2015/04/15 by Jan Bouwe van den Berg, Robert Ghrist, van der Vorst, R.C.A.M. +2 · 1 citation
Mathematics · #Algebra over a field #Braid #Braid group #Braid theory #Cellular homology #Floer homology #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Mathematics #Morse homology #Pure mathematics #Relative homology #Symplectic geometry
paper · pdf · doi:10.1016/j.jde.2015.03.022
openalex publication_date 2015/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on R/Z×D2. The periodic flow-lines define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a new invariant for such classes, the braid Floer homology. This refinement of Floer homology, originally used for the Arnol'd Conjecture, yields a Morse-type forcing theory for periodic points of area-preserving diffeomorphisms of the 2-disc based on braiding.Contributions of this paper include (1) a monotonicity lemma for the behavior of the nonlinear Cauchy-Riemann equations with respect to algebraic lengths of braids; (2) establishment of the topological invariance of the resulting braid Floer homology; (3) a shift theorem describing the effect of twisting braids in terms of shifting the braid Floer homology; (4) computation of examples; and (5) a forcing theorem for the dynamics of Hamiltonian disc maps based on braid Floer homology.