Skip to main content
Log in

Differentiable group actions on homotopy spheres

I. Differential structure and the knot invariant

  • Published:
Inventiones mathematicae Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Alexander, J. P., Hamrick, G. C., Vick, J. W.: Involutions on homotopy spheres. Invent. Math.24, 35–50 (1974)

    Google Scholar 

  2. Anderson, D. W., Brown, E. H., Peterson, F.: The structure of the spin cobordism ring. Ann. Math.86, 271–298 (1967)

    Google Scholar 

  3. Becker, J. C., Schultz, R. E.: Equivariant function spaces and stable homotopy theory I. Comment. Math. Helv.49, 1–34 (1974)

    Google Scholar 

  4. Boardman, J. M., Vogt, R. M.: Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Math.347. Berlin-Heidelberg-New York: Springer 1973

    Google Scholar 

  5. Bredon, G.: Aπ * structure forΘ * and applications to transformation groups. Ann. Math.86, 434–448 (1967)

    Google Scholar 

  6. Bredon, G.: Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46. New York: Academic Press 1972

    Google Scholar 

  7. Bredon, G.: Classification of regular actions of classical groups with three orbit types, mimeographed, Rutgers University, 1973 (Revised edition to appear in Ann. of Math. Studies)

  8. Browder, W.: Surgery and the theory of differentiable transformation groups. Proceedings of the conference on transformation groups (New Orleans, 1967), pp. 1–46. New York: Springer 1968

    Google Scholar 

  9. Browder, W.: Surgery on simply connected manifolds. In: Ergebnisse der Mathematik, Bd. 65. New York: Springer 1972

    Google Scholar 

  10. Browder, W., Petrie, T.: Semifree and quasifreeS 1 actions on homotopy spheres. Essays on topology and related topics (Memoires dediés à G. de Rham), pp. 136–146. New York: Springer 1970

    Google Scholar 

  11. Browder, W., Petrie, T.: Diffeomorphisms of manifolds and semifree actions on homotopy spheres. Bull. Amer. Math. Soc.77, 160–163 (1971)

    Google Scholar 

  12. Brumfiel, G.: On the homotopy groups of BPL and PL/0-I. Ann. Math.87, 291–311 (1968)

    Google Scholar 

  13. Brumfiel, G.: DifferentiableS 1 actions on homotopy spheres, mimeographed. Berkeley: University of California 1969

    Google Scholar 

  14. Cappell, S.: Uhitary nilpotent groups and HermitianK-theory I. Bull. Amer. Math. Soc.80, 1117–1122 (1974)

    Google Scholar 

  15. Cappell, S., Shaneson, J.: The codimension two placement problem and homology equivalent manifolds. Ann. Math.99, 277–348 (1974)

    Google Scholar 

  16. Connolly, F.: Linking numbers and surgery. Topology12, 389–410 (1973)

    Google Scholar 

  17. Davis, M., Hsiang, W.-C., Hsiang, W.-Y.: to appear

  18. Dold, A., Lashof, R.: Principal quasifibration and fibre homotopy equivalence of bundles. Illinois J. Math.3, 285–305 (1959)

    Google Scholar 

  19. Douady, A., Hérault, L.: Arrondisement des variétès à coins (appendix to a paper by A. Borel and J.-P. Serre). Comment. Math. Helv.48, 484–489 (1973)

    Google Scholar 

  20. Friedlander, E. M.: The etale homotopy theory of a geometric fibration. Manuscripta Math.10, 209–244 (1973)

    Google Scholar 

  21. Hirzebruch, F., Mayer, K.:O(n)-Mannigfaltigkeiten, Exotische Sphären, und Singularitäten. Lecture Notes in Math.57. New York: Springer 1968

    Google Scholar 

  22. Holzsager, R.: Stable splitting ofK(G, 1). Proc. Amer. Math. Soc.31, 305–306 (1972)

    Google Scholar 

  23. Hsiang, W.-C., Hsiang, W.-Y: Differentiable actions of compact connected classical groups I. Amer. J. Math.89, 705–786 (1967);ibid. Hsiang, W.-C., Hsiang, W.-Y: Differentiable actions of compact connected classical groups II. Ann. Math.92, 189–223 (1970)

    Google Scholar 

  24. Hsiang, W.-C., Hsiang, W.-Y.: The degree of symmetry of homotopy spheres. Ann. Math.89, 52–67 (1969)

    Google Scholar 

  25. Hsiang, W.-Y.: On the unknottedness of the fixed point set of differentiable circle group actions on spheres—P. A. Smith conjecture. Bull. Amer. Math. Soc.70, 678–680 (1964)

    Google Scholar 

  26. Hsiang, W.-Y.: On the degree of symmetry and the structure of highly symmetric manifolds. Tamkang J. Math.2, 1–22 (1971)

    Google Scholar 

  27. Jones, L.: The converse to the fixed point theorem of P. A. Smith: I. Ann. Math.94, 52–68 (1971)

    Google Scholar 

  28. Jones, L.:Ibid.. Indiana University Math. J.22, 309–325 (1972); correction24, 1001–1003 (1975)

    Google Scholar 

  29. Jones, L.: Patch spaces. Ann. Math.97, 306–343 (1973); correction102. 183–185 (1975)

    Google Scholar 

  30. Kervaire, M., Milnor, J.: Groups of homotopy spheres. Ann. Math.78, 514–537 (1963)

    Google Scholar 

  31. Lawson, H. B., Yau, S.-T.: Scalar curvature, nonabelian group actions, and the degree of symmetry of exotic spheres. Comment. Math. Helv.49, 232–244 (1974)

