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Homotopy invariance of relative eta-invariants and $C^{*}$-algebra $K$-theory


Author: Navin Keswani
Journal: Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 18-26
MSC (1991): Primary 19K56
DOI: https://doi.org/10.1090/S1079-6762-98-00042-0
Published electronically: April 1, 1998
MathSciNet review: 1613055
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Abstract: We prove a close cousin of a theorem of Weinberger about the homotopy invariance of certain relative eta-invariants by placing the problem in operator $K$-theory. The main idea is to use a homotopy equivalence $h:M \to M'$ to construct a loop of invertible operators whose ``winding number" is related to eta-invariants. The Baum-Connes conjecture and a technique motivated by the Atiyah-Singer index theorem provides us with the invariance of this winding number under twistings by finite-dimensional unitary representations of $\pi _{1}(M)$.


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  • [APS1] M. Atiyah, V. K. Patodi, and I. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43-69. MR 53:1655a
  • [APS2] -, Spectral asymmetry and Riemannian geometry. II, Math. Proc. Camb. Phil. Soc. 78 (1975), 405-432. MR 53:1655b
  • [B] B. Blackadar, K-theory for operator algebras, MSRI Publications, vol. 5, Springer-Verlag, New York, 1986. MR 88g:46082
  • [BCH] P. Baum, A. Connes, and N. Higson, Classifying space for proper $G$-actions and K-theory of group $C^{*}$-algebras, Proceedings of a Special Session on $C^{*}$-algebras, Contemporary Mathematics, volume 167, American Mathematical Society, Providence, RI, 1994, pp. 241-291. MR 96c:46070
  • [BD] P. Baum and R. Douglas, $K$-homology and index theory, Proc. Symp. Pure Math. 38 (1982), 117-173. MR 84d:58075
  • [G] P. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, 2nd ed., CRC Press, 1995. MR 98b:58156
  • [HS] P. de la Harpe and G. Skandalis, Déterminant associé à une trace sur une algèbre de Banach, Ann. Inst. Fourier (Grenoble) 34 (1984), 241-260. MR 87i:46146a
  • [HK] N. Higson and G. Kasparov, Operator $K$-theory for groups which act properly and isometrically on Hilbert space, ERA Amer. Math. Soc. 3 (1997), 131-142. CMP 98:05
  • [HR] N. Higson and J. Roe, Mapping surgery to analysis, Preprint.
  • [K1] G. Kasparov, Topological invariants of elliptic operators. I: K-homology, Math. USSR Izv. 9 (1975), 751-792. MR 58:7603
  • [K2] -, Equivariant $KK$-theory and the Novikov conjecture, Inven. Math. 91 (1988), 147-201. MR 88j:58123
  • [KM] J. Kaminker and J. Miller, Homotopy invariance of the analytic index of signature operators over $C^{*}$-algebras, J. Operator Theory 14 (1985), 113-127. MR 87b:58082
  • [Kes] N. Keswani, Homotopy invariance of relative eta-invariants and $C^{*}$-algebra $K$-theory, Ph.D. Thesis, 1997, University of Maryland at College Park.
  • [M] V. Mathai, On the homotopy invariance of reduced eta and other signature type invariants, Preprint.
  • [N] W. Neumann, Signature related invariants of manifolds. I: monodromy and $\gamma $-invariants, Topology 18 (1979), 147-172. MR 80k:57048
  • [deR] G. de Rham, S. Maumary, and M. A. Kervaire, Torsion et type simple d'homotopie, Lecture Notes in Mathematics, vol. 48, Springer-Verlag, 1967. MR 36:5943
  • [R] J. Roe, Elliptic operators, topology, and asymptotic methods, Longman Scientific & Technical, Harlow, Essex, England, and Wiley, New York, 1988. MR 89j:58126
  • [Ros1] J. Rosenberg, Analytic Novikov for topologists, Novikov Conjectures, Index Theorems and Rigidity, vol.1 (S. Ferry, A. Ranicki, and J. Rosenberg, eds.), London Math. Soc. Lecture Notes, vol. 226, Cambrige Univ. Press, Cambridge, 1995, pp. 338-372. MR 97b:58138
  • [Ros2] -, Recent progress in algebraic $K$-theory and its relationship with topology and analysis, Lecture notes prepared for the Joint Summer Research Conference on Algebraic $K$-theory, Seattle, July 1997, $K$-theory preprint server.
  • [W] S. Weinberger, Homotopy invariance of eta invariants, Proc. Nat. Acad. Sci. 85 (1988), 5362-5365. MR 90a:58168

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Additional Information

Navin Keswani
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
Email: navin@math.psu.edu

DOI: https://doi.org/10.1090/S1079-6762-98-00042-0
Keywords: Eta-invariants, $K$-theory
Received by editor(s): January 28, 1998
Published electronically: April 1, 1998
Additional Notes: The author would like to thank Nigel Higson for his guidance with this project.
Communicated by: Masamichi Takesaki
Article copyright: © Copyright 1998 American Mathematical Society

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