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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

 

Nonlinear analogs of linear group actions on spheres


Author: Reinhard Schultz
Journal: Bull. Amer. Math. Soc. 11 (1984), 263-285
MSC (1980): Primary 57S15, 57S17, 57S25; Secondary 55M35
MathSciNet review: 752788
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  • 1. J. F. Adams, Graeme Segal's Burnside ring conjecture, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 201-210. MR 640946
  • 2. A. Assadi, Finite group actions on simply connected manifolds and CW complexes, Mem. Amer. Math. Soc. No. 257 (1982). MR 643341
    2A. A. Assadi, Extension of finite group actions from submanifolds of a disk, Conf. Current Trends in Algebraic Topology (Univ. of Western Ontario, London, 1981), Canad. Math. Soc. Conf. Proc., Vol. 2, Part 2, 1982, pp. 45-66. MR 686139

  • 3. A. Assadi and W. Browder, On the existence and classification of extensions of actions of finite groups on submanifolds of disks and spheres, Trans. Amer. Math. Soc (to appear). MR 800249
  • 4. M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II: Applications, Ann. of Math. (2) 88 (1968), 451-491. MR 232406
  • 5. M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546-604. MR 236952
  • 6. R. H. Bing, A homeomorphism between the 3-sphere and the sum of two solid horned spheres, Ann. of Math. (2) 56 (1952), 354-362. MR 49549
  • 7. S. Bochner, Compact groups of differentiable transformations, Ann. of Math. (2) 46 (1945), 372-381. MR 13161
  • 8. A. Borel (Ed.), Seminar on transformation groups, Ann. of Math. Stud. No. 46., Princeton Univ. Press, Princeton, N. J., 1960. MR 116341
  • 9. G. Bredon, Exotic actions on spheres, Proc. Conf. Transformation Groups (New Orleans, 1967), Springer-Verlag, 1968, pp. 47-76. MR 266239
  • 10. G. Bredon, A π Θ* and applications to transformation groups, Ann. of Math. (2) 86 (1967), 434-448.
  • 11. G. Bredon, Representations at fixed points of smooth actions of compact groups, Ann. of Math. (2) 89 (1969), 515-532. MR 246324
  • 12. G. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR 413144
  • 13. G. Bredon, Regular O( n) -manifolds, suspension of knots, and knot periodicity, Bull. Amer. Math. Soc. 79 (1973), 87-91. MR 310901
  • 14. G. Bredon, Biaxial actions of classical groups, unpublished manuscript, Rutgers Univ., 1973.
  • 15. Th. Bröcker and K. Jänich, Introduction to differential topology (translated by C. B. and M. J. Thomas), Cambridge Univ. Press, New York, 1982. MR 674117
  • 16. W. Browder, Surgery and the theory of differentiable transformation groups, Proc. Conf. Transformation Groups (New Orleans, 1967), Springer, New York, 1968, pp. 1-46. MR 261629
  • 17. W. Browder and T. Petrie, Diffeomorphisms of manifolds and semifree actions on homotopy spheres, Bull. Amer. Math. Soc. 77 (1971), 160-163. MR 273636
  • 18. S. Cappell and J. Shaneson, Nonlinear similarity, Ann. of Math. (2) 113 (1981), 315-355. MR 607895
  • 19. S. Cappell and J. Shaneson, Fixed points of periodic differentiable maps, Invent. Math. 68 (1982), 1-19. MR 666635
  • 20. S. Cappell and S. Weinberger (to appear).
  • 21. E. C. Cho, s-Smith equivalence for representations of generalized quaternion groups, Ph. D. Thesis, Rutgers Univ., 1984; summarized in Abstracts Amer. Math. Soc. 4 (1983), 546.
