Pseudofree representations and 2-pseudofree actions on spheres

Laitinen, Erkki; Traczyk, Paweł

doi:10.1090/S0002-9939-1986-0831405-5

Pseudofree representations and $2$-pseudofree actions on spheres
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by Erkki Laitinen and Paweł Traczyk

Proc. Amer. Math. Soc. 97 (1986), 151-157

DOI: https://doi.org/10.1090/S0002-9939-1986-0831405-5

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Abstract:

We characterize $1$-pseudofree real representations of finite groups. We apply this to show that the representations at fixed points of a $2$-pseudofree smooth action of a finite group on a sphere of dimension $\geqslant 5$ are topologically equivalent. Moreover with one possible exception, the sphere is $G$-homeomorphic to a linear representation sphere.

References

Finite collineation groups

Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
W. C. Hsiang and William Pardon, Orthogonal transformations for which topological equivalence implies linear equivalence, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 459–461. MR 648535, DOI 10.1090/S0273-0979-1982-15016-9
W. C. Hsiang and William Pardon, When are topologically equivalent orthogonal transformations linearly equivalent?, Invent. Math. 68 (1982), no. 2, 275–316. MR 666164, DOI 10.1007/BF01394060
B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
Sören Illman, Representations at fixed points of actions of finite groups on spheres, Current trends in algebraic topology, Part 2 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 135–155. MR 686142
John McKay, The non-abelian simple groups $G,$ $G\ <\ 10^{6}$ — character tables, Comm. Algebra 7 (1979), no. 13, 1407–1445. MR 539357, DOI 10.1080/00927877908822410
J. H. Lindsey II, Finite linear groups of degree six, Canadian J. Math. 23 (1971), 771–790. MR 289665, DOI 10.4153/CJM-1971-086-x
Ib Madsen and Mel Rothenberg, Classifying $G$ spheres, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 223–226. MR 656199, DOI 10.1090/S0273-0979-1982-15015-7
Ted Petrie, Pseudoequivalences of $G$-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 169–210. MR 520505
Ted Petrie, Three theorems in transformation groups, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 549–572. MR 561238

Représentations linéaires des groupes finis

Elliott Stein, Surgery on products with finite fundamental group, Topology 16 (1977), no. 4, 473–493. MR 474336, DOI 10.1016/0040-9383(77)90054-4
Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740