Recognition of linear actions on spheres

Author:
Sören Illman

Journal:
Trans. Amer. Math. Soc. **274** (1982), 445-478

MSC:
Primary 57S17; Secondary 57Q30, 57S25

DOI:
https://doi.org/10.1090/S0002-9947-1982-0675064-4

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 fixed-point 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 fixed-point set is nonempty and all other fixed-point 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 fixed-point situation is simple. Our results generalize, for actions of finite groups, a well-known theorem of Connell, Montgomery and Yang, and are the first to also cover the case where codimension 2 fixed-point situations occur.

**[1]**G. E. Bredon,*Introduction to compact transformation groups*, Academic Press, New York, 1972. MR**0413144 (54:1265)****[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. 251-267. MR**520546 (80e:55001)****[3]**M. Brown,*The monotone union of**-cells is an open**-cell*, Proc. Amer. Math. Soc.**12**(1961), 812-814. MR**0126835 (23:A4129)****[4]**J. C. Cantrell,*Almost locally flat embedding of*in , Bull. Amer. Math. Soc.**69**(1963), 716-718. MR**0154288 (27:4237)****[5]**E. H. Connell, D. Montgomery and C. T. Yang,*Compact groups in*, Ann. of Math. (2)**80**(1964), 94-103. MR**0162885 (29:189)****[6]**-, (Correction to [**5**]), Ann. of Math. (2)**81**(1965), 194. MR**0168698 (29:5956)****[7]**C. H. Giffen,*The generalized Smith conjecture*, Amer. J. Math.**88**(1966), 187-198. MR**0198462 (33:6620)****[8]**J. F. P. Hudson,*Piecewise linear topology*, Benjamin, New York, 1969. MR**0248844 (40:2094)****[9]**S. Illman,*Smooth equivariant triangulations of**-manifolds for**a finite group*, Math. Ann.**233**(1978), 199-220. MR**0500993 (58:18474)****[10]**-,*Approximation of**-maps by maps in equivariant general position and imbeddings of**-complexes*, Trans. Amer. Math. Soc.**262**(1980), 113-157. 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. 267-311. MR**520507 (80j:57038)****[13]**T. B. Rushing,*Topological embeddings*, Academic Press, New York, 1973. MR**0348752 (50:1247)****[14]**L. C. Siebenmann,*On detecting open collars*, Trans. Amer. Math. Soc.**142**(1969), 201-227. MR**0246301 (39:7605)****[15]**E. H. Spanier,*Algebraic topology*, McGraw-Hill, New York, 1966. MR**0210112 (35:1007)****[16]**J. Stallings,*The piecewise-linear structure of euclidean space*, Proc. Cambridge Philos. Soc.**58**(1962), 481-488. MR**0149457 (26:6945)****[17]**-,*On topologically unknotted spheres*, Ann. of Math. (2)**77**(1963), 490-503. MR**0149458 (26:6946)****[18]**A. G. Wasserman,*Equivariant differential topology*, Topology**8**(1969), 127-150. MR**0250324 (40:3563)****[19]**J. H. C. Whitehead,*Simplicial spaces, nuclei and**-groups*, Proc. London Math. Soc. (2)**45**(1939), 243-327.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0675064-4

Keywords:
Smooth actions on spheres,
linear actions on spheres,
equivariant engulfing

Article copyright:
© Copyright 1982
American Mathematical Society