  • 22. E. H. Connell, D. Montgomery and C. T. Yang, Compact groups in RO, Ann. of Math. (2) 80 (1964), 94-103; correction, ibid. 81 (1965), 194. MR 162885
  • 23. P. E. Conner, Differentiable periodic maps, 2nd ed., Lecture Notes in Math., vol. 738, Springer-Verlag, 1979. (First Edition written jointly with E. E. Floyd.) MR 548463
  • 24. M. Davis, Multiaxial actions on manifolds, Lecture Notes in Math., vol. 643, Springer-Verlag, 1978. MR 488195
  • 25. M. Davis, W. C. Hsiang and W. Y. Hsiang, Differential actions of compact simple Lie groups on homotopy spheres and Euclidean spaces, Proc. Sympos. Pure Math., vol. 32, part 1, Amer. Math. Soc. Providence, R. I., 1978, pp. 313-324. MR 520508
  • 26. M. Davis and W. C. Hsiang, Concordance classes of regular U(n) and Sp(n) actions on homotopy spheres, Ann. of Math. (2) 105 (1977), 325-341. MR 438368
  • 27. M. Davis, W. C. Hsiang and J. Morgan, Concordance classes of regular O(n) actions on homotopy spheres, Acta Math. 144 (1980), 153-221. MR 573451
  • 28. G. de Rham, Reidemeister's torsion invariants and rotations of S, Differential Analysis (Bombay Colloq., 1963), Oxford Univ. Press, New York, 1964, pp. 27-36.
  • 29. T. tom Dieck, Semi-linear group actions on spheres: Dimension functions, Algebraic Topology Aarhus 1978 (Proc. Sympos.), Lecture Notes in Math., vol. 763, Springer-Verlag, 1979, pp. 448-457. MR 561232
  • 30. T. tom Dieck, Homotopy representations of the torus, Arch. Math. (Basel) 38 (1982), 459-469. MR 666921
  • 31. T. tom Dieck and T. Petrie, Homotopy representations of finite groups, Inst. Haute Études Sci. Publ. Math. 56 (1982), 129-170. MR 686044
  • 32. R. Dotzel and G. Hamrick, p-group actions on homology spheres, Invent. Math. 62 (1981), 437-442. MR 604837
  • 33. K. H. Dovermann, Even dimensional s-Smith equivalent representations (Proc. Topology Conf. Aarhus, 1982), Lecture Notes in Math., vol. 1051, Springer-Verlag, 1984, pp. 587-602. MR 764601
  • 34. K. H. Dovermann and T. Petrie, Smith equivalence for odd order cyclic groups (in preparation); summarized in Abstracts Amer. Math. Soc. 4 (1983), 476-477.
  • 35. K. H. Dovermann and M. Rothenberg, An equivariant surgery sequence and equivariant diffeomorphism and homeomorphism classification, Topology Sympos. (Siegen, 1979), Lecture Notes in Math., vol. 788, Springer-Verlag, pp. 257-280; for detailed version see preprint, Purdue Univ., 1983. MR 585663
  • 36. A. Edmonds and R. Lee, Cyclic group actions on Euclidean space, Topology 14 (1975), 339-345. MR 413146
  • 37. A. Edmonds, Transformation groups and low dimensional manifolds (Proc. Conf. Group Actions, Boulder, Colo., 1983), Contemporary Math., Amer. Math. Soc. (to appear). MR 780973
  • 38. D. Erle and W. C. Hsiang, On certain unitary and symplectic actions with three orbit types, Amer. J. Math. 94 (1972), 289-308. MR 305428
  • 39. J. Ewing, Spheres as fixed point sets, Quart. J. Math. Oxford (2) 27 (1976), 445-455. MR 431233
  • 40. E. E. Floyd, Examples of fixed point sets of periodic maps, Ann. of Math. (2) 55 (1952), 167-171. MR 45379
  • 41. M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357-453. MR 679066
  • 42. C. H. Giffen, The generalized Smith conjecture, Amer. J. Math. 88 (1966), 187-198. MR 198462
  • 43. C. McA. Gordon, On the higher dimensional Smith conjecture, Proc. London Math. Soc. (3) 29 (1974), 98-110. MR 356073
  • 44. M. W. Hirsch, Differential topology, Graduate Texts in Math., vol. 33, Springer-Verlag, 1976. MR 448362
  • 45. W. C. Hsiang and W. Y. Hsiang, The degree of symmetry of homotopy spheres, Ann. of Math. (2) 89 (1969), 52-67. MR 239621
  • 46. W. C. Hsiang and W. Y. Hsiang, Differentiable actions of compact connected classical groups. I, Amer. J. Math. 89 (1967), 705-786. MR 217213
  • 47. W. C. Hsiang and W. Y. Hsiang, Differentiable actions of compact connected Lie groups. II, III, Ann. of Math. (2) 92 (1970), 189-223; ibid. 99 (1974), 220-256.
