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Transactions of the American Mathematical Society

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On the existence and classification of extensions of actions on submanifolds of disks and spheres

Authors: Amir Assadi and William Browder
Journal: Trans. Amer. Math. Soc. 291 (1985), 487-502
MSC: Primary 57S17; Secondary 57R65, 57S25
MathSciNet review: 800249
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Abstract: Given a $ G$-action $ \psi :G \times W \to W$ and an embedding $ W \subset {D^n}$, when is it possible to find a $ G$-action $ \phi :G \times {D^n} \to D$ such that $ {D^n} - W$ is $ G$-free?

Sufficient conditions of cohomological nature for the existence of such extensions are given and the extensions are classified. This leads to the characterization of the stationary point sets and classification of semifree actions on disks up to $ G$-diffeomorphism under suitable dimension hypotheses.

References [Enhancements On Off] (What's this?)

  • [As1] A. H. Assadi, Finite group actions on simply-connected manifolds and CW complexes, Mem. Amer. Math. Soc. No. 257 (1982). MR 643341 (83e:57028)
  • [As2] -, Extensions of finite group actions from submanifolds of disks and spheres, Proc. Topology Conf. (London, Ontario, 1981), Current Trends in Topology, Amer. Math. Soc., Providence, R. I., 1982.
  • [At] M. F. Atiyah, Characters and cohomology of finite groups, Inst. Haùtes Etudes Sci. Publ. Math. 9 (1961). MR 0148722 (26:6228)
  • [Bre1] G. E. Bredon, Equivariant cohomology theories, Lecture Notes in Math., vol. 34, Springer-Verlag, 1967. MR 0214062 (35:4914)
  • [Bre2] -, Introduction to compact transformation groups, Academic Press, New York, 1972. MR 0413144 (54:1265)
  • [Bro] W. Browder, Surgery on simply-connected manifolds, Springer-Verlag, 1972. MR 0358813 (50:11272)
  • [Dov] K. H. Dovermann, Diffeomorphic classification of finite group actions on disks, Topology Appl. 16 (1983), 122-133. MR 712858 (85g:57020)
  • [I] S. Illman, Smooth equivariant triangulations of $ G$-manifolds for $ G$ a finite group, Math. Ann. 233 (1978), 199-220. MR 0500993 (58:18474)
  • [J] L. Jones, Converse to the fixed-point theorem of P. A. Smith. I, Ann. of Math. (2) 95 (1971), 52-68. MR 0295361 (45:4427)
  • [M1] J. W. Milnor, Lectures on the $ h$-cobordism theorem, Princeton Math. Notes, Princeton, N. J., 1965.
  • [M2] -, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358-426. MR 0196736 (33:4922)
  • [O] R. Oliver, Fixed-point set of group actions of group actions on finite acyclic complexes, Comment Math. Helv. 50 (1975), 155-177. MR 0375361 (51:11556)
  • [OP] R. Oliver and T. Petrie, $ G$-surgery in the homotopy category and $ {K_0}(\mathbf{Z}G)$, Geometric Applications of Homotopy Theory. I, Lecture Notes in Math., vol. 657, Springer-Verlag, Berlin and New York, 1978, pp. 361-375. MR 513559 (81c:57033)
  • [R] D. S. Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959), 700-712. MR 0104721 (21:3474)
  • [RS] M. Rothenberg and J. Sondow, Non-linear smooth representations of compact Lie groups, preprint. MR 568660 (81d:57033)
  • [S] R. Swan, Periodic resolutions for finite groups, Ann. of Math. (2) 72 (1960), 267-291. MR 0124895 (23:A2205)
  • [W] C. T. C. Wall, Surgery on compact manifolds, Academic Press, New York, 1970. MR 0431216 (55:4217)
  • [W2] -, Classification problems in differential topology, Topology 4 (1966), 73-94.

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Article copyright: © Copyright 1985 American Mathematical Society

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