Abstract
Let (N, g) be a closed Riemannianmanifold of dimension 2m − 1 and let Γ → Ñ → N be a Galois covering of N. We assumethat Γ is of polynomial growth with respect to a word metric and that ΔÑ is L 2-invertible in degree m. By employing spectral sections with asymmetry property with respect to the ⋆-Hodge operator, we define the higher eta invariant associatedwith the signature operator on Ñ, thus extending previous work of Lott. If π1(M)→ \({\tilde M}\) →M is the universal cover of a compact orientable even-dimensionalmanifold with boundary (∂M = N)then, under the above invertibility assumption on Δ∂ \({\tilde M}\), andalways employing symmetric spectral sections, we define acanonical Atiyah–Patodi–Singer index class, in K 0(C * r (Γ)), for the signature operator of\({\tilde M}\). Using the higherAPS index theory developed in [6], we express the Chern character ofthis index class in terms of a local integral and of the higher etainvariant defined above, thus establishing a higher APS index theoremfor the signature operator on Galois coverings. We expect the notion ofa symmetric spectral section for the signature operator to have widerimplications in higher index theory for signatures operators.
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Atiyah, M. F., Patodi, V. and Singer, I.: Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69; 78 (1975), 405–432.
Atiyah, M. F. and Singer, I.: The index of elliptic operators IV, Ann. Math 91 (1971), 119–138.
Bismut, J.-M. and Cheeger, J.: Family index for manifolds with boundary, superconnections and cones I, II, J. Funct. Anal. 89 (1990), 313; 90 (1990), 306–354.
Connes, A. and Moscovici, H.: Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29(3) (1990), 345–388.
Leichtnam, E. and Piazza, P.: The b-pseudodifferential calculus on Galois coverings and a higher Atiyah–Patodi–Singer index theorem, Mé m. S.M.F. 68 (1997).
Leichtnam, E. and Piazza, P.: Spectral sections and higher Atiyah–Patodi–Singer index theory on Galois coverings, GAFA 8 (1998), 17–58.
Leichtnam, E. and Piazza, P.: Higher eta invariants and the Novikov conjecture on manifolds with boundary, C.R. Acad. Sci. Sér. I 327 (1998), 497–502.
Leichtnam, E. and Piazza, P.: Homotopy invariance of twisted higher signatures on manifolds with boundary, Bull. Soc. Math. France 127 (1999), 307–331.
Lott, J.: Superconnections and higher index theory, GAFA 2 (1992), 421–454.
Lott, J., Higher eta invariants, K-Theory 6 (1992), 191–233.
Lott, J.: The zero-in-the-spectrum question, Enseign. Math. 42 (1996), 341–376.
Lott, J.: Signatures and higher signatures on S1-quotients, Preprint, 1998 (http://www.math.lsa.umich.edu/Qlott).
Lusztig, G.: Novikov's higher signature and families of elliptic operators, J. Differential Geom. 7 (1971), 229–256.
Melrose, R.: The Atiyah–Patodi–Singer Index Theorem, A. and K. Peters, 1993.
Melrose, R. and Piazza, P.: Families of Dirac operators, boundaries and the b-calculus, J. Differential Geom. 46(1) (1997), 99–180.
Melrose, R. and Piazza, P.: An index theorem for families of Dirac operators on odd dimensional manifolds with boundary, J. Differential Geom. 46(2) (1997), 287–334.
Mishenko and Fomenko: The index of elliptic operators over C_-algebras, Izv. Akad. Nauk SSR, Ser. Mat. 43 (1979), 831–859.
Piazza, P.: Dirac operators, heat kernels and microlocal analysis. Part 1, Rend. Circolo Mat. Palermo 49 (1997), 187–201.
Roe, J.: Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (1993).
Weinberger, S.: Higher _-invariants, Contemp. Math. Proceedings of the Tel-Aviv Topology Conference, 1997, to appear.
Wu, F.: The noncommutative spectral flow. Preprint, 1997 (http://www.ksu.edu/Qfangbing).
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Leichtnam, E., Piazza, P. A Higher Atiyah–Patodi–Singer Index Theorem for the Signature Operator on Galois Coverings. Annals of Global Analysis and Geometry 18, 171–189 (2000). https://doi.org/10.1023/A:1006649505610
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DOI: https://doi.org/10.1023/A:1006649505610