  • 48. W. Y. Hsiang, A survey of regularity theorems in differentiable compact transformation groups, Proc. Conf. Transformation Groups (New Orleans, 1967), Springer-Verlag, 1968, pp. 77-124. MR 256416
  • 49. W. Y. Hsiang, On generalizations of the theorem of A. Borel and their applications in the study of topological actions, Topology of Manifolds (Proc. Univ. Georgia Conf., 1969), Markham, Chicago, 1970, pp. 274-290.
  • 50. W. Y. Hsiang, On the degree of symmetry and the structure of highly symmetric manifolds, Tamkang J. Math. 2 (1971), 1-22. MR 309140
  • 51. W. Y. Hsiang, Cohomology theory of topological transformation groups, Ergebnisse Math. Grenzgebiete, Bd. 85, Springer-Verlag, 1975. MR 423384
  • 52. S. Illman, Recognition of linear actions on spheres, Trans. Amer. Math. Soc. 274 (1982), 445-478. MR 675064
  • 53. L. Jones, The converse to the fixed point theorem of P. A. Smith. I, Ann. of Math. (2) 94 (1971), 52-68. MR 295361
  • 54. L. Jones, The converse to the fixed point theorem of P. A. Smith. II, Indiana Univ. Math. J. 22 (1972), 309-325; correction, ibid. 24 (1975), 1001-1003. MR 339249
  • 55. D. S. Kahn and S. B. Priddy, The transfer and stable homotopy, Math. Proc. Cambridge Philos. Soc. 83 (1978), 103-111. MR 464230
  • 56. M. Kervaire and J. Milnor, Groups of homotopy spheres, Ann. of Math. (2) 77 (1963), 504-537. MR 148075
  • 57. J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15-50. MR 180981
  • 58. W. H. Lin, On conjectures of Mahowald, Segal, and Sullivan, Math. Proc. Cambridge Philos. Soc. 87 (1980), 449-458. MR 556925
  • 59. W. H. Lin et al., Calculation of Lin's Ext groups, Math. Proc. Cambridge Philos. Soc. 87 (1980), 459-469. MR 569195
  • 60. P. Löffler, Homotpielineare Z, Topology 20 (1981), 291-312. MR 608602
  • 61. T. Matumoto, On G-CW complexes and a theorem of J. H. C. Whitehead, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 (1971), 363-374. MR 345103
  • 62. A. Meyerhoff and T. Petrie, Quasiequivalence of G-modules, Topology 15 (1976), 69-75. MR 391141
  • 63. J. Milnor, Introduction to algebraic K-theory, Ann. of Math. Stud., No. 72, Princeton Univ. Press, Princeton, N. J., 1971. MR 349811
  • 64. J. Milnor and J. Stasheff, Characteristic classes, Ann. of Math. Stud., No. 76, Princeton Univ. Press, Princeton, N. J., 1974. MR 440554
  • 65. D. Montgomery and C. T. Yang, Differentiable transformation groups on homotopy spheres, Michigan Math. J. 14 (1967), 33-46. MR 217815
  • 66. J. W. Morgan, Actions de groupes finis sur S3: La conjecture de P. A. Smith (d'après Thurston et Meeks-Yau), Séminaire Bourbaki, Vol. 1980/81, Lecture Notes in Math., vol. 901, Springer-Verlag, 1981, pp. 277-289. [The results summarized in this article represent the work of several mathematicians. There are plans to publish a single volume containing most of the separate contributions.] MR 647502
  • 67. J. R. Munkres, Elementary differential topology, rev. ed., Ann. of Math. Stud., No. 54, Princeton Univ. Press, Princeton, N. J., 1966. MR 198479
  • 68. S. P. Novikov, On manifolds with free abelian fundamental groups and their applications, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 207-246; English transl., Amer. Math. Soc. Transl (2) 71 (1968), 1-42. MR 196765
  • 69. R. Oliver, Fixed point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155-177. MR 375361
  • 70. R. Oliver, Weight systems for SO(3) actions, Ann. of Math. (2) 110 (1980), 227-241. MR 549488
  • 71. R. Oliver and T. Petrie, G-CW surgery and K0(Z[G]), Math. Z. 179 (1982), 11-42. MR 643044
  • 72. K. Pawalowski, Fixed points of cyclic group actions on disks, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), 1011-1015. MR 528987
  • 73. K. Pawalowski, On some problems in compact transformation groups, summarized in Math. Forschungsinst. Oberwolfach Tagungsbericht 35/1982, pp. 11-12.
