Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T01:18:08.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Smith equivalence of representations

Published online by Cambridge University Press:  24 October 2008

Ted Petrie
Ted Petrie
Affiliation:
Rutgers University, New Brunswick, New Jersey
Get access Rights & Permissions [Opens in a new window]

Extract

An old question of P. A. Smith asks: If a finite group G acts smoothly on a closed homotopy sphere Σ with fixed set ΣG consisting of two points p and q, are the tangential representations Tp Σ and Tq Σ of G at p and q equal? Put another way: Describe the representations (V, W) of G which occur as (Tp ΣTq Σ) for Σ a sphere with smooth action of G and ΣG = p ∪ q. Under these conditions we say V and W are Smith equivalent (21) and write V ~ W. A stronger equivalence relation is also interesting. We say representations V and W are s-Smith equivalent if (V, W) = (Tp Σ, Tq Σ) and Σ is a semi-linear G sphere (23), i.e. ΣK is a homotopy sphere for all K and ΣG = p ∪ q. In this case we write V ≈ W.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 1 , July 1983 , pp. 61 - 99
Copyright
Copyright © Cambridge Philosophical Society 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Atiyah, M. F.K-theory (Benjamin, 1967).Google Scholar
(2)Atiyah, M. F. and Bott, R.A Lefschetz fixed point formula for elliptic complexes I. Ann. Math. 86 (1967), 374–407.CrossRefGoogle Scholar
(3)Atiyah, M. F. and Singer, I. M.The index of elliptic operators III. Ann. Math. 87 (1968), 546–604.CrossRefGoogle Scholar
(4)Atiyah, M. F. and Segal, G.The index of elliptic operators II. Ann. Math. 87 (1968), 531–545.CrossRefGoogle Scholar
(5)Atiyah, M. F. and Tall, D.Group representations, λ-rings and the J-homomorphism. Topology 8 (1969), 253–297.CrossRefGoogle Scholar
(6)Bak, A. The computation of surgery groups of finite groups with abelian 2 hyperelementary subgroups. Proc. Conf.Alg. K-theory. Lecture Notes in Math. 551 (Springer-Verlag, 1976), 384–409.Google Scholar
(7)Bredon, G.Representations at fixed points of smooth actions of compact groups. Ann. Math. 89 (1969), 515–532.CrossRefGoogle Scholar
(8)Cappell, S. E. and Shaneson, J. L. Fixed points of periodic differentiable maps. (Preprint.)Google Scholar
(9)Dovermann, K. H.Even dimensional s-Smith equivalent representations. Proceedings of 1982 Aarhus Topology Conference (To appear).Google Scholar
(10)Dovermann, K. and Petrie, T.G surgery II. Memoirs Amer. Math. Soc. 260 (1982).Google Scholar
(11)Dovermann, K.An induction theorem for equivariant surgery. Amer.J. Math. (To appear).Google Scholar
(12)Dovermann, K.Artin relation for smooth representations. Proc. Nat. Acad. Sci. 77 (1980), 5620–5621.CrossRefGoogle ScholarPubMed
(13)Meyerhoff, A. and Petrie, T.Quasi equivalence of G modules. Topology 15 (1976), 69–75.CrossRefGoogle Scholar
(14)Milnor, J.Whitehead torsion. Bull. Amer. Math. Soc. 72 (1966), 358–426.CrossRefGoogle Scholar
(15)Petrie, T. Three theorems in transformation groups. Lecture Notes in Mathematics, no. 763 (Springer-Verlag, 1979), 549–572.Google Scholar
(16)Petrie, T. Isotropy representations of actions on spheres. (To appear.)Google Scholar
(17)Petrie, T.The Atiyah-Singer invariant, the Wall groups Ln (π, 1) and the function (tex + 1)/ (tex − 1). Ann. Math. 92 (1970), 174–187.CrossRefGoogle Scholar
(18)Petrie, T.Pseudoequivalences of G manifolds. Proc. Symp. Pure Math. XXXII (AMS, 1978), 129–210.Google Scholar
(19)Petrie, T.One fixed point actions on spheres I. Adv. Math. 46 (1982), 3–14.CrossRefGoogle Scholar
(20)Petrie, T.One fixed point actions on spheres II. Adv. Math. 46 (1982), 15–70.CrossRefGoogle Scholar
(21)Petrie, T.The equivariant J homomorphism and Smith equivalence of representations. Can. Math. Society, Conf. Proc. 2 (1982), (To appear).Google Scholar
(22)Petrie, T. and Randall, J.Transformation groups on manifolds. Dekker Lecture Series (To appear).Google Scholar
(23)Rothenberg, M.Torsion invariants and finite transformation groups. Proc. Symp. Pure Math. no. XXXII (A.M.S., 1978), 267–311.CrossRefGoogle Scholar
(24)Sanchez, C.Actions of groups of odd order on compact, orientable manifolds. Proc. Amer. Math. Soc. 54 (1976), 445–448.CrossRefGoogle Scholar
(25)Schultz, R.Spherelike G-manifolds with exotic equivariant tangent bundles. Adv. Math. Supp. Studies 5 (1979), 1–38.Google Scholar
(26)Siegel, A. Thesis, Rutgers (1982).Google Scholar
(27)Smith, P. A.New results and old problems in finite transformation groups. Bull. Amer. Math. Soc. (1960), 401–415.CrossRefGoogle Scholar
(28)Wall, T.Surgery on compact manifolds (Academic Press, 1970).Google Scholar
(29)Wall, T.Classification of Hermitian forms VI. Ann. Math. 103 (1976), 1–80.CrossRefGoogle Scholar
(30)Wasserman, A.Equivariant differential topology. Topology 8 (1969), 127–150.CrossRefGoogle Scholar
(31)Kawakubo, K.Compact Lie group actions and fiber homotopy type. J. Math. Soc. Japan 33, (1981), 295–321.Google Scholar