Abstract
We apply the geometric analog of the analytic surgery group of Higson and Roe to the relative \(\eta \)-invariant. In particular, by solving a Baum–Douglas type index problem, we give a “geometric” proof of a result of Keswani regarding the homotopy invariance of relative \(\eta \)-invariants. The starting point for this work is our previous constructions in “Realizing the analytic surgery group of Higson and Roe geometrically, Part I: The geometric model”.
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Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry I. Math. Proc. Camb. Philos. Soc. 77, 43–69 (1975)
Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry II. Math. Proc. Camb. Philos. Soc. 78, 405–432 (1975)
Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry III. Math. Proc. Camb. Philos. Soc. 79, 71–99 (1976)
Antonini, P., Azzali, S., Skandalis, G.: Flat bundles, von Neumann algebras and \(K\)-theory with \(R/Z\)-coefficients. J. \(K\)-Theory 13(2), 275–303 (2014)
Basu, D.: \(K\)-theory with R/Z coefficients and von Neumann algebras. \(K\)-theory 36, 327–343 (2005)
Baum, P., Douglas, R.: \(K\)-homology and index theory. In: Kadison, R (ed.) Proceedings of Symposia in Pure Math Operator Algebras and Applications of AMS, vol. 38, pp. 117–173, Providence RI (1982)
Baum, P., Douglas, R.: Index theory, bordism, and \(K\)-homology. Contemp. Math. 10, 1–31 (1982)
Baum, P., van Erp, E.: \(K\)-homology and index theory on contact manifolds. Acta Math. 213(1), 1–48 (2014)
Baum, P., Higson, N., Schick, T.: On the equivalence of geometric and analytic \(K\)-homology. Pure Appl. Math. Q. 3, 1–24 (2007)
Benameur, M.-T., Piazza, P.: Index, \(\eta \) and \(\rho \) invariants on foliated bundles. Astérisque 327, 199–284 (2009)
Benameur, M.-T., Roy, I.: Leafwise homotopies and Hilbert-Poincare complexes I. J. Noncommut. Geom. 8(3), 789–836 (2014)
Benameur, M.-T., Roy, I.: The Higson–Roe exact sequence and \(l^2\)-\(\eta \) invariants. J. Funct. Anal. 268(4), 974–1031 (2015)
Blackadar, B.: Operator algebras. Theory of \(C^{*}\)-algebras and von Neumann algebras. Encyclopaedia of Mathematical Sciences, 122. Operator Algebras and Non-commutative Geometry, III. Springer, Berlin, pp. xx+517 (2006)
Booß-Bavnbek, B., Wojciechowski, K.P.: Elliptic boundary problems for Dirac operators. In: Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, pp. xviii+307 (1993)
Deeley, R.J.: \(\mathbb{R}/\mathbb{Z}\)-valued index theory via geometric \(K\)-homology. Münster J. of Math. 7, 697–729 (2014)
Deeley, R.J.: Analytic and topological index maps with values in the \(K\)-theory of mapping cones. arXiv:1302.4296
Deeley, R., Goffeng, M.: Realizing the analytic surgery group of Higson and Roe geometrically, Part I: The geometric model. To appear in J. Homotopy Relat. Struct. arXiv:1308.5990
Fomenko, A.T., Miscenko, A.S.: The index of elliptic operators over \(C^{*}\)-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 43(4), 831–859, 967 (1979)
Guentner, E.: \(K\)-homology and the index theorem, Index theory and operator algebras (Boulder, CO, 1991), pp. 47–66. Contemp. Math., 148, Amer. Math. Soc., Providence, RI (1993)
Higson, N., Roe, J.: Mapping surgery to analysis I: analytic signatures. \(K\)-theory 33, 277–299 (2005)
Higson, N., Roe, J.: Mapping surgery to analysis II: Geometric signatures. \(K\)-theory 33, 301–325 (2005)
Higson, N, Roe, J.: Mapping surgery to analysis III: Exact sequences. \(K\)-theory 33, 325–346 (2005)
Higson, N., Roe, J.: \(K\)-homology, assembly and rigidity theorems for relative \(\eta \)-invariants. Pure Appl. Math. Q. (Special Issue: In honor of Michael Atiyah and Isadore Singer) 6(2), 555–601 (2010)
Hilsum, M.: Bordism invariance in \(KK\)-theory. Math. Scand. 107(1), 73–89 (2010)
Karoubi, M.: \(K\)-theory: an introduction. Springer, Berlin Heidelberg New York (1978)
Keswani, N.: Geometric \(K\)-homology and controlled paths. N. Y. J. Math. 5, 53–81 (1999)
Keswani, N.: Relative \(\eta \)-invariants and \(C^{*}\)-algebra \(K\)-theory. Topology 39, 957–983 (2000)
Keswani, N.: Von Neumann \(\eta \)-theory. J. Lond. Math. Soc. (2) 62(3), 771–783 (2000)
Leichtnam, E., Piazza, P.: Dirac index classes and the noncommutative spectral flow. J. Funct. Anal. 200(2), 348–400 (2003)
Leichtnam, E., Piazza, P.: Spectral sections and higher Atiyah–Patodi–Singer index theory on Galois coverings. Geom. Funct. Anal. 8(1), 17–58 (1998)
Mathai, V.: On the homotopy invariance of reduced \(\eta \) and other signature type invariants, preprint
Melrose, R.B., Piazza, P.: Families of Dirac operators, boundaries and the b-calculus. J. Differ. Geom. 46(1), 99–180 (1997)
Neumann, W.: Signature related invariants of manifolds I: monodromy and \(\gamma \)-invariants. Topology 18, 147–172 (1979)
Piazza, P., Schick, T.: Bordism, rho-invariants and the Baum–Connes conjecture. J. Noncommut. Geom. 1(1), 27–111 (2007)
Ramachandran, M.: Von Neumann index theorems for manifolds with boundary. J. Differ. Geom. 38, 315–349 (1993)
Raven, J.: An equivariant bivariant chern character, PhD Thesis, Pennsylvania State University, 2004. (available online at the Pennsylvania State Digital Library)
Schick, T.: \(L^2\)-index theorems, \(KK\)-theory, and connections. N. Y. J. Math. 11, 387–443 (2005)
Wahl, C.: Higher rho-invariants and the surgery structure set. J. Topol. 6(1), 154–192 (2013)
Weinberger, S.: Homotopy invariance of \(\eta \)-invariants. Proc. Nat. Acad. Sci. 85, 5362–5365 (1988)
Wu, F.: The noncommutative spectral flow, unpublished preprint (1997)
Acknowledgments
The authors wish to express their gratitude towards Heath Emerson, Nigel Higson, and Thomas Schick for discussions. They also thank the Courant Centre of Göttingen, the Leibniz Universität Hannover and the Graduiertenkolleg 1463 (Analysis, Geometry and String Theory) for facilitating this collaboration. The authors also thank the referee for a number of useful suggestions.
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Deeley, R.J., Goffeng, M. Realizing the analytic surgery group of Higson and Roe geometrically part II: relative \(\eta \)-invariants. Math. Ann. 366, 1319–1363 (2016). https://doi.org/10.1007/s00208-016-1364-7
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DOI: https://doi.org/10.1007/s00208-016-1364-7