Braid Floer homology

Abstract

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.