## Geometry & Topology

### Instanton Floer homology with Lagrangian boundary conditions

#### 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

#### 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

#### 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