Recognition of linear actions on spheres
Author:
Sören Illman
Journal:
Trans. Amer. Math. Soc. 274 (1982), 445478
MSC:
Primary 57S17; Secondary 57Q30, 57S25
MathSciNet review:
675064
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Abstract: Let be a finite group acting smoothly on a homotopy sphere . We wish to establish necessary and sufficient conditions for the given action on to be topologically equivalent to a linear action. That is, we want to be able to decide whether or not there exists a homeomorphism , where denotes the unit sphere in an orthogonal representation space for . In order for a action on to be topologically equivalent to a linear action it is clearly necessary that: (i) For each subgroup of the fixedpoint set is homeomorphic to a sphere, or empty. (ii) For any subgroups and , of the pair is homeomorphic to a standard pair , where each , is a standard subsphere of . In this paper we consider the case where the fixedpoint set is nonempty and all other fixedpoint sets have dimension at least 5. In giving efficient sufficient conditions we do not need the full strength of condition (ii). We only need: (ii) For any subgroups and , of such that , the pair is homeomorphic to a standard pair , where each , is a standard subsphere of . Our main results are then that, in the case when is abelian, conditions (i) and (ii) are necessary and sufficient for a given action on to be topologically equivalent to a linear action, and in the case of an action of an arbitrary finite group the same holds under the additional assumption that any simultaneous codimension 1 and 2 fixedpoint situation is simple. Our results generalize, for actions of finite groups, a wellknown theorem of Connell, Montgomery and Yang, and are the first to also cover the case where codimension 2 fixedpoint situations occur.
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 [2]
 W. Browder and W. C. Hsiang, Some problems on homotopy theory, manifolds and transformation groups, Proc. Sympos. Pure Math., Vol. 32, Part 2 (Algebraic and Geometric Topology), Amer. Math. Soc., Providence, R. I., 1978, pp. 251267. MR 520546 (80e:55001)
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 J. C. Cantrell, Almost locally flat embedding of in , Bull. Amer. Math. Soc. 69 (1963), 716718. MR 0154288 (27:4237)
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 E. H. Connell, D. Montgomery and C. T. Yang, Compact groups in , Ann. of Math. (2) 80 (1964), 94103. MR 0162885 (29:189)
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 , (Correction to [5]), Ann. of Math. (2) 81 (1965), 194. MR 0168698 (29:5956)
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 C. H. Giffen, The generalized Smith conjecture, Amer. J. Math. 88 (1966), 187198. MR 0198462 (33:6620)
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 J. F. P. Hudson, Piecewise linear topology, Benjamin, New York, 1969. MR 0248844 (40:2094)
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 S. Illman, Smooth equivariant triangulations of manifolds for a finite group, Math. Ann. 233 (1978), 199220. MR 0500993 (58:18474)
 [10]
 , Approximation of maps by maps in equivariant general position and imbeddings of complexes, Trans. Amer. Math. Soc. 262 (1980), 113157. MR 583849 (81k:57019)
 [11]
 , Equivariant engulfing and recognition of linear actions on spheres, Proc. Internat. Conf. on Topology and its Appl., Moscow, 1979 (to appear).
 [12]
 M. Rothenberg, Torsion invariants and finite transformation groups, Proc. Sympos. Pure Math., Vol. 32, Part 1 (Algebraic and Geometric Topology), Amer. Math. Soc., Providence, R. I., 1978, pp. 267311. MR 520507 (80j:57038)
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 T. B. Rushing, Topological embeddings, Academic Press, New York, 1973. MR 0348752 (50:1247)
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 L. C. Siebenmann, On detecting open collars, Trans. Amer. Math. Soc. 142 (1969), 201227. MR 0246301 (39:7605)
 [15]
 E. H. Spanier, Algebraic topology, McGrawHill, New York, 1966. MR 0210112 (35:1007)
 [16]
 J. Stallings, The piecewiselinear structure of euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481488. MR 0149457 (26:6945)
 [17]
 , On topologically unknotted spheres, Ann. of Math. (2) 77 (1963), 490503. MR 0149458 (26:6946)
 [18]
 A. G. Wasserman, Equivariant differential topology, Topology 8 (1969), 127150. MR 0250324 (40:3563)
 [19]
 J. H. C. Whitehead, Simplicial spaces, nuclei and groups, Proc. London Math. Soc. (2) 45 (1939), 243327.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206750644
PII:
S 00029947(1982)06750644
Keywords:
Smooth actions on spheres,
linear actions on spheres,
equivariant engulfing
Article copyright:
© Copyright 1982 American Mathematical Society
