Finite groups acting on spheres

Miller, John David

doi:10.1090/S0002-9939-1972-0288783-8

Finite groups acting on spheres
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Proc. Amer. Math. Soc. 32 (1972), 289-293 Request permission

Abstract:

This paper classifies smooth actions of a finite group G acting on a sphere of sufficiently high dimension with precisely two fixed points (and the action free elsewhere) by relating the group action to an element of the group’s Whitehead group. Essentially, if that element is zero, the action corresponds to a free action on a sphere of dimension one less. If not, what we call the “double” of the action is. An example is constructed to show that the various cases can occur.

References

Hyman Bass and M. Pavaman Murthy, Grothendieck groups and Picard groups of abelian group rings, Ann. of Math. (2) 86 (1967), 16–73. MR 219592, DOI 10.2307/1970360
Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. MR 0106454
John Milnor, Groups which act on $S^n$ without fixed points, Amer. J. Math. 79 (1957), 623–630. MR 90056, DOI 10.2307/2372566
J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 196736, DOI 10.1090/S0002-9904-1966-11484-2
Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104

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