Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T15:08:47.015Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-cocycle twists and Atiyah–Patodi–Singer index theory

Published online by Cambridge University Press:  22 August 2018

SARA AZZALI  and
CHARLOTTE WAHL
SARA AZZALI
Affiliation:
Institut für Mathematik, Universität Potsdam, Karl–Liebknecht Str. 24-25, 14476 Golm, Germany. e-mail: [email protected]
CHARLOTTE WAHL
Affiliation:
Leibniz–Archiv, Waterloostr. 8, 30169 Hannover, Germany. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct η- and ρ-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah–Patodi–Singer index theorem in this setting, as well as its higher generalisation. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 3 , November 2019 , pp. 437 - 487
Copyright
Copyright © Cambridge Philosophical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Atiyah, M. F., Donnelly, H. and Singer, I. M. Signature defects of cusps, and values of L-functions. Ann. Math. 118 (1983), 131–177.Google Scholar
[2] Azzali, S. and Wahl, C. Spectral flow, index and the signature operator. J. Top. Anal. 3 (2011), 37–67.Google Scholar
[3] Baum, P. and Karoubi, M. On the Baum–Connes conjecture in the real case. Q. J. Math 55 (2004), 231–235.Google Scholar
[4] Benameur, M.–T. and Mathai, V.. Index type invariants for twisted signature complexes and homotopy invariance. Math. Proc. Camb. Phil. Soc 156 (2014), 473–503.Google Scholar
[5] Benameur, M.–T. and Mathai, V.. Conformal invariants of twisted Dirac operators and positive scalar curvature. J. Geom. Phys. 70 (2013), 39–47; corrigendum ibid. 76 (2014), 263–264.Google Scholar
[6] Benameur, M.–T. and Mathai, V.. Spectral sections, twisted rho invariants and positive scalar curvature. J. Noncommut. Geom. 9 (2015), 821–850.Google Scholar
[7] Botvinnik, B. and Gilkey, P. B. The eta invariant and metrics of positive scalar curvature. Math. Ann. 302 (1995), 507–517.Google Scholar
[8] Bunke, U. A K-theoretic relative index theorem and Callias-type Dirac operators. Math. Ann. 303 (1995), 241–279.Google Scholar
[9] Carey, A. L. and Phillips, J. Spectral flow in Fredholm modules, eta invariants and the JLO cocycle. K-Theory 31 (2004), 135–194.Google Scholar
[10] Carey, A. L., Phillips, J., Rennie, A. and Sukochev, F. A.. The local index formula in semifinite von Neumann algebras II: the even case. Adv. Math. 202 (2006), 517–554.Google Scholar
[11] Cheeger, J. and Gromov, M. On the characteristic numbers of complete manifolds of bounded curvature and finite volume. Differential geometry and complex analysis (Springer Verlag 1985), 115–154.Google Scholar
[12] Cheeger, J. and Gromov, M. Bounds on the von Neumann dimension of L 2-cohomology and the Gauss–Bonnet theorem for open manifolds. J. Diff. Geom. 21 (1985), 1–34.Google Scholar
[13] Connes, A. and Moscovici, H. Cyclic cohomology, the Novikov conjecture and hyperbolic groups. Topology 29 (1990), 345–388.Google Scholar
[14] Echterhoff, S., Lück, W., Phillips, N.C. and Walters, S.. The structure of crossed products of irrational rotation algebras by finite subgroups of SL2(${\mathbb Z}$). J. Reine Angew. Math. 639 (2010), 173–221.Google Scholar
[15] Gajer, P. Riemannian metrics of positive scalar curvature on compact manifolds with boundary. Ann. Global Anal. Geom. 5 (1987), 179–191.Google Scholar
[16] Gromov, M. Kähler hyperbolicity and L 2-Hodge theory. J. Diff. Geom. 33 (1991), 263–292.Google Scholar
[17] Gromov, M. Positive curvature, macroscopic dimension, spectral gaps and higher signatures. Functional analysis on the eve of the 21st century 2 (Progr. Math. 132) (Birkhäuser 1996), 1–213.Google Scholar
[18] Gromov, M. and Lawson, H.B. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math. IHES 58 (1983), 83–196.Google Scholar
[19] Gruber, M. J. Noncommutative Bloch theory. J. Math. Phys. 42 (2001), 2438–2465.Google Scholar
[20] Hanke, B. and Schick, T. The strong Novikov conjecture for low degree cohomology. Geom. Dedicata 135 (2008), pp. 119–127.Google Scholar
[21] Hurder, S. Eta invariants and the odd index theorem for coverings. Contemp. Math. 105 (1990), 47–82.Google Scholar
[22] Jolissaint, P. K-theory of reduced C*-algebras and rapidly decreasing functions on groups. K-theory 2 (1989), 723–735.Google Scholar
