Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-09T18:36:27.602Z Has data issue: false hasContentIssue false
  • English
  • Français

An index theorem for end-periodic operators

Part of: Partial differential equations on manifolds; differential operators Parabolic equations and systems

Published online by Cambridge University Press:  07 September 2015

Tomasz Mrowka ,
Daniel Ruberman  and
Nikolai Saveliev
Tomasz Mrowka
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA email mrowka@mit.edu
Daniel Ruberman
Affiliation:
Department of Mathematics, MS 050, Brandeis University, Waltham, MA 02454, USA email ruberman@brandeis.edu
Nikolai Saveliev
Affiliation:
Department of Mathematics, University of Miami, PO Box 249085, Coral Gables, FL 33124, USA email saveliev@math.miami.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We extend the Atiyah, Patodi, and Singer index theorem for first-order differential operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. This theorem provides a natural complement to Taubes’ Fredholm theory for general end-periodic operators. Our index theorem is expressed in terms of a new periodic eta-invariant that equals the Atiyah–Patodi–Singer eta-invariant in the cylindrical setting. We apply this periodic eta-invariant to the study of moduli spaces of Riemannian metrics of positive scalar curvature.

Keywords

index theoremFredholm operatorperiodic endpositive scalar curvature

MSC classification

Primary: 58J20: Index theory and related fixed point theorems 58J28: Eta-invariants, Chern-Simons invariants 58J35: Heat and other parabolic equation methods 35K05: Heat equation

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 2 , February 2016 , pp. 399 - 444
Copyright
© The Authors 2015 

