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The nontriviality of the first rational homology group of some connected invariant subsets of periodic transformations


Authors: Amir Assadi and Dan Burghelea
Journal: Proc. Amer. Math. Soc. 88 (1983), 701-707
MSC: Primary 57S17; Secondary 57S10
DOI: https://doi.org/10.1090/S0002-9939-1983-0702303-X
Erratum: Proc. Amer. Math. Soc. 94 (1985), 187.
MathSciNet review: 702303
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Abstract: This note was inspired by some results of P. A. Smith [S]. One proves that for any periodic map of a manifold $ M$ and any codimension two invariant submanifold $ P$ of $ M$ containing part of the stationary point set, connected invariant subsets of the complement of $ P$ must carry nontrivial one-dimensional rational cycles, provided that $ M$ satisfies some simple homological conditions (Theorem A). This fact has interesting consequences in transformation group theory.


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

  • [A] A. Assadi, Finite group actions on simply-connected manifolds and $ CW$-complexes, Mem. Amer. Math. Soc. No. 257 (1982).
  • [B] G. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR 0413144 (54:1265)
  • [K1] R. Kirby, Codimension two locally flat embeddings have normal bundles, Topology of Manifolds (J. Cantrell and C. H. Edwards, eds.), Markham, Chicago, Ill., 1970. MR 0273625 (42:8502)
  • [K2] J. M. Kister, Differential periodic actions on $ {E^\infty }$ without fixed points, Amer. J. Math. 85 (1963), 316-319. MR 0154278 (27:4227)
  • [S] P. A. Smith, New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401-415. MR 0125581 (23:A2880)

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DOI: https://doi.org/10.1090/S0002-9939-1983-0702303-X
Article copyright: © Copyright 1983 American Mathematical Society

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