Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T21:35:30.547Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher spectral flow and an entire bivariant JLO cocycle

Published online by Cambridge University Press:  13 November 2012

Moulay-Tahar Benameur  and
Alan L. Carey
Moulay-Tahar Benameur
Affiliation:
UMR 7122, LMAM, Université Paul Verlaine-Metz, [email protected]
Alan L. Carey
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, ACT. 0200, [email protected]
Get access

Abstract

For a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 1 , February 2013 , pp. 183 - 232
Copyright
Copyright © ISOPP 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Atiyah, M. F.K-theory, Lecture notes by D. W. Anderson W. A. Benjamin, Inc., New York-Amsterdam 1967.Google Scholar
2.Atiyah, M. F. and Singer, I. M.The index of elliptic operators. IV. Ann. of Math. 93 (2) (1971), 119–138.Google Scholar
3.Baum, P. and Douglas, R. G.K homology and index theory. Operator algebras and applications, Part I (Kingston, Ont., 1980), pp. 117–173, Proc. Sympos. Pure Math. 38, Amer. Math. Soc., Providence, R.I., 1982.Google Scholar
4.Benameur, M.-T. and Gorokhovsky, A., Local index theorem for projective families. (English), arXiv:1007.3667v1 [math.DG], Fields Institute Communications 61 (2011), 1–27.Google Scholar
5.Benameur, M.-T. and Heitsch, J. L., Index theory and non-commutative geometry. I. Higher families index theory. K-Theory 33 (2) (2004), 151–183.CrossRefGoogle Scholar
6.Benameur, M.-T. and Heitsch, J. L., The twisted higher harmonic signature for foliations. (English) J. Differ. Geom. 87 (3) (2011), 389–468.Google Scholar
7.Benameur, M.-T. and Piazza, P., Index, eta and rho invariants on foliated bundles, Astérisque 327 (2009), 199–284.Google Scholar
8.Berline, N.; Getzler, E. and Vergne, M., Heat kernels and Dirac operators. Corrected reprint of the 1992 original. Grundlehren Text Editions. Springer-Verlag, Berlin, 2004.Google Scholar
9.Bismut, J.-M., The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs. Invent. Math. 83 (1) (1985), 91–151.CrossRefGoogle Scholar
10.J., Block, J., Fox, Asymptotic pseudodifferential operators and index theory, Contemp. Math. 105 (1990), 1–45.Google Scholar
11.Bourbaki, N., Espaces vectoriels topologiques. Chapitres 1–5 (French) [Topological vector spaces. Chapters 1–5], Éléments de mathématique. [Elements of mathematics] New edition. Masson, Paris, 1981.Google Scholar
12.Carey, A. L. and Phillips, J., Spectral flow in Fredholm modules, eta invariants and the JLO cocycle. K-Theory 31 (2004), 135–194.Google Scholar
13.Connes, A., Entire cyclic cohomology of Banach algebras and characters of θ-summable Fredholm modules. K-Theory 1 (6) (1988), 519–548.CrossRefGoogle Scholar
14.Connes, A., Noncommutative geometry. Academic Press, Inc., San Diego, CA, 1994.Google Scholar
15.Dai, X. and Zhang, W., Higher spectral flow. J. Funct. Anal. 157 (2) (1998), 432–469.CrossRefGoogle Scholar
16.Getzler, E. and Szenesz, A., On the Chern character of a theta-summable Fredholm module. J. Funct. Anal. 84 (2) (1989), 343–357.Google Scholar
17.Getzler, E., The odd Chern character in cyclic homology and spectral flow. Topology 32 (3) (1993), 489–507.Google Scholar
18.Gorokhovsky, A., Bivariant Chern character and longitudinal index. J. of Funct. Analysis 237 (2006), 105–134.Google Scholar
19.Jaffe, A.; Lesniewski, A. and Osterwalder, K.Quantum K-theory. I. The Chern character. Comm. Math. Phys. 118 (1) (1988), 1–14.Google Scholar
20.Kasparov, G. G., Topological invariants of elliptic operators. I. K-homology. (Russian) Math. USSR-Izv. 9 (1975), no. 4, 751–792 (1976).; translated from Izv. Akad. Nauk SSSR Ser. Mat. 39 (4) (1975), 796–838(Russian).CrossRefGoogle Scholar
21.Lance, E. C., Hilbert C*-Modules, Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
22.Leichtnam, E. and Piazza, P., Dirac index classes and the noncommutative spectral flow. J. Funct. Anal. 200 (2003), 348–400.Google Scholar
23.Melrose, R. B. and Piazza, P., Families of Dirac operators, boundaries and the b-calculus. J. Differential Geom. 46 (1) (1997), 99–180.Google Scholar
24.Meyer, R., Local and analytic cyclic homology. EMS Tracts in Mathematics 3. European Mathematical Society (EMS), Zürich, 2007.Google Scholar
25.Nistor, V., A bivariant Chern character for p-summable quasihomomorphisms. K-Theory 5 (3) (1991), 193–211.Google Scholar
26.Perrot, D., A bivariant Chern character for families of spectral triples. Comm. Math. Phys. 231 (2002), no. 1, 45–95.Google Scholar
27.Quillen, D., Algebra cochains and cyclic cohomology. Inst. Hautes Études Sci. Publ. Math. 68 (1988), 139–174 (1989).Google Scholar
28.Reed, M. and Simon, B., Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.Google Scholar
29.Vassout, S.Unbounded pseudodifferential calculus on Lie groupoids. J. Funct. Anal. 236 (1) (2006), 161–200.Google Scholar
30.Wu, F., A bivariant Chern-Connes character and the higher Γ-index theorem. K-Theory 11 (1) (1997), 35–82.Google Scholar