  • 74. T. Petrie, Free metacyclic group actions on homotopy spheres, Ann. of Math. (2) 94 (1971), 108-124. MR 293667
  • 75. T. Petrie, Equivariant quasiequivalence, transversality, and normal cobordism, Proc. Internat. Congress Math. (Vancouver, B. C, 1974), Canad. Math. Soc. Toronto, 1975, pp. 537-541. MR 431234
  • 76. T. Petrie, Three theorems in transformation groups, Algebraic Topology Aarhus 1978 (Proc. Sympos.), Lecture Notes in Math., vol. 763, Springer-Verlag, 1979, pp. 549-572. MR 561238
  • 77. T. Petrie, One fixed point actions on spheres. I, Adv. in Math. 46 (1982), 3-14. MR 676986
  • 78. T. Petrie, The equivariant J. homomorphism and Smith equivalence of representations, Conf. Current Trends in Algebraic Topology (Univ. of Western Ontario, London, 1981), Canad. Math. Soc. Conf. Proc., Vol. 2, Part 2, 1982, pp. 223-234. MR 686148
  • 79. T. Petrie, Smith equivalence of representations, Math. Proc. Cambridge Philos. Soc. 94 (1983), 61-99. MR 704803
  • 80. T. Petrie, Spherical isotropy representations, Inst. Hautes Études Sci. Publ. Math, (to appear). MR 823175
  • 81. T. Petrie, Isotropy representations of group actions on disks (in preparation).
  • 82. D. S. Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959), 700-712. MR 104721
  • 83. M. Rothenberg, Differentiable group actions on spheres, Proc. Adv. Study Inst, on Algebraic Topology (Aarhus, 1970), Various Publications Ser., Vol. 13, Mat. Inst., Aarhus Univ., 1970, pp. 455-475. MR 301764
  • 84. M. Rothenberg, Torsion invariants and finite transformation groups, Proc. Sympos. Pure Math., vol. 32, part 1, Amer. Math. Soc., Providence, R. I., 1978, pp. 267-311. MR 520507
  • 85. M. Rothenberg and J. Sondow, Nonlinear smooth representations of compact Lie groups, Pacific J. Math. 84 (1980), 427-444. (A preliminary version of this paper circulated as a Univ. of Chicago preprint in 1969.) MR 568660
  • 86. C. Sánchez, Actions of groups of odd order on compact orientable manifolds, Proc. Amer. Math. Soc. 54 (1976), 445-448. MR 407871
  • 87. R. Schultz, Improved estimates for the degree of symmetry of certain homotopy spheres, Topology 10 (1971), 227-235. MR 283822
  • 88. R. Schultz, Semifree circle actions and the degree of symmetry of homotopy spheres, Amer. J. Math. 93 (1971), 829-839. MR 287548
  • 89. R. Schultz, Semifree circle actions with twisted fixed point sets, Proc. Second Conf. Compact Transformation Groups (Univ. of Massachusetts, Amherst, 1971), Part I, Lecture Notes in Math., vol. 298, Springer-Verlag, 1972, pp. 102-116. MR 362382
  • 90. R. Schultz, Homotopy sphere pairs admitting semifree differentiable actions, Amer. J. Math. 96 (1974), 308-323. MR 368053
  • 91. R. Schultz, Closed curves and circle homomorphisms in groups of diffeomorphisms, Fund. Math. 95 (1977), 141-146. MR 436176
  • 92. R. Schultz, Spherelike G-manifolds with exotic equivariant tangent bundles, Studies in Algebraic Topology, Academic Press, New York, 1979, pp. 1-39. MR 527243
  • 93. R. Schultz, Differentiable actions on homotopy spheres. I: Differential structure and the knot invariant, Invent. Math. 31 (1975), 105-128. MR 405471
  • 94. R. Schultz, Differentiable actions on homotopy spheres. II: Ultrasemifree actions, Trans. Amer. Math. Soc. 268 (1981), 255-297. MR 632531
  • 95. R. Schultz, Differentiable actions on homotopy spheres. III: Invariant subspheres and smooth suspensions, Trans. Amer. Math. Soc. 275 (1983), 729-750. MR 682728
  • 96. R. Schultz, Differentiable actions on homotopy spheres. IV: Formulas for certain normal invariants (in preparation).
  • 97. R. Schultz, Differentiable actions on homotopy spheres. V: Semifree actions on exotic spheres (in preparation).