References

Atiyah, M., Elliptic operators, discrete groups and von Neumann algebras, in Colloque ‘Analyse et Topologie’ en l’Honneur de Henri Cartan (Orsay, 1974), Asterisque, vol. 32–33 (Société Mathématique de France, Paris, 1976), 43–72.Google Scholar
Atiyah, M., Bott, R. and Patodi, V. K., On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330.CrossRefGoogle Scholar
Atiyah, M., Patodi, V. and Singer, I., Spectral asymmetry and Riemannian geometry: I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69.CrossRefGoogle Scholar
Atiyah, M., Patodi, V. and Singer, I., Spectral asymmetry and Riemannian geometry: II, Math. Proc. Cambridge Philos. Soc. 78 (1975), 405–432.CrossRefGoogle Scholar
Atiyah, M., Patodi, V. and Singer, I., Spectral asymmetry and Riemannian geometry: III, Math. Proc. Cambridge Philos. Soc. 79 (1976), 71–99.CrossRefGoogle Scholar
Atiyah, M. and Singer, I., The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604.CrossRefGoogle Scholar
Berline, N., Getzler, E. and Vergne, M., Heat kernels and Dirac operators (Springer, Berlin, 1992).CrossRefGoogle Scholar
Booß-Bavnbek, B. and Wojciechowski, K., Elliptic boundary problems for Dirac operators (Birkhäuser, Boston, MA, 1993).CrossRefGoogle Scholar
Botvinnik, B. and Gilkey, P., The eta invariant and metrics of positive scalar curvature, Math. Ann. 302 (1995), 507–517.CrossRefGoogle Scholar
Bunke, U., Relative index theory, J. Funct. Anal. 105 (1992), 63–76.CrossRefGoogle Scholar
Cappell, S. and Shaneson, J., There exist inequivalent knots with the same complement, Ann. of Math. (2) 103 (1976), 349–353.CrossRefGoogle Scholar
Cappell, S. and Shaneson, J., Some new four-manifolds, Ann. of Math. (2) 104 (1976), 61–72.CrossRefGoogle Scholar
Cheng, S. Y., Li, P. and Yau, S. T., On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math. 103 (1981), 1021–1063.CrossRefGoogle Scholar
Coulhon, T. and Duong, X.-T., Riesz transforms for 1⩽p⩽2, Trans. Amer. Math. Soc. 351 (1999), 1151–1169.CrossRefGoogle Scholar
Davies, E. B., Pointwise bounds on the space and time derivatives of heat kernels, J. Operator Theory 21 (1989), 367–378.Google Scholar
Donnelly, H., Asymptotic expansions for the compact quotients of properly discontinuous group actions, Illinois J. Math. 23 (1979), 485–496.CrossRefGoogle Scholar
Donnelly, H., Essential spectrum and heat kernel, J. Funct. Anal. 75 (1987), 362–381.CrossRefGoogle Scholar
Donnelly, H., Local index theorem for families, Michigan Math. J. 35 (1988), 11–20.CrossRefGoogle Scholar
Douglas, R. and Wojciechowski, K., Adiabatic limits of the 𝜂-invariants. The odd-dimensional Atiyah–Patodi–Singer problem, Comm. Math. Phys. 142 (1991), 139–168.CrossRefGoogle Scholar
Dragomir, S. and Ornea, L., Locally conformal Kähler geometry, Progress in Mathematics, vol. 155 (Birkhäuser, Boston, MA, 1998).CrossRefGoogle Scholar
Dungey, N., Some gradient estimates on covering manifolds, Bull. Pol. Acad. Sci. Math. 52 (2004), 437–443.CrossRefGoogle Scholar
Gajer, P., Riemannian metrics of positive scalar curvature on compact manifolds with boundary, Ann. Global Anal. Geom. 5 (1987), 179–191.CrossRefGoogle Scholar
Gauduchon, P., Hermitian connections and Dirac operators, Boll. Unione Mat. Ital. (7) 11 (1997), 257–288; suppl.Google Scholar
Gilkey, P. B., The eta invariant and the K-theory of odd dimensional spherical space forms, Invent. Math. 79 (1984), 1421–1453.Google Scholar
Gilkey, P. B., The geometry of spherical space form groups (World Scientific, Teaneck, NJ, 1989); with an appendix by A. Bahri and M. Bendersky.CrossRefGoogle Scholar
Gilkey, P. B., On the index of geometrical operators for Riemannian manifolds with boundary, Adv. Math. 102 (1993), 129–183.CrossRefGoogle Scholar
Grigoryan, A., Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold, J. Funct. Anal. 127 (1995), 363–389.CrossRefGoogle Scholar
Gromov, M. and Lawson, H. B. Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), 423–434.CrossRefGoogle Scholar
Gromov, M. and Lawson, H. B. Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. Inst. Hautes Études Sci. 58 (1983), 83–196.CrossRefGoogle Scholar
Hitchin, N., Harmonic spinors, Adv. Math. 14 (1974), 1–55.CrossRefGoogle Scholar
Inoue, M., On surfaces of class VII0, Invent. Math. 24 (1974), 269–310.CrossRefGoogle Scholar
Lawson, H. B. and Michelsohn, M.-L., Spin geometry (Princeton University Press, Princeton, NJ, 1989).Google Scholar
