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Semifree actions on finite groups on homotopy spheres
Author:
John Ewing
Journal:
Trans. Amer. Math. Soc. 245 (1978), 431-442
MSC:
Primary 57S25; Secondary 57R85
MathSciNet review:
511421
Full-text PDF Free Access
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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.
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F. Atiyah and I.
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A.
I. Borevich and I.
R. Shafarevich, Number theory, Translated from the Russian by
Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press,
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William
Browder and Ted
Petrie, Diffeomorphisms of manifolds and
semifree actions on homotopy spheres., Bull.
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(42 #8513), http://dx.doi.org/10.1090/S0002-9904-1971-12646-0
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John
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Larry
Joel Goldstein, Analytic number theory, Prentice-Hall Inc.,
Englewood Cliffs, N.J., 1971. MR 0498335
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Melvin
Rothenberg, Differentiable group actions on spheres, (Univ.
Aarhus, Aarhus, 1970) Mat. Inst., Aarhus Univ., Aarhus, 1970,
pp. 455–475. MR 0301764
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Reinhard
Schultz, Homotopy sphere pairs admitting semifree differentiable
actions, Amer. J. Math. 96 (1974), 308–323. MR 0368053
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Reinhard
Schultz, Rational ℎ-cobordism invariants for lens space
bundles, Quart. J. Math. Oxford Ser. (2) 25 (1974),
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MR
0234452 (38 #2769)
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Kai
Wang, Semifree actions on homotopy
spheres, Trans. Amer. Math. Soc. 211 (1975), 321–337. MR 0377951
(51 #14120), http://dx.doi.org/10.1090/S0002-9947-1975-0377951-X
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Joseph
A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co.,
New York, 1967. MR 0217740
(36 #829)
- [1]
- M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. 87 (1968), 546-604. MR 0236952 (38:5245)
- [2]
- Z. I. Borevich and I. R. Shafarevich, Number theory, Academic Press, New York, 1966. MR 0195803 (33:4001)
- [3]
- W. Browder and T. Petrie, Diffeomorphisms of manifolds and semifree actions on homotopy spheres, Bull. Amer. Math. Soc. 77 (1971), 160-163. MR 0273636 (42:8513)
- [4]
- J. H. Ewing, Spheres as fixed point sets, Quart. J. Math. Oxford Ser. 27 (1976), 445-455. MR 0431233 (55:4234)
- [5]
- L. J. Goldstein, Analytic number theory, Prentice-Hall, Englewood Cliffs, N.J., 1971. MR 0498335 (58:16471)
- [6]
- M. Rothenberg, Differentiable group actions on spheres, Proc. Advanced Study Inst. Algebraic Topology, Aarhus, 1970, 455-475. MR 0301764 (46:919)
- [7]
- R. Schultz, Homotopy sphere pairs admitting semifree differentiable actions, Amer. J. Math. 96 (1974), 308-323. MR 0368053 (51:4295)
- [8]
- -, Rational h-corbordism invariants for lens space bundles, Quart. J. Math. Oxford Ser. 25 (1974), 497-511. MR 0372898 (51:9102)
- [9]
- -, Corrigenda, Quart. J. Math. Oxford Ser. 28 (1977), 128.
- [10]
- -, Spherelike G-manifolds with exotic equivariant tangent bundles (preprint).
- [11]
- G. B. Segal, Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129-151. MR 0234452 (38:2769)
- [12]
- K. Wang, Semifree actions on homotopy spheres, Trans. Amer. Math. Soc. 211 (1975), 321-337. MR 0377951 (51:14120)
- [13]
- J. A. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967. MR 0217740 (36:829)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9947-1978-0511421-3
PII:
S 0002-9947(1978)0511421-3
Article copyright:
© Copyright 1978 American Mathematical Society
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