree of symmetry of homotopy spheres, Amer. J. Math. 93 (1971), 829–839. MR 287548, DOI 10.2307/2373473
Reinhard Schultz, Homotopy decompositions of equivariant function spaces. I, Math. Z. 131 (1973), 49–75. MR 407866, DOI 10.1007/BF01213825
Reinhard Schultz, Circle actions on homotopy spheres bounding plumbing manifolds, Proc. Amer. Math. Soc. 36 (1972), 297–300. MR 309138, DOI 10.1090/S0002-9939-1972-0309138-3
Reinhard Schultz, Homotopy sphere pairs admitting semifree differentiable actions, Amer. J. Math. 96 (1974), 308–323. MR 368053, DOI 10.2307/2373635
Reinhard Schultz, Circle actions on homotopy spheres bounding generalized plumbing manifolds, Math. Ann. 205 (1973), 201–210. MR 380852, DOI 10.1007/BF01349230
Reinhard Schultz, Closed curves and circle homomorphisms in groups of diffeomorphisms, Fund. Math. 95 (1977), no. 2, 141–146. MR 436176, DOI 10.4064/fm-95-2-141-146
Reinhard Schultz, Differentiable group actions on homotopy spheres. I. Differential structure and the knot invariant, Invent. Math. 31 (1975), no. 2, 105–128. MR 405471, DOI 10.1007/BF01404111
Reinhard Schultz, Circle actions on homotopy spheres not bounding spin manifolds, Trans. Amer. Math. Soc. 213 (1975), 89–98. MR 380853, DOI 10.1090/S0002-9947-1975-0380853-6
Reinhard Schultz, Spherelike $G$-manifolds with exotic equivariant tangent bundles, Studies in algebraic topology, Adv. in Math. Suppl. Stud., vol. 5, Academic Press, New York-London, 1979, pp. 1–38. MR 527243
Reinhard Schultz, Smooth actions of small groups on exotic spheres, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 155–160. MR 520503
Reinhard Schultz, Isotopy classes of periodic diffeomorphisms on spheres, Algebraic topology, Waterloo, 1978 (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1978) Lecture Notes in Math., vol. 741, Springer, Berlin, 1979, pp. 334–354. MR 557176
Reinhard Schultz, Differentiable group actions on homotopy spheres. II. Ultrasemifree actions, Trans. Amer. Math. Soc. 268 (1981), no. 2, 255–297. MR 632531, DOI 10.1090/S0002-9947-1981-0632531-6

Outline of almost isovariant homotopy smoothing theory

G. B. Segal, Equivariant stable homotopy theory, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 59–63. MR 0423340
Julius L. Shaneson, Surgery on four-manifolds and topological transformation groups, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 298, Springer, Berlin, 1972, pp. 441–453. MR 0365606

Actions of cyclic groups on spheres

12

ibid.

Neal W. Stoltzfus, Equivariant concordance of invariant knots, Trans. Amer. Math. Soc. 254 (1979), 1–45. MR 539906, DOI 10.1090/S0002-9947-1979-0539906-5

Characteristic subspheres of differentiable group actions on homotopy spheres

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