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Periodicity theorems in equivariant surgery

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Equivariant Surgery Theories and Their Periodicity Properties

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References for Chapter III

  1. J. P. Alexander, P. E. Conner, and G. C. Hamrick, “Odd Order Group Actions and Witt Classification of Inner Products,” Lecture Notes in Mathematics Vol. 625, Springer, Berlin-Heidelberg-New York, 1977.

    Book  MATH  Google Scholar 

  2. M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. 87 (1968), 546–604.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Bredon, “Introduction to Compact Transformation Groups,” Pure and Applied Mathematics Vol. 46, Academic Press, New York, 1972.

    MATH  Google Scholar 

  5. W. Browder, “Surgery on Simply Connected Manifolds,” Ergeb. der Math. (2) 65, Springer, New York, 1972.

    Book  MATH  Google Scholar 

  6. _____, “Surgery and group actions,” lectures at A.M.S. Symposium on Algebraic and Geometric Topology, Stanford University, 1976.

    Google Scholar 

  7. W. Browder and F. Quinn, A surgery theory for G-manifolds and stratified sets, in “Manifolds-Tokyo, 1973,” (Conf. Proc. Univ. of Tokyo, 1973), University of Tokyo Press, Tokyo, 1975, pp. 27–36.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. P. E. Conner and E. E. Floyd, “Differentiable Periodic Maps,” Ergeb. der Math. Bd. 33, Springer, Berlin-Göttingen-Heidelberg, 1963.

    MATH  Google Scholar 

  10. M. Davis, Smooth G-manifolds as collections of fiber bundles, Pac. J. Math. 77 (1978), 315–363.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Davis and W. C. Hsiang, Concordance classes of regular U(n) and Sp(n) actions on homotopy spheres, Ann. of Math. 105 (1977), 325–341.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Davis, W. C. Hsiang, and J. Morgan, Concordance classes of regular O(n) actions on homotopy spheres, Acta Math. 144 (1980), 153–221.

    Article  MathSciNet  MATH  Google Scholar 

  13. T. tom Dieck, Orbittypen und äquivariante Homologie II, Arch. Math. (Basel) 26 (1975), 650–662.

    Article  MathSciNet  MATH  Google Scholar 

  14. K. H. Dovermann, “Addition of equivariant surgery obstructions,” Ph.D. Thesis, Rutgers University, 1978 (Available from University Microfilms, Ann Arbor, Mich.: Order Number DEL79-10380.)—Summarized in Dissertation Abstracts International 39 (1978/1979), 5406.

    Google Scholar 

  15. _____, Addition of equivariant surgery obstructions, in “Algebraic Topology, Waterloo 1978 (Conference Proceedings),” Lecture Notes in Mathematics Vol. 741, Springer, Berlin-Heidelberg-New York, 1979, pp. 244–271.

    Chapter  Google Scholar 

  16. _____, ℤ2 surgery theory, Michigan Math. J. 28 (1981), 267–287.

    Article  MathSciNet  MATH  Google Scholar 

  17. K. H. Dovermann and T. Petrie, G-Surgery II, Mem. Amer. Math. Soc. 37 (1982), No. 260.

    Google Scholar 

  18. _____, An induction theorem for equivariant surgery (G-Surgery III), Amer. J. Math. 105 (1983), 1369–1403.

    Article  MathSciNet  MATH  Google Scholar 

  19. K. H. Dovermann, T. Petrie, and R. Schultz, Transformation groups and fixed point data, Proceedings of the A.M.S. Summer Research Conference on Group Actions (Boulder, Colorado, 1983), Contemp. Math. 36 (1985), 161–191.

    MathSciNet  MATH  Google Scholar 

  20. K. H. Dovermann and M. Rothenberg, An Equivariant Surgery Sequence and Equivariant Diffeomorphism and Homeomorphism Classification, Mem. Amer. Math. Soc. 71 (1988), No. 379.

    Google Scholar 

  21. K. H. Dovermann and R. Schultz, Surgery on involutions with middle dimensional fixed point set, Pac. J. Math. 130 (1988), 275–297.

    Article  MathSciNet  MATH  Google Scholar 

  22. F. Hirzebruch, Involutionen auf Mannigfaltigkeiten, in “Proceedings of the Conference on Transformation Groups (Tulane, 1967),” Springer, Berlin-Heidelberg-New York, 1968, pp. 148–166.

    MATH  Google Scholar 

  23. K. Jänich, On the classification of O n -manifolds, Math. Ann. 176 (1978), 53–76.

    Article  MathSciNet  MATH  Google Scholar 

  24. K. Jänich and E. Ossa, On the signature of an involution, Topology 8 (1969), 27–30.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. C. Kirby and L. C. Siebenmann, “Foundational Essays on Topological Manifolds, Smoothings, and Triangulations,” Annals of Mathematics Studies Vol. 88, Princeton University Press, Princeton, 1977.

    Book  MATH  Google Scholar 

  26. W. Lellmann, Orbiträume von G-Mannigfaltigkeiten und stratifizierte Mengen, Diplomarbeit, Universität Bonn, 1975.

    Google Scholar 

  27. P. Löffler, Homotopielineare p Operationen auf Sphären, Topology 20 (1981), 291–312.

    Article  MathSciNet  Google Scholar 

  28. W. Lück and I. Madsen, Equivariant L-theory I, Aarhus Univ. Preprint Series (1987/1988), No. 8; [same title] II, Aarhus Univ. Preprint Series (1987/1988), No. 16 (to appear in Math. Zeitschrift).

