Bulletin of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-9485(e) ISSN 0273-0979(p)

Nonlinear analogs of linear group actions on spheres

Author(s): Reinhard Schultz
Journal: Bull. Amer. Math. Soc. 11 (1984), 263-285.
MSC (1980): Primary 57S15, 57S17, 57S25; Secondary 55M35
MathSciNet review: 752788
Retrieve article in: PDF

References:

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14.: G. Bredon, Biaxial actions of classical groups, unpublished manuscript, Rutgers Univ., 1973.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.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|