Geometriae Dedicata

, Volume 148, Issue 1, pp 263–289 | Cite as

A Bass–Heller–Swan formula for pseudoisotopies

Original Paper

Abstract

The Bass–Heller–Swan formula is a basic calculational tool in pseudoisotopy K-theory. We describe the Nil-groups and the Bass–Heller–Swan splitting for the group of the pseudoisotopies of a closed manifold. We use the methods of controlled topology used in the Bass–Heller–Swan splitting in K-theory.

Keywords

Pseudoisotopy Relaxation Nil-groups 

Mathematics Subject Classification (2010)

57N37 19D10 

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References

  1. 1.
    Anderson D.R., Hsiang W.-C.: The functors K i and pseudoisotopies of polyhedra. Ann. Math. 105, 201–223 (1977)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bass H.: Algebraic K-Theory. W. A. Benjamin, New York, Amsterdam (1968)Google Scholar
  3. 3.
    Bass H., Heller A., Swan R.: The whitehead group of a polynomial extension. Inst. Hautes Études Sci. Publ. Math. 22, 61–79 (1964)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Burghelea D.: On the decomposition of the automorphisms group of M × S 1. Rev. Roumaine Math. Pures Appl. 22, 17–30 (1977)MATHMathSciNetGoogle Scholar
  5. 5.
    Burghelea D., Lashof R.: Stability of concordances and the suspension homomorphism. Ann. Math. 105, 449–472 (1977)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Burghelea, D., Lashof, R., Rothenberg, M.: Groups of Automorphisms of Manifolds. Lecture Notes in Math., no. 473, Springer Verlag, New York, (1975)Google Scholar
  7. 7.
    Chapman, T.A.: Lectures on Hilbert cube manifolds. C.B.M.S. Regional Conference Series in Mathematics 28, Am. Math. Soc. (1976a)Google Scholar
  8. 8.
    Chapman T.A.: Concordances of Hilbert cube manifolds. Trans. Am. Math. Soc. 219, 253–268 (1976b)MATHGoogle Scholar
  9. 9.
    Chapman T.A.: Approximation results in Hilbert cube manifolds. Trans. Am. Math. Soc. 262, 303–334 (1980)MATHGoogle Scholar
  10. 10.
    Chapman, T.A.: Approximation results in topological manifolds. Mem. Am. Math. Soc. 34, iii+64pp (1981)Google Scholar
  11. 11.
    Chapman, T. A.: Controlled Simple Homotopy Theory and Applications. Springer Lecture Notes in Math. 1009, Springer, New York, (1983)Google Scholar
  12. 12.
    Farrell F.T.: The obstruction to fibering a manifold over the circle. Indiana Univ. Math. J. 21, 315–346 (1971)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Grayson, D.: Higher algebraic K-theory II (after Daniel Quillen). In: Algebraic K-Theory, Proceedings Conference on Northwestern Univ., Evanston, Ill., vol. 551, pp. 217–240, Lecture Notes in Math., Springer, Berlin (1976)Google Scholar
  14. 14.
    Hatcher, A.: Concordance spaces, higher simple homotopy theory, and applications. In: Algebraic and Geometric Topology (Proceedings Symposium on Pure Mathamatics, Standford Univ., Standford, Calif. 1976), Part 1, pp. 3–21, Proceedings Symposium on Pure Mathamatics, XXXII, Am. Math. Soc., Providence, R.I., (1978)Google Scholar
  15. 15.
    Hatcher, A., Wagoner, J.: Pseudo-isotopies of compact manifolds. Asterisque 6, Soc. Math. France, (1973)Google Scholar
  16. 16.
    Hsiang, W.C.: Decomposition formula of Laurent extension in algebraic K-theory and the role of codimension 1 submanifold in topology. In: Algebraic K-Theory, II: “Classical” Algebraic K-Theory and Connections with Arithmetic (Proceedings Conference on Seattle Research Center, Battelle Memorial Inst., 1972), pp. 308–327. Lecture Notes in Math., vol. 342, Springer, Berlin, (1973)Google Scholar
