2021/06/11 by Kenji Fukaya, Fukaya, Kenji · 1 citation
Mathematics · Medicine · #53D37 #53D40 #Advanced Combinatorial Mathematics #Botulinum Toxin and Related Neurological Disorders #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG) #math.DG #math.DS #math.QA #math.SG #msc:53D37 #msc:53D40
paper · pdf · doi:10.48550/arxiv.2106.06378
73 pages, 19 figures
arxiv created 2021/06/11 · openalex publication_date 2021/06/11 · arxiv updated 2021/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we introduce and study a distance, Gromov-Hausdorff distance, which measures how two filtered A A∞ categories are far away each other. In symplectic geometry the author associated a filteredA∞ category, Fukaya category, to a finite set of Lagrangian submanifolds. The Gromov-Hausdorff distance then gives a new invariant of a finite set of Lagrangian submanifolds. One can estimate it by the Hofer distance of Hamiltonian diffeomorphisms needed to send one Lagrangain submanifold to the other. A motivation to introduce Gromov-Hausdorff distance is to obtain a certain completion of Fukaya category. If we have a sequence of sets of Lagrangian submanifolds, which is a Cauchy sequence in the sense of Hofer metric, then the associated filtered A infinity categories also form a Cauchy sequence in Gromov-Hausdorff distance. In this paper we develop a theory to obtain an inductive limit of such a sequence of filtered A∞ categories. In other words, we give an affirmative answer to [Fu5] Conjecture 15.34.