  • 98. R. Schultz, Differentiable actions on homotopy spheres. VI: Circle actions on exotic spheres through dimension 30 (in preparation).
  • 99. R. Schultz, Smooth actions of small groups on exotic spheres, Proc. Sympos. Pure Math., vol. 32, part 1, Amer. Math. Soc., Providence, R. I., 1978, pp. 155-161. MR 520503
  • 100. R. Schultz, Isotopy classes of periodic diffeomorphisms of spheres, Algebraic Topology (Proc. Waterloo, 1978), Lecture Notes in Math., vol. 741, Springer-Verlag, 1979. MR 557176
  • 101. R. Schultz, Differentiability and the P. A. Smith theorems for spheres. I: Actions of prime order groups, Conf. Current Trends in Algebraic Topology (Univ. of Western Ontario, London, 1981), Canad. Math. Soc. Conf. Proc., Vol. 2, Part 2, 1982, pp. 235-273. MR 686149
  • 102. R. Schultz, Differentiability and the P. A. Smith theorems for spheres. II: Actions of larger groups (in preparation).
  • 103. R. Schultz, Exotic spheres admitting smooth circle actions with codimension four stationary sets, Proc. Northwestern Homotopy Theory Conf. (Evanston, Ill., 1982), Contemporary Math., vol. 19, Amer. Math. Soc., Providence, R. I., 1983, pp. 339-368. MR 711061
  • 104. R. Schultz, Upper bounds for the toral symmetry of certain homotopy spheres (Proc. Topology Conf., Aarhus, 1982), vol. 1051, Lecture Notes in Math., Springer-Verlag, 1984, pp. 645-659. MR 764606
  • 105. R. Schultz, Almost isovariant homotopy smoothings of compact G-manifolds (in preparation).
  • 106. R. Schultz, Almost extremal torus actions on homotopy spheres (in preparation).
  • 107. R. Schultz, Homotopy invariants for elementary abelian group actions on exotic spheres (in preparation).
  • 108. R. Schultz, Transformation groups and exotic spheres (Proc. Conf. Group Actions, Boulder, Colo., 1983), Contemporary Math, (to appear). MR 780966
  • 109. R. Schultz (Editor), Problems submitted to the A.M.S. Summer Research Conference on group actions on manifolds (Proc. Conf. Group Actions, Boulder, Colo., 1983), Contemporary Math. (to appear). MR 780951
  • 110. G. Segal, Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129-151. MR 234452
  • 111. A. Siegel, Fixed point representations of group actions on spheres, Ph.D. Thesis, Rutgers Univ., 1982.
  • 112. P. A. Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66(1960), 401-415. MR 125581
  • 113. R. Stong, Complex and oriented equivariant bordism, Topology of Manifolds (Proc. Univ. of Georgia Conf., 1969), Markham, Chicago, 1970, pp. 291-316. MR 273644
  • 114. D. Y. Suh, s-Smith equivalence for finite abelian groups, Ph.D. Thesis, Rutgers Univ., 1984; summarized in Abstracts Amer. Math. Soc. 4 (1983), 548.
  • 115. R. W. Sullivan, Differentiable actions of classical groups on spheres, Topology 9 (1970), 155-167. MR 261631
  • 116. R. W. Sullivan, Linear models for compact group actions on spheres, Topology 13 (1974), 77-87. MR 339245
  • 117. R. Swan, Periodic resolutions for finite groups, Ann. of Math. (2) 72 (1960), 267-291. MR 124895
  • 118. H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Stud., No. 49, Princeton Univ. Press, Princeton, N. J., 1962. MR 143217
  • 119. Y. D. Tsai, Isotropy representations of nonabelian groups on disks, Ph.D. Thesis, Rutgers Univ., 1984; summarized in Abstracts Amer. Math. Soc. 4 (1983), 545.
  • 120. C. T. C. Wall, Surgery on compact manifolds, London Math. Soc. Monographs, vol. 1, Academic Press, New York, 1970. MR 431216
  • 121. S. Weinberger, Homologically trivial group actions. I, (preprint).
  • 122. J. A. Wolf, Spaces of constant curvature, 4th. ed., Publish or Perish, Berkeley, Calif., 1977.

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DOI: http://dx.doi.org/10.1090/S0273-0979-1984-15290-X
PII: S 0273-0979(1984)15290-X