Li, P. and Yau, S.-T., On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153–201.CrossRefGoogle Scholar
Lichnerowicz, A., Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7–9.Google Scholar
Marques, F. C., Deforming three-manifolds with positive scalar curvature, Ann. of Math. (2) 176 (2012), 815–863.CrossRefGoogle Scholar
Mazzeo, R. and Pollack, D., Gluing and moduli for noncompact geometric problems, in Geometric theory of singular phenomena in partial differential equations (Cortona, 1995), Symposia Mathematica, vol. XXXVIII (Cambridge University Press, Cambridge, 1998), 17–51.Google Scholar
Mazzeo, R., Pollack, D. and Uhlenbeck, K., Moduli spaces of singular Yamabe metrics, J. Amer. Math. Soc. 9 (1996), 303–344.CrossRefGoogle Scholar
Melrose, R., The Atiyah–Patodi–Singer index theorem, Research Notes in Mathematics, vol. 4 (A K Peters, Wellesley, MA, 1993).CrossRefGoogle Scholar
Miller, J., The Euler characteristic and finiteness obstruction of manifolds with periodic ends, Asian J. Math. 10 (2006), 679–713.CrossRefGoogle Scholar
Miyazaki, T., On the existence of positive scalar curvature metrics on non-simply-connected manifolds, J. Fac. Sci. Univ. Tokyo 30 (1984), 549–561.Google Scholar
Mrowka, T., Ruberman, D. and Saveliev, N., Seiberg–Witten equations, end-periodic Dirac operators, and a lift of Rohlin’s invariant, J. Differential Geom. 88 (2011), 333–377.CrossRefGoogle Scholar
Mrowka, T., Ruberman, D. and Saveliev, N., Index theory of the de Rham complex on manifolds with periodic ends, Algebr. Geom. Topol. 14 (2014), 3689–3700.CrossRefGoogle Scholar
Nicolaescu, L., Eta invariants of Dirac operators on circle bundles over Riemann surfaces and virtual dimensions of finite energy Seiberg–Witten moduli spaces, Israel J. Math. 114 (1999), 61–123.CrossRefGoogle Scholar
Okonek, C. and Teleman, A., Seiberg–Witten invariants for 4-manifolds with b + = 0, in Complex analysis and algebraic geometry (de Gruyter, Berlin, 2000), 347–357.Google Scholar
Roe, J., An index theorem on open manifolds. I, J. Differential Geom. 27 (1988), 87–113.Google Scholar
Roe, J., An index theorem on open manifolds. II, J. Differential Geom. 27 (1988), 115–136.Google Scholar
Roe, J., Elliptic operators, topology and asymptotic methods, Pitman Research Notes in Mathematics Series, vol. 395, second edition (Longman, Harlow, 1998).Google Scholar
Rosenberg, J., C ∗ -algebras, positive scalar curvature and the Novikov conjecture. II, in Geometric methods in operator algebras (Kyoto, 1983), Pitman Research Notes in Mathematics Series, vol. 123 (Longman, Harlow, 1986), 341–374.Google Scholar
Rosenberg, J., Manifolds of positive scalar curvature: a progress report, in Surveys in differential geometry, Vol. XI, Surveys in Differential Geometry, vol. 11 (International Press, Somerville, MA, 2007), 259–294.Google Scholar
Rosenberg, J. and Stolz, S., Metrics of positive scalar curvature and connections with surgery, Surveys on Surgery Theory, vol. 2 (Princeton University Press, Princeton, NJ, 2001), 353–386.Google Scholar
Rourke, C. P. and Sanderson, B. J., Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69 (Springer, New York, 1972).CrossRefGoogle Scholar
Ruberman, D., Positive scalar curvature, diffeomorphisms and the Seiberg–Witten invariants, Geom. Topol. 5 (2001), 895–924 (electronic).CrossRefGoogle Scholar
Ruberman, D. and Saveliev, N., Dirac operators on manifolds with periodic ends, J. Gökova Geom. Topol. GGT 1 (2007), 33–50.Google Scholar
Schoen, R. and Yau, S. T., On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), 159–183.CrossRefGoogle Scholar
Taubes, C., Gauge theory on asymptotically periodic 4-manifolds, J. Differential Geom. 25 (1987), 363–430.CrossRefGoogle Scholar
Taubes, C., The stable topology of self-dual moduli spaces, J. Differential Geom. 29 (1989), 163–230.CrossRefGoogle Scholar
Taubes, C., Asymptotic spectral flow for Dirac operators, Comm. Anal. Geom. 15 (2007), 569–587.CrossRefGoogle Scholar
Teleman, A., Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds, Geom. Topol. 11 (2007), 1681–1730.CrossRefGoogle Scholar
Tricerri, F., Some examples of locally conformal Kähler manifolds, Rend. Semin. Mat. Univ. Politec. Torino 40 (1982), 81–92.Google Scholar
Witten, E., Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), 769–796.CrossRefGoogle Scholar