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Elliptic Theory on Manifolds with Corners: II. Homotopy Classification and K-Homology

  • Vladimir Nazaikinskii
  • Anton Savin
  • Boris Sternin
Part of the Trends in Mathematics book series (TM)

Abstract

We establish the stable homotopy classification of elliptic pseudo-differential operators on manifolds with corners and show that the set of elliptic operators modulo stable homotopy is isomorphic to the K-homology group of some stratified manifold. By way of application, generalizations of some recent results due to Monthubert and Nistor are given.

Keywords

Manifold with corners elliptic operator stable homotopy K-homology stratified manifold 

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References

  1. [1]
    M.F. Atiyah, Global theory of elliptic operators, Proc. of the Int. Symposium on Functional Analysis (Tokyo), University of Tokyo Press, 1969, pp. 21–30.Google Scholar
  2. [2]
    B. Blackadar, K-theory for operator algebras, Mathematical Sciences Research Institute Publications, no. 5, Cambridge University Press, 1998, Second edition.Google Scholar
  3. [3]
    U. Bunke, Index theory, eta forms, and Deligne cohomology. Preprint arXiv: math.DG/0201112.Google Scholar
  4. [4]
    A. Connes, Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994.zbMATHGoogle Scholar
  5. [5]
    N. Higson and J. Roe, Analytic K-homology, Oxford University Press, Oxford, 2000.zbMATHGoogle Scholar
  6. [6]
    G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Inv. Math. 91 (1988), no. 1, 147–201.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    T. Krainer, Elliptic boundary problems on manifolds with polycylindrical ends, J. Funct. Anal., 244 (2007), no. 2, 351–386.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    R. Lauter and S. Moroianu, The index of cusp operators on manifolds with corners, Ann. Global Anal. Geom. 21 (2002), no. 1, 31–49.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    P.-Y. Le Gall and B. Monthubert, K-theory of the indicial algebra of a manifold with corners, K-Theory 23 (2001), no. 2, 105–113.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    P. Loya, The index of b-pseudodifferential operators on manifolds with corners, Ann. Global Anal. Geom. 27 (2005), no. 2, 101–133.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    G. Luke, Pseudodifferential operators on Hilbert bundles, J. Diff. Equations 12 (1972), 566–589.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    R. Melrose and V. Nistor, K-theory of C *-algebras of b-pseudodifferential operators, Geom. Funct. Anal. 8 (1998), no. 1, 88–122.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    R. Melrose and P. Piazza, Analytic K-theory on manifolds with corners, Adv. in Math. 92 (1992), no. 1, 1–26.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    B. Monthubert, Groupoids and pseudodifferential calculus on manifolds with corners, J. Funct. Anal. 199 (2003), no. 1, 243–286.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    B. Monthubert and V. Nistor, A topological index theorem for manifolds with corners, arXiv: math.KT/0507601, 2005.Google Scholar
  16. [16]
    V. Nistor, An index theorem for gauge-invariant families: The case of solvable groups, Acta Math. Hungarica 99 (2003), no. 2, 155–183.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    A. Savin, Elliptic operators on singular manifolds and K-homology, K-theory 34 (2005), no. 1, 71–98.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    V.E. Nazaikinskii, A.Yu. Savin, and B.Yu. Sternin, On the homotopy classification of elliptic operators on stratified manifolds, Izvestiya: Mathematics, 71 (2007), no. 6, 91–118.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Vladimir Nazaikinskii
    • 1
  • Anton Savin
    • 2
  • Boris Sternin
    • 2
  1. 1.Institute for Problems in MechanicsRussian Academy of SciencesMoscowRussia
  2. 2.Independent University of MoscowMoscowRussia

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