    Google Scholar 

  29. I. Madsen, L. Taylor, and B. Williams, Tangential homotopy equivalence, Comment. Math. Helv. 55 (1980), p. 445–484.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Masuda and T. Petrie, Lectures on transformation groups and Smith equivalence, Proceedings of the A.M.S. Summer Research Conference on Group Actions (Boulder, Colorado, 1983), Contemp. Math. 36 (1985), 193–244.

    MathSciNet  MATH  Google Scholar 

  31. J. Milnor and J. Stasheff, “Characteristic Classes,” Annals of Mathematics Studies Vol. 76, Princeton University Press, Princeton, 1974.

    Book  MATH  Google Scholar 

  32. J. Morgan, A Product Formula for Surgery Obstructions, Mem. Amer. Math. Soc. 14 (1978), No. 201.

    Google Scholar 

  33. M. Morimoto, Bak groups and equivariant surgery, K-Theory 2 (1989), 456–483.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Nicas, Induction Theorems for Groups of Homotopy Manifold Structures, Memoirs Amer. Math. Soc. 39 (1982). No. 267.

    Google Scholar 

  35. T. Petrie, G-Surgery I-A survey, in “Algebraic and Geometric Topology (Conference Proceedings, Santa Barbara, 1977),” Lecture Notes in Mathematics Vol. 644, Springer, Berlin-Heidelberg-New York, 1978, pp. 197–223.

    MATH  Google Scholar 

  36. A. Ranicki, The total surgery obstruction, in “Algebraic Topology,” (Sympos. Proc., Aarhus, 1978) Lecture Notes in Mathematics, Springer, Berlin-Heidelberg-New York, 1979, pp. 271–316.

    Google Scholar 

  37. _____, The algebraic theory of surgery I: Foundations, Proc. London Math. Soc. (3) 40 (1980), 87–192.

    Article  MathSciNet  MATH  Google Scholar 

  38. _____, The algebraic theory of surgery II: Applications to Topology, Proc. London Math. Soc. (3) 40 (1980), 193–283.

    Article  MathSciNet  MATH  Google Scholar 

  39. _____, Exact sequences in the algebraic theory of surgery,” Princeton Mathematical Notes No. 26, Princeton University Press, Princeton, N. J., 1981.

    MATH  Google Scholar 

  40. M. Rothenberg, Differentiable group actions on spheres, in “Proc. Adv. Study Inst. on Algebraic Topology (Aarhus, 1970),” Various Publications Series Vol. 13, Matematisk Institut, Aarhus Universitet, 1970, pp. 455–475.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. _____, Homotopy invariants and G-manifolds: A look at the past 15 years, Proceedings of the A.M.S. Summer Research Conference on Group Actions (Boulder, Colorado, 1983), Contemp. Math. 36 (1985), 17–83.

    Google Scholar 

  43. _____, An infinite exact sequence in equivariant surgery, Mathematisches Forschungsinstitut Oberwolfach Tagungsbericht 14/1985 (Surgery and L-theory), 4–5.

    Google Scholar 

  44. J. Shaneson, Wall’s surgery obstruction groups for G × Z, Ann. of Math. 90 (1969), 296–334.

    Article  MathSciNet  MATH  Google Scholar 

  45. _____, Product formulas for L n (π), Bull. Amer. Math. Soc. 76 (1970), 787–791.

    Article  MathSciNet  MATH  Google Scholar 

  46. S. Sullivan, On the Hauptvermutung for manifolds, Bull. Amer. Math. Soc. 73 (1967), 598–600.

    Article  MathSciNet  MATH  Google Scholar 

  47. R. Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240–284.

    Article  MathSciNet  MATH  Google Scholar 

  48. A. Verona, “Stratified Mappings-Structure and Triangulability,” Lecture Notes in Mathematics Vol. 1102, Springer, Berlin-Heidelberg-New York, 1984.

    Book  MATH  Google Scholar 

  49. C. T. C. Wall, “Surgery on Compact Manifolds,” London Math. Soc. Monographs Vol. 1, Academic Press, London and New York, 1970.

    MATH  Google Scholar 

  50. S. Weinberger, The topological classification of stratified spaces, preprint, University of Chicago, 1989.

    Google Scholar 

  51. R. E. Williamson, Surgery in M × N with π 1 (M) ≠ 1, Bull. Amer. Math. Soc. 75 (1969), 582–585.

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Yan, Periodicity in equivariant surgery and applications, Ph.D. Thesis, University of Chicago, in preparation.

    Google Scholar 

  53. T. Yoshida, Surgery obstructions of twisted products, J. Math. Okayama Univ. 24 (1982), 73–97.

    MATH  Google Scholar 

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Dovermann, K.H., Schultz, R. (1990). Periodicity theorems in equivariant surgery. In: Equivariant Surgery Theories and Their Periodicity Properties. Lecture Notes in Mathematics, vol 1443. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092358

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