    Google Scholar 

  32. Lee, R.: Nonexistence of free differentiable actions ofS 1 and ℤ2 on homotopy spheres. Proceedings of the Conference on Transformation Groups (New Orleans, 1967), pp. 208–209. New York: Springer 1968

    Google Scholar 

  33. Levine, J.: A classification of differentiable knots. Ann. Math.82, 15–50 (1965)

    Google Scholar 

  34. Levine, J.: Self-equivalences ofS n×S k. Trans. Amer. Math. Soc.143, 523–543 (1969)

    Google Scholar 

  35. Levine, J.: Semi-free circle actions on spheres. Invent. math.22, 161–186 (1973)

    Google Scholar 

  36. Lynch, P.: FramedG-homology spheres, Ph. D. Thesis, Brandeis University, 1971

  37. Milnor, J.: Remarks concerning spin manifolds, differential and combinatorial topology (A symposium in honor of M. Morse), Princeton Mathematical Series No. 27, 55–62. Princeton: Princeton University Press 1965

    Google Scholar 

  38. Mosher, R., Tangora, M.: Cohomology operations and applications in homotopy theory. New York: Harper and Row 1968

    Google Scholar 

  39. Rothenberg, M.: Differentiable group actions on spheres. Proceedings of the advanced study institute on algebraic topology (Aarhus, 1970), 455–475. Mathematical Institute, Aarhus University, Aarhus, 1970

    Google Scholar 

  40. Rothenberg, M., Sondow, J.: Nonlinear smooth representations of compact Lie groups. Mimeographed, University of Chicago, 1969

  41. Rourke, C. P.: The Hauptvermuting according to Sullivan. Mimeographed, Institute for Advanced Study, 1968

  42. Schultz, R.: The nonexistence of freeS 1 actions on some homotopy spheres. Proc. Amer. Math. Soc.27, 595–597 (1971)

    Google Scholar 

  43. Schultz, R.: Improved estimates for the degree of symmetry of certain homotopy spheres. Topology10, 227–235 (1971)

    Google Scholar 

  44. Schultz, R.: Semifree circle actions and the degree of symmetry of homotopy spheres. Amer. J. Math.93, 829–839 (1971)

    Google Scholar 

  45. Schultz, R.: Composition constructions on diffeomorphisms ofS p×S q. Pacific J. Math.42, 739–754 (1972)

    Google Scholar 

  46. Schultz, R.: Circle actions on homotopy spheres bounding plumbing manifolds. Proc. Amer. Math. Soc.36, 297–300 (1972)

    Google Scholar 

  47. Schultz, R.: Circle actions on homotopy spheres bounding generalized plumbing manifolds. Math. Ann.205, 201–210 (1973)

    Google Scholar 

  48. Schultz, R.: Homotopy sphere pairs admitting semifree differentiable actions. Amer. J. Math.96, 308–323 (1974)

    Google Scholar 

  49. Schultz, R.: Differentiable ℤ p actions on homotopy spheres. Bull. Amer. Math. Soc.80, 961–964 (1974)

    Google Scholar 

  50. Schultz, R.: Circle actions on homotopy spheres not bounding spin manifolds. Trans. Amer. Math. Soc. to appear

  51. Steenrod, N., Epstein, D. B. A.: Cohomology operations. Annals of mathematics studies No. 50. Princeton: Princeton University Press 1962

    Google Scholar 

  52. Sullivan, D.: Triangulating homotopy equivalences, Ph.D. Thesis, Princeton University, 1965

  53. Sullivan, D.: Smoothing homotopy equivalences. Mimeographed, University of Warwick, 1966

  54. Sullivan, D.: Geometric topology I. Localization, periodicity, and Galois symmetry. Mimeographed, M.I.T., 1970

  55. Sullivan, D.: Genetics of homotopy theory and the Adams conjecture. Ann. Math.100, 1–79 (1974)

    Google Scholar 

  56. Toda, H.:p-primary components of homotopy groups IV. Compositions and toric constructions. Mem. College Sci. Kyoto University32, 297–332 (1959)

    Google Scholar 

  57. Toda H.: Composition methods in homotopy groups of spheres. Ann. of Math. Studies No. 49. Princeton: Princeton University Press 1962

    Google Scholar 

  58. Wall, C. T. C.: Surgery on compact manifolds. London Mathematical Society Monographs No. 1. New York: Academic Press 1970

    Google Scholar 

  59. Schultz, R.: Homotopy decompositions of equivariant function spaces I. Spaces of principal bundle maps. Math. Z.131, 49–75 (1973)

    Google Scholar 

  60. Quinn, F.: Semifree group actions and surgery onPL homology manifolds. Geometric Topology (Proceedings of the Geometric Topology Conference, Park City, Utah, 1974). Lecture Notes in Math.438, 395–414. Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Partially supported by NSF Grants GP-19530A1/A2, GP-36418X, and MPS74-03609.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schultz, R. Differentiable group actions on homotopy spheres. Invent Math 31, 105–128 (1976). https://doi.org/10.1007/BF01404111

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01404111

Keywords

Navigation