Homotopy invariance of relative eta-invariants and 𝐶*-algebra 𝐾-theory

Keswani, Navin

doi:10.1090/S1079-6762-98-00042-0

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $\pi _{1}(M)$.

M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 397797, DOI https://doi.org/10.1017/S0305004100049410
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 397797, DOI https://doi.org/10.1017/S0305004100049410
Bruce Blackadar, $K$-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867
Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions and $K$-theory of group $C^\ast $-algebras, $C^\ast $-algebras: 1943–1993 (San Antonio, TX, 1993) Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291. MR 1292018, DOI https://doi.org/10.1090/conm/167/1292018
Paul Baum and Ronald G. Douglas, $K$ homology and index theory, Operator algebras and applications, Part I (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 117–173. MR 679698
Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1396308
P. de la Harpe and G. Skandalis, Déterminant associé à une trace sur une algébre de Banach, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 1, 241–260 (French, with English summary). MR 743629

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.

N. Higson and J. Roe, Mapping surgery to analysis, Preprint.

G. G. Kasparov, Topological invariants of elliptic operators. I. $K$-homology, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 4, 796–838 (Russian); Russian transl., Math. USSR-Izv. 9 (1975), no. 4, 751–792 (1976). MR 0488027

G. G. Kasparov, Equivariant $KK$-theory and the Novikov conjecture, Invent. Math. 91 (1988), no. 1, 147–201. MR 918241, DOI https://doi.org/10.1007/BF01404917

Jerome Kaminker and John G. Miller, Homotopy invariance of the analytic index of signature operators over $C^\ast $-algebras, J. Operator Theory 14 (1985), no. 1, 113–127. MR 789380

N. Keswani, Homotopy invariance of relative eta-invariants and $C^{*}$-algebra $K$-theory, Ph.D. Thesis, 1997, University of Maryland at College Park.

V. Mathai, On the homotopy invariance of reduced eta and other signature type invariants, Preprint.

Walter D. Neumann, Signature related invariants of manifolds. I. Monodromy and $\gamma $-invariants, Topology 18 (1979), no. 2, 147–172. MR 544156, DOI https://doi.org/10.1016/0040-9383%2879%2990033-8

G. de Rham, S. Maumary, and M. A. Kervaire, Torsion et type simple d’homotopie, Lecture Notes in Mathematics, No. 48, Springer-Verlag, Berlin-New York, 1967 (French). Exposés faits au séminaire de Topologie de l’Université de Lausanne. MR 0222893

John Roe, Elliptic operators, topology and asymptotic methods, Pitman Research Notes in Mathematics Series, vol. 179, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. MR 960889

Jonathan Rosenberg, Analytic Novikov for topologists, Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993) London Math. Soc. Lecture Note Ser., vol. 226, Cambridge Univ. Press, Cambridge, 1995, pp. 338–372. MR 1388305, DOI https://doi.org/10.1017/CBO9780511662676.013

---, 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.

Shmuel Weinberger, Homotopy invariance of $\eta $-invariants, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), no. 15, 5362–5363. MR 952817, DOI https://doi.org/10.1073/pnas.85.15.5362