Atiyah-Floer Conjecture: a Formulation, a Strategy to Prove and Generalizations

2017/07/12 by Aliakbar Daemi, Kenji Fukaya, Daemi, Aliakbar +1 · 1 citation
Mathematics · Medicine · #53D37 #53D40 #57M27 #57R57 #57R58 #Botulinum Toxin and Related Neurological Disorders #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1707.03924

openalex publication_date 2017/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic manifolds. Soon after that, Atiyah conjectured that the two theories should be related to each other and Lagrangian Floer homology of certain Lagrangians in the moduli space of flat connections on Riemann surfaces should recover instanton Floer homology. However, the space of flat connections on a Riemann surface is singular and the first step to address this conjecture is to make sense of Lagrangian Floer homology on this space. In this note, we formulate a possible approach to resolve this issue. A strategy to construct the desired isomorphism in the Atiyah-Floer conjecture is also sketched. We also use the language of A infty-categories to state generalizations of the Atiyah-Floer conjecture.

Citations

• Symplectic Surgery and the Spinc–Dirac Operator

Cited by

• Floer homotopy theory for monotone Lagrangians

Related

• Instanton Floer homology with Lagrangian boundary conditions
• Floer homology on the extended moduli space
• A Symplectic Instanton Homology via Traceless Character Varieties
• Categorification of invariants in gauge theory and sypmplectic geometry
• Lagrangians, SO(3)-instantons and the Atiyah-Floer Conjecture