Geometry & Topology

Instanton Floer homology with Lagrangian boundary conditions

Dietmar Salamon and Katrin Wehrheim

Full-text: Open access

Abstract

In this paper we define instanton Floer homology groups for a pair consisting of a compact oriented 3–manifold with boundary and a Lagrangian submanifold of the moduli space of flat SU(2)–connections over the boundary. We carry out the construction for a general class of irreducible, monotone boundary conditions. The main examples of such Lagrangian submanifolds are induced from a disjoint union of handle bodies such that the union of the 3–manifold and the handle bodies is an integral homology 3–sphere. The motivation for introducing these invariants arises from our program for a proof of the Atiyah–Floer conjecture for Heegaard splittings. We expect that our Floer homology groups are isomorphic to the usual Floer homology groups of the closed 3–manifold in our main example and thus can be used as a starting point for an adiabatic limit argument.

Article information

Source
Geom. Topol., Volume 12, Number 2 (2008), 747-918.

Dates
Received: 19 July 2006
Accepted: 10 December 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800063

Digital Object Identifier
doi:10.2140/gt.2008.12.747

Mathematical Reviews number (MathSciNet)
MR2403800

Zentralblatt MATH identifier
1166.57018

Subjects
Primary: 57R58: Floer homology
Secondary: 58J32: Boundary value problems on manifolds

Keywords
3-manifold with boundary Atiyah-Floer conjecture

Citation

Salamon, Dietmar; Wehrheim, Katrin. Instanton Floer homology with Lagrangian boundary conditions. Geom. Topol. 12 (2008), no. 2, 747--918. doi:10.2140/gt.2008.12.747. https://projecteuclid.org/euclid.gt/1513800063


Export citation

References

  • S Agmon, A Douglis, L Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959) 623–727
  • S Agmon, L Nirenberg, Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Comm. Pure Appl. Math. 20 (1967) 207–229
  • M F Atiyah, New invariants of $3$– and $4$–dimensional manifolds, from: “The mathematical heritage of Hermann Weyl (Durham, NC, 1987)”, (R O Wells, Jr, editor), Proc. Sympos. Pure Math. 48, Amer. Math. Soc. (1988) 285–299
  • M F Atiyah, R Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) 523–615
  • M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43–69
  • W Ballmann, J Brüning, G Carron, Dirac systems, in preparation
  • B Booss-Bavnbek, K Furutani, The Maslov index: a functional analytical definition and the spectral flow formula, Tokyo J. Math. 21 (1998) 1–34
  • B Booss-Bavnbek, C Zhu, Weak symplectic functional analysis and general spectral flow formula
  • S K Donaldson, The orientation of Yang–Mills moduli spaces and $4$–manifold topology, J. Differential Geom. 26 (1987) 397–428
  • S K Donaldson, Floer homology groups in Yang–Mills theory, Cambridge Tracts in Mathematics 147, Cambridge University Press (2002) With the assistance of M Furuta and D Kotschick
  • S K Donaldson, P B Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York (1990)
  • S Dostoglou, D A Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. $(2)$ 139 (1994) 581–640
  • N Dunford, JTSchwarz, Linear operators II: spectral theory, Interscience (1964)
  • A Floer, An instanton-invariant for $3$–manifolds, Comm. Math. Phys. 118 (1988) 215–240
  • K Froyshov, On Floer homology and four-manifolds with boundary, PhD thesis, Oxford (1994)
  • K Fukaya, Floer homology for $3$–manifolds with boundary I (1997) Available at \setbox0{\makeatletter\@url {http://www.math.kyoto-u.ac.jp/
  • A A Kirillov, A D Gvishiani, Theorems and problems in functional analysis, Problem Books in Mathematics, Springer, New York (1982) Translated from the Russian by H H McFaden
  • P Kirk, M Lesch, The $\eta$–invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary, Forum Math. 16 (2004) 553–629
  • P B Kronheimer, Four-manifold invariants from higher-rank bundles, J. Differential Geom. 70 (2005) 59–112
  • P Kronheimer, T Mrowka, Monopoles and Three-Manifolds, New Mathematical Monographs 10, Cambridge University Press (2007)
  • D McDuff, D Salamon, $J$–holomorphic curves and symplectic topology, Amer. Math. Soc. Colloquium Publ. 52, Amer. Math. Soc. (2004)
  • J W Milnor, Topology from the differentiable viewpoint, Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va. (1965)
  • T S Mrowka, K Wehrheim, $L^2$–topology and Lagrangians in the space of connections over a Riemann surface, in preparation
  • J Robbin, Y Ruan, D A Salamon, The moduli space of regular stable maps, to appear in Math. Zeit.
  • J Robbin, D Salamon, The Maslov index for paths, Topology 32 (1993) 827–844
  • J Robbin, D Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995) 1–33
  • D Salamon, Spin geometry and Seiberg–Witten invariants, in preparation
  • D Salamon, Lagrangian intersections, $3$–manifolds with boundary, and the Atiyah–Floer conjecture, from: “Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994)”, Birkhäuser, Basel (1995) 526–536
  • D Salamon, Lectures on Floer homology, from: “Symplectic geometry and topology (Park City, UT, 1997)”, IAS/Park City Math. Ser. 7, Amer. Math. Soc. (1999) 143–229
  • C H Taubes, Casson's invariant and gauge theory, J. Differential Geom. 31 (1990) 547–599
  • C H Taubes, Unique continuation theorems in gauge theories, Comm. Anal. Geom. 2 (1994) 35–52
  • K K Uhlenbeck, Connections with $L\sp{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982) 31–42
  • K Wehrheim, Banach space valued Cauchy–Riemann equations with totally real boundary conditions, Commun. Contemp. Math. 6 (2004) 601–635
  • K Wehrheim, Uhlenbeck compactness, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich (2004)
  • K Wehrheim, Anti-self-dual instantons with Lagrangian boundary conditions. I. Elliptic theory, Comm. Math. Phys. 254 (2005) 45–89
  • K Wehrheim, Anti-self-dual instantons with Lagrangian boundary conditions. II. Bubbling, Comm. Math. Phys. 258 (2005) 275–315
  • K Wehrheim, Lagrangian boundary conditions for anti-self-dual instantons and the Atiyah–Floer conjecture, Conference on Symplectic Topology, J. Symplectic Geom. 3 (2005) 703–747