Semifree actions on finite groups on homotopy spheres

Ewing, John

doi:10.1090/S0002-9947-1978-0511421-3

Semifree actions on finite groups on homotopy spheres
HTML articles powered by AMS MathViewer

by John Ewing

Trans. Amer. Math. Soc. 245 (1978), 431-442

DOI: https://doi.org/10.1090/S0002-9947-1978-0511421-3

PDF | Request permission

Abstract:

We show that for any finite group the group of semifree actions on homotopy spheres of some fixed even dimension is finite, provided that the dimension of the fixed point set is greater than 2. The argument shows that for such an action the normal bundle to the fixed point set is equivariantly, stably trivial.

References

M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
William Browder and Ted Petrie, Diffeomorphisms of manifolds and semifree actions on homotopy spheres, Bull. Amer. Math. Soc. 77 (1971), 160–163. MR 273636, DOI 10.1090/S0002-9904-1971-12646-0
John Ewing, Spheres as fixed point sets, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 108, 445–455. MR 431233, DOI 10.1093/qmath/27.4.445
Larry Joel Goldstein, Analytic number theory, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0498335
Melvin Rothenberg, Differentiable group actions on spheres, Proc. Advanced Study Inst. on Algebraic Topology (Aarhus, 1970) Aarhus Univ., Matematisk Inst., Aarhus, 1970, pp. 455–475. MR 0301764
Reinhard Schultz, Homotopy sphere pairs admitting semifree differentiable actions, Amer. J. Math. 96 (1974), 308–323. MR 368053, DOI 10.2307/2373635
Reinhard Schultz, Rational $h$-cobordism invariants for lens space bundles, Quart. J. Math. Oxford Ser. (2) 25 (1974), 497–511. MR 372898, DOI 10.1093/qmath/25.1.497

28

Spherelike G-manifolds with exotic equivariant tangent bundles

Graeme Segal, Equivariant $K$-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151. MR 234452, DOI 10.1007/BF02684593
Kai Wang, Semifree actions on homotopy spheres, Trans. Amer. Math. Soc. 211 (1975), 321–337. MR 377951, DOI 10.1090/S0002-9947-1975-0377951-X
Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740