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Flat bundles, von Neumann algebras and K-theory with ℝ/ℤ-coefficients

Published online by Cambridge University Press:  24 February 2014

Paolo Antonini ,
Sara Azzali  and
Georges Skandalis
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Abstract

Let M be a closed manifold and α: π1 (M) → Un a representation. We give a purely K-theoretic description of the associated element in the K-theory group of M with ℝ/ℤ-coefficients ([α] ∈ K1 (M; ℝ/ℤ)). To that end, it is convenient to describe the ℝ/ℤ-K-theory as a relative K-theory of the unital inclusion of ℂ into a finite von Neumann algebra B. We use the following fact: there is, associated with α, a finite von Neumann algebra B together with a flat bundle ℰ → M with fibers B, such that Eα ⊗ ℰ is canonically isomorphic with ℂn ⊗ ℰ, where Eα denotes the flat bundle with fiber ℂn associated with α. We also discuss the spectral flow and rho type description of the pairing of the class [α] with the K-homology class of an elliptic selfadjoint (pseudo)-differential operator D of order 1.

Keywords

II1-factorsR/Z-K-theoryflat bundlesrelative eta invariantPrimary: 46L80Secondary: 58J2819K56
Type
Research Article
Information
Journal of K-Theory , Volume 13 , Issue 2 , April 2014 , pp. 275 - 303
Copyright
Copyright © ISOPP 2014 

