Mathematische Annalen

, Volume 366, Issue 3–4, pp 1513–1559 | Cite as

Realizing the analytic surgery group of Higson and Roe geometrically part III: higher invariants

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Abstract

We construct an isomorphism between the geometric model and Higson-Roe’s analytic surgery group, reconciling the constructions in the previous papers in the series on “Realizing the analytic surgery group of Higson and Roe geometrically” with their analytic counterparts. Following work of Lott and Wahl, we construct a Chern character on the geometric model for the surgery group; it is a “delocalized Chern character”, from which Lott’s higher delocalized \(\rho \)-invariants can be retrieved. Following work of Piazza and Schick, we construct a geometric map from Stolz’ positive scalar curvature sequence to the geometric model of Higson-Roe’s analytic surgery exact sequence.

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Notes

Acknowledgments

The authors wish to express their gratitude towards Karsten Bohlen, Heath Emerson, Nigel Higson, Paolo Piazza, Thomas Schick and Charlotte Wahl for discussions. They also thank the Courant Centre of Göttingen, the Leibniz Universität Hannover, the Graduiertenkolleg 1463 (Analysis, Geometry and String Theory) and Université Blaise Pascal Clermont-Ferrand for facilitating this collaboration.

References

  1. 1.
    Baum, P., Connes, A.: Chern Character for Discrete Groups, A féte of Topology, pp. 163–232. Academic Press, Boston (1988)Google Scholar
  2. 2.
    Baum, P., Douglas, R.: \(K\)-Homology and Index Theory. Operator Algebras and Applications (R. Kadison editor), vol. 38 of Proceedings of Symposia in Pure Math., pp. 117–173. AMS, Providence RI (1982)Google Scholar
  3. 3.
    Baum, P., Douglas, R.: Index theory, bordism, and \(K\)-homology. Contemp. Math. 10, 1–31 (1982)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Baum, P., van Erp, E.: \(K\)-homology and index theory on contact manifolds. Acta Math. 213(1), 1–48 (2014)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Baum, P., Higson, N., Schick, T.: On the equivalence of geometric and analytic \(K\)-homology. Pure Appl. Math. Q. 3, 1–24 (2007)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Burghelea, D.: The cyclic homology of the group rings. Comment. Math. Helv. 60(3), 354–365 (1985)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Connes, A.: Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math. No. 62, 257–360 (1985)MathSciNetGoogle Scholar
  8. 8.
    Connes, A.: Noncommutative Geometry. Academic Press Inc., San Diego (1994)MATHGoogle Scholar
  9. 9.
    Connes, A., Moscovici, H.: Cyclic cohomology, the Novikov conjecture and hyperbolic groups. Topology 29, 345–388 (1990)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Cuntz, J., Skandalis, G., Tsygan, B.: Cyclic homology in non-commutative geometry, Encyclopaedia of Mathematical Sciences, 121. Operator Algebras and Non-commutative Geometry II. Springer-Verlag, Berlin (2004)MATHCrossRefGoogle Scholar
  11. 11.
    Deeley, R., Goffeng, M.: Realizing the analytic surgery group of Higson and Roe geometrically, Part I: The geometric model. J. Homotopy Relat. Struct. (to appear)Google Scholar
  12. 12.
    Deeley, R., Goffeng, M.: Realizing the analytic surgery group of Higson and Roe geometrically, Part II: Relative \(\eta \)-invariants. Math. Ann. (to appear)Google Scholar
  13. 13.
    Guentner, E.: \(K\)-homology and the index theorem, Index theory and operator algebras (Boulder, CO, 1991), pp. 47–66. Contemp. Math., 148, Amer. Math. Soc., Providence, RI (1993)Google Scholar
  14. 14.
    Gorokhovsky, A., Moriyoshi, H., Piazza, P.: A note on the higher Atiyah-Patodi-Singer index theorem on Galois coverings. J. Noncommutative Geom. (to appear)Google Scholar
  15. 15.
    Haagerup, U.: An example of nonnuclear C*-algebra which has the metric approximation property. Inv. Math. 50, 279–293 (1979)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Hanke, B., Pape, D., Schick, T.: Codimension two index obstructions to positive scalar curvature. Annales de l’Institut Fourier. (to appear)Google Scholar
  17. 17.
    Helemskii, A.Y.: The Homology of Banach and Topological Algebras, Translated from the Russian by Alan West. Mathematics and its Applications (Soviet Series), vol. 41. Kluwer Academic Publishers Group, Dordrecht (1989)Google Scholar
  18. 18.
    Higson, N., Roe, J.: On the Coarse Baum-Connes Conjecture, Novikov Conjectures, Index Theorems and Rigidity, vol. 2 (Oberwolfach, 1993), pp. 227–254, London Math. Soc. Lecture Note Ser., 227. Cambridge Univ. Press, Cambridge (1995)Google Scholar
  19. 19.
    Higson, N., Roe, J.: Analytic \(K\)-Homology. Oxford University Press, Oxford (2000)MATHGoogle Scholar
  20. 20.
    Higson, N., Roe, J.: Mapping Surgery to Analysis I: Analytic Signatures. \(K\)-Theory 33, pp. 277–299 (2005)Google Scholar
  21. 21.
    Higson, N., Roe, J.: Mapping Surgery to Analysis II: Geometric Signatures. \(K\)-Theory 33, pp. 301–325 (2005)Google Scholar