[23] Jolissaint, P. Rapidly decreasing functions in reduced C*-algebras of groups. Trans. Am. Math. Soc. 317 (1990), 167–196.Google Scholar
[24] Keswani, N. Von Neumann eta-invariants and C*-algebra K-theory. J. Lond. Math. Soc. 62 (2000), 771–783.Google Scholar
[25] Köthe, G. Topological vector spaces 2 Grundlagen math. Wiss. 237 (Springer-Verlag 1979).Google Scholar
[26] Leichtnam, E., Lück, W. and Kreck, M. On the cut-and-paste property of higher signatures of closed oriented manifolds. Topology 41 (2002), 725–744.Google Scholar
[27] Leichtnam, E. and Piazza, P. The b-pseudodifferential calculus on Galois coverings and a higher Atiyah–Patodi–Singer index theorem. Mém. Soc. Math. Fr. 68 (1997).Google Scholar
[28] Leichnam, E. and Piazza, P. On higher eta-invariants and metrics of positive scalar curvature. K-Theory 24 (2001), pp. 341–359.Google Scholar
[29] Leichtnam, E. and Piazza, P. Dirac index classes and the noncommutative spectral flow. J. Funct. Anal. 200 (2003), 348–400.Google Scholar
[30] Leichtnam, E. and Piazza, P. Homotopy invariance of twisted higher signatures on manifolds with boundary. Bull. Soc. Math. France 127 (1999), 307–331.Google Scholar
[31] Lott, J. Higher eta-invariants. K-Theory 6 (1992), 191–233.Google Scholar
[32] Lott, J. Superconnections and higher index theory. Geom. Funct. Anal. 2 (1992), 421–454.Google Scholar
[33] Lott, J. Diffeomorphisms and noncommutative analytic torsion. Mem. Amer. Math. Soc. 673 (1999).Google Scholar
[34] Marcolli, M. Solvmanifolds and noncommutative tori with real multiplication. Comm. Number Theory Phys. 2 (2008), 421–476.Google Scholar
[35] Marcolli, M. and Mathai, V. Twisted index theory on good orbifolds I. Noncommutative Bloch theory, Commun. Contemp. Math. 1 (1999), pp. 553–587.Google Scholar
[36] Marcolli, M. and Mathai, V. Twisted index theory on good orbifolds II. Fractional quantum numbers. Comm. Math. Phys. 217 (2001), 55–87.Google Scholar
[37] Mathai, V. K-Theory of twisted group C*-algebras and positive scalar curvature. Contemp. Math. 231 (1999), 203–225.Google Scholar
[38] Mathai, V. The Novikov conjecture for low degree cohomology classes. Geom. Dedicata 99 (2003), 1–15.Google Scholar
[39] Mathai, V. Heat kernels and the range of the trace on completions of twisted group algebras. Contemp. Math. 398 (2006), 321–346.Google Scholar
[40] Milnor, J. Lectures on the h -Cobordism Theorem (Princeton University Press 1965).Google Scholar
[41] Miyazaki, T. On the existence of positive scalar curvature metrics on non-simply-connected manifolds. J. Fac. Sci., Univ. Tokyo, Sect. I A 30 (1984), 549–561.Google Scholar
[42] Piazza, P. and Schick, T. Bordism, rho-invariants and the Baum–Connes conjecture. J. Noncommut. Geom. 1 (2007), 27–111.Google Scholar
[43] Piazza, P. and Schick, T. Groups with torsion, bordism and rho-invariants. Pacitic. J. Math. 232 (2007), 355–378.Google Scholar
[44] Ramachandran, M. Von Neumann index theorems for manifolds with boundary. J. Diff. Geom. 38 (1993), 315–349.Google Scholar
[45] Rosenberg, J. and Stolz, S. A “stable'' version of the Gromov–Lawson conjecture. Contemp. Math. 181 (1995), pp. 405–418.Google Scholar
[46] Rosenberg, J. and Stolz, S. Metrics of positive scalar curvature and connections with surgery. Surveys on surgery theory 2 Ann. Math. Stud. 149 (Princeton University Press 2001), pp. 353–386.Google Scholar
[47] Sauer, R. and Schick, T. Homotopy invariance of almost flat Betti numbers. Enseign. Math. (2) 54 (Guido's book of conjectures) (2008), 158–160.Google Scholar
[48] Stolz, S. Simply connected manifolds of positive scalar curvature. Ann. Math. 136 (year), 511–540.Google Scholar
[49] Trèves, F. Topological Vector Spaces, Distributions and Kernels (Dever Publications 2006).Google Scholar
[50] Wahl, C. Spectral flow and winding number in von Neumann algebras. J. Inst. Math. Jussieu 7 (2008), 589–619.Google Scholar
[51] Wahl, C. The Atiyah–Patodi–Singer index theorem for Dirac operators over C*-algebras. Asian J. Math. 17 (2013), 265–320.Google Scholar
[52] Walsh, M. Metrics of positive scalar curvature and generalised Morse functions I. Mem. Amer. Math. Soc. 983 (2011).Google Scholar
[53] Wu, F. A bivariant Chern–Connes character and the higher Γ-index theorem. K-theory 11 (1997), 35–82.Google Scholar