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References

REFERENCES

An. Antonini, P., The Atiyah Patodi Singer signature formula for measured foliations, Ph.D Thesis, La Sapienza, Romehttp://arxiv.org/abs/0901.0143.Google Scholar
Ati. Atiyah, M.F., Elliptic operators discrete groups and von Neumann algebras. Astérisque 32–33 (1976), 43–72.Google Scholar
APS1. Atiyah, M.F., Patodi, V.K., Singer, I.M.. Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–49.CrossRefGoogle Scholar
APS2. Atiyah, M.F., Patodi, V.K., Singer, I.M., Spectral asymmetry and Riemannian geometry. II. Math. Proc. Cambridge Philos. Soc. 78 (1975), 405–432.CrossRefGoogle Scholar
APS3. Atiyah, M.F., Patodi, V.K., Singer, I.M., Spectral asymmetry and Riemannian geometry. III. Math. Proc. Cambridge Philos. Soc. 79 (1976), 71–99.CrossRefGoogle Scholar
Ba. Basu, D., K-theory with ℝ/ℤ coefficients and von Neumann algebras, K-Theory 36(3–4) (2005), 327–343.Google Scholar
BGV. Berline, N., Getzler, E., Vergne, M., Heat kernels and Dirac operators, Springer-Verlag, New York 1992.Google Scholar
BF. Bismut, J.M. and Freed, D.S., The analysis of elliptic families II: Dirac operators, eta invariants, and the holonomy theorem of Witten, Commun. Math. Phys. 107 (1986), 103–163.Google Scholar
B1. Blackadar, B., K-theory for operator algebras, second edition, MSRI publications, Cambridge University Press.Google Scholar
B11. Blackadar, B., Operator algebras: theory of C*-Algebras and non Neumann algebras, Encyclopaedia on operator algebras and non-commutative geometry 13, Springer 2006.Google Scholar
Bo. Bohn, M., On Rho invariants of fibre bundles, http://arxiv.org/abs/0907.3530 [math.GT].Google Scholar
CP1. Carey, A., Phillips, J., Spectral flow in Fredholm modules, eta invariants and the JLO cocycle, K-Theory 31(2) (2004), 135–194.CrossRefGoogle Scholar
CG. Cheeger, J., Gromov, M., On the characteristic numbers of complete manifolds of bounded curvature and finite volume, in Differential geometry and complex analysis, Springer, Berlin 1985, 115–154.Google Scholar
CM. Connes, A., Moscovici, H., Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29(3) (1990), 345–388.CrossRefGoogle Scholar
CS. Connes, A., Skandalis, G., The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20(6) (1984).Google Scholar
De. Deeley, R.J., ℝ/ℤ-valued index theory via geometric K-homology, http://arxiv.org/abs/1206.5662 [math.KT].Google Scholar
DHK. Douglas, R.G., Hurder, S., Kaminker, J., Eta invariants and von Neumann algebras, Bull. Amer. Math. Soc 21(1) (1989).Google Scholar
DHK2. Douglas, R.G., Hurder, S., Kaminker, J., Cyclic cocycles, renormalization and etainvariants, Inv. Math. 103(1) (1991), 101–180.Google Scholar
G. Getzler, E., The odd Chern character in cyclic homology and spectral flow, Topology 32(3) (1993), 489–507.Google Scholar
Gi2. Gilkey, P. B., The residue of the global eta function at the origin, Adv. Math. 40 (1981), 290–307.CrossRefGoogle Scholar
Gi. Gilkey, P. B., The eta invariant and secondary characteristic classes of locally flat bundles, in Algebraic and differential topology – global differential geometry, Teubner–Texte Math. 70, Leipzig (1984), 49–87.Google Scholar
Gr. Gromov, M., Kähler hyperbolicity and L 2-Hodge theory, J. Diff. Geom. 33 (1991), 263–292.Google Scholar
Hu. Hurder, S., Eta invariants and the odd index theorem for coverings. Geometric and topological invariants of elliptic operators, Contemp. Math. 105, Amer. Math. Soc., Providence, RI, (1990), 47–82.Google Scholar
Ka. Karoubi, M., Homologie Cyclique et K-Théorie, Astérisque 149 (1987).Google Scholar
Ka2. Karoubi, M., K-theory. An introduction. Grundlehren der Mathematischen Wissenschaften 226. Springer-Verlag, Berlin-New York, 1978.Google Scholar
Ka3. Karoubi, M., Théorie générale des classes caractéristiques secondaires. K-Theory 4 (1990), 55–87.Google Scholar
Ka4. Karoubi, M., Classes caractéristiques de fibrés feuilletés, holomorphes ou algébriques. K-Theory 8 (1994), 153–211.Google Scholar
KP. Kaminker, J., Perera, V., Type II Spectral Flow and the Eta Invariant, Canad. Math. Bull. 43(1) (2000), 69–73.CrossRefGoogle Scholar
Ku. Kucerovsky, D., The KK-product of unbounded modules, K-Theory 11(1) (1997), 17–34.Google Scholar
LP. Leichtnam, E., Piazza, P., Cut-and-Paste on foliated bundles, Contemp. Mathem. 366 (2005), 351–392 “Spectral Geometry of Manifolds with Boundary”, Amer. Math. Soc., Providence, RI.Google Scholar
Lo. Lott, J., ℝ/ℤ-index theory, Comm. Anal. Geom. 2(2) (1994), 279–311.Google Scholar
Lü. Luck, W., The relation between the Baum–Connes Conjecture and the Trace Conjecture, Invent. Math. 149 (2002), 123–152Google Scholar
MZ1. Ma, X., Zhang, W., η-Invariant and flat vector bundles, Chin. Ann. Math 27B(1) (2006), 57–72.Google Scholar
MZ2. Ma, X., Zhang, W., η-Invariant and flat vector bundles II, in “Inspired by S. S. Chern. A memorial volume in honor of a great mathematician.” Hackensack, NJ: World Scientific. Nankai Tracts in Mathematics 11 (2006), 335–350.Google Scholar
Mi. Miščenko, A. S., The theory of elliptic operators over C*-algebras, Dokl. Akad. Nauk SSSR 239(6) (1978), 1289–1291.Google Scholar
MF. Miščenko, A. S., Fomenko, A. T., The index of elliptic operators over C*-algebras. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 43(4) (1979), 831–859, 967.Google Scholar
MN. Moriyoshi, I., Natsume, T., The Godbillon-Vey cyclic cocycle and longitudinal Dirac operators, Pacific J. of Math. 172(2) (1996), 483–539.CrossRefGoogle Scholar
PS. Piazza, P., Schick, T., Bordism, rho-invariants and the Baum-Connes conjecture. J. of Noncommutative Geometry 1 (2007), 27–111.Google Scholar
Pu. Putnam, I. F., An excision theorem for the K–theory of C* algebras, J. Operator Theory 38 (1997), 151–171.Google Scholar
Ra. Raeburn, I., K-theory and K-homology relative to a II∞-factor. Proc. Amer. Math. Soc. 71(2) (1978), 294–298.Google Scholar
Ram. Ramachandran, M., Von Neumann index theorems for manifolds with boundary, J. Diff. Geom. 38 (1993), 315–349.Google Scholar
Sc. Schick, T., L 2-index theorems, KK-theory, and connections, New York J. Math. 11 (2005), 387–443.Google Scholar
Si. Singer, I.M., Some remarks on operator theory and index theory, in “K-Theory and Operator Algebras”, Lect. Notes Math. 575 (1977), 128–138.Google Scholar
Va. Vaillant, B., Index theory for coverings, http://arxiv.org/abs/0806.4043.Google Scholar
Wo. Wojciechowski, K., A note on the space of pseudodifferential projections with the same principal symbol, J. Operator Theory 15(2) (1986), 207–216.Google Scholar