  17. 17.
    Hughes C.B.: Spaces of approximate fibrations on Hilbert cube manifolds. Compositio Math. 56, 131–151 (1985a)MATHMathSciNetGoogle Scholar
  18. 18.
    Hughes C.B.: Bounded homotopy equivalences of Hilbert cube manifolds. Trans. Am. Math. Soc. 287, 621–643 (1985b)MATHGoogle Scholar
  19. 19.
    Hughes C.B.: Approximate fibrations on topological manifolds. Michigan Math. J. 32, 167–183 (1985c)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hughes C.B.: Delooping controlled pseudo-isotopies of Hilbert cube manifolds. Topol. Appl. 26, 175–191 (1987)MATHCrossRefGoogle Scholar
  21. 21.
    Hughes C.B.: Controlled homotopy topological structures. Pac. J. Math. 133, 69–97 (1988)MATHGoogle Scholar
  22. 22.
    Hughes, C.B.: Problem List, Workshop on Nil Phenomena in Topology. Vanderbilt University, April 14–15, http://www.math.vanderbilt.edu/~hughescb/Shanks2007/ProblemSession3.pdf (2007)
  23. 23.
    Hughes, C.B., Prassidis, S.: Control and relaxation over the circle. Mem. Am. Math. Soc. 145(691), x+96pp (2000)Google Scholar
  24. 24.
    Hughes, C.B., Ranicki, A.: Ends of Complexes. Cambridge Tracts in Mathamatics, vol. 123, Cambridge Univ. Press, Cambridge (1996)Google Scholar
  25. 25.
    Hughes C.B., Taylor L., Williams E.B.: Bundle theories for topological manifolds. Trans. Am. Math. Soc. 319, 1–65 (1990)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Hughes C.B., Taylor L., Williams E.B.: Manifold approximate fibrations are approximate bundles. Forum Math. 3, 309–325 (1991)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Hughes, C.B., Taylor, L., Williams, E.B.: Splitting forget control maps, preprint (2009)Google Scholar
  28. 28.
    Hütenmann T., Klein J., Vogell W., Waldhausen F., Williams B.: The “fundamental theorem” of the algebraic K-theory of spaces. I. J. Pure Appl. Algebra 160, 21–52 (2001)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Igusa, K.: What happens to Hatcher and Wagoner’s formulas for π 0 C(M) when the first Postnikov invariant of M is nontrivial? Algebraic K-theory, number theory, geometry and analysis (Bielefeld, 1982), 104–172, Lecture Notes in Math., 1046, Springer, Berlin (1984)Google Scholar
  30. 30.
    Igusa, K.: The stability theorem for smooth pseudoisotopies. K-Theory 2, vi+355 pp (1988)Google Scholar
  31. 31.
    Kinsey L.C.: Pseudoisotopies and submersions of a compact manifold to the circle. Topology 26, 67–78 (1987)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Prassidis S.: The Bass–Heller–Swan formula for the equivariant topological Whitehead group. K-theory 5, 395–448 (1992)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Ranicki A.: Lower K- and L-Theory. London Math. Soc. Lect. Notes Ser., vol. 178. Cambridge University Press, New York (1992)Google Scholar
  34. 34.
    Siebenmann, L.C.: The obstruction to finding a boundary of an open manifold of dimension greater than five. Thesis, Princeton University (1965)Google Scholar
  35. 35.
    Siebenmann L.C.: A total whitehead torsion obstruction to fibering over the circle. Comment. Math. Helv. 45, 1–48 (1970)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Weiss, M., Williams, B.: Automorphisms of Manifolds. Surveys on Surgery Theory, vol. 2, 165–220, Ann. Math. Stud., 149, Princeton University Press, Princeton, NJ, (2001)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsCanisius CollegeBuffaloUSA

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