  22. 22.
    Higson, N., Roe, J.: Mapping Surgery to Analysis III: Exact Sequences. \(K\)-Theory 33, pp. 325–346 (2005)Google Scholar
  23. 23.
    Higson, N., Roe, J.: \(K\)-homology, assembly and rigidity theorems for relative eta invariants, Pure Appl. Math. Q. 6 (2010), no. 2, Special Issue: In honor of Michael Atiyah and Isadore Singer, pp. 555–601Google Scholar
  24. 24.
    Hilsum, M.: Index classes of Hilbert modules with boundary. Preprint (2001)Google Scholar
  25. 25.
    Hilsum, M.: Bordism invariance in \(KK\)-theory. Math. Scand. 107(1), 73–89 (2010)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Ji, R.: Smooth dense subalgebras of reduced group \(C^{*}\)-algebras, Schwartz cohomology of groups, and cyclic cohomology. J. Funct. Anal. 107(1), 1–33 (1992)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Kasparov, G.: Equivariant \(KK\)-theory and the Novikov conjecture. Invent. Math. 91, 147–201 (1998)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Karoubi, M.: Homologie cyclique et \(K\)-théorie, Astérisque No. 149, p. 147 (1987)Google Scholar
  29. 29.
    Keswani, N.: Geometric \(K\)-homology and controlled paths. N. Y. J. Math. 5, 53–81 (1999)MathSciNetMATHGoogle Scholar
  30. 30.
    Keswani, N.: Relative eta-invariants and \(C^{*}\)-algebra \(K\)-theory. Topology 39, 957–983 (2000)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Leichtnam, E., Piazza, P.: On higher eta-invariants and metrics of positive scalar curvature. K-Theory 24, 341–359 (2001)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Leichtnam, E., Piazza, P.: Dirac index classes and the noncommutative spectral flow. J. Funct. Anal. 200(2), 348–400 (2003)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Leichtnam, E., Piazza, P.: Spectral sections and higher Atiyah-Patodi-Singer index theory on Galois coverings. Geom. Funct. Anal. 8(1), 17–58 (1998)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Lesch, M., Moscovici, H., Pflaum, M.J.: Connes-Chern character for manifolds with boundary and \(\eta \)-cochains. Mem. Am. Math. Soc. 220, 1036 (2012)MathSciNetMATHGoogle Scholar
  35. 35.
    Lott, J.: Superconnections and higher index theory. Geom. Funct. Anal. 2(4), 421–454 (1992)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Lott, J.: Higher \(\eta \) invariants. K-Theory. 6(3), 191–233 (1992)Google Scholar
  37. 37.
    Moriyoshi, H.: Chern character for proper \(\Gamma \)-manifolds, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988), pp. 221–234, Proc. Sympos. Pure Math., 51, Part 2, Amer. Math. Soc., Providence, RI (1990)Google Scholar
  38. 38.
    Moriyoshi, H., Piazza, P.: Eta cocycles, relative pairings and the Godbillon-Vey index theorem. Geom. Funct. Anal. 22(6), 1708–1813 (2012)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Piazza, P., Schick, T.: Bordism, rho-invariants and the Baum-Connes conjecture. J. Noncommun. Geom. 1(1), 27–111 (2007)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Piazza, P., Schick, T.: Rho-classes, index theory and Stolz’ positive scalar curvature sequence. J. Topol. 7(4), 965–1004 (2014)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Piazza, P., Schick, T.: The surgery exact sequence, \(K\)-theory and the signature operator. Ann. K-Theory 1–2, 109–154 (2016)Google Scholar
  42. 42.
    Puschnigg, M.: New holomorphically closed subalgebras of \(C^{*}\)-algebras of hyperbolic groups. Geom. Funct. Anal. 20(1), 243–259 (2010)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Raven, J.: An equivariant bivariant Chern character. PhD Thesis, Pennsylvania State University. Available online at the Pennsylvania State Digital Library (2004)Google Scholar
  44. 44.
    Roe, J.: Index theory, coarse geometry, and topology of manifolds. CBMS Regional Conference Series in Mathematics 90, (1996)Google Scholar
  45. 45.
    Schick, T.: The topology of positive scalar curvature. In: Proceedings of the ICM 2014 (Seoul). arXiv:1405.4220
  46. 46.
    Stolz, S.: Positive scalar curvature metrics: Existence and classification questions. In: Proceedings of the International congress of mathematicians, ICM 94, vol. 1, pp. 625–636. Birkhäuser, Boston (1995)Google Scholar
  47. 47.
    Taylor, J.L.: Homology and cohomology for topological algebras. Adv. Math. 9, 137–182 (1972)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Wahl, C.: The Atiyah-Patodi-Singer index theorem for Dirac operator over \(C^{*}\)-algebras. Asian J. Math. 17(2), 265–319 (2013)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Wahl, C.: Higher \(\rho \)-invariants and the surgery structure set. J. Topol. 6(1), 154–192 (2013)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Xie, Z., Yu, G.: Positive scalar curvature, higher \(\rho \)-invariants and localization algebras. Adv. Math. 262, 823–866 (2014)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HawaiiHonoluluUSA
  2. 2.Insitut für Analysis, Leibniz Universität HannoverHannoverGermany

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