Braid Floer homology

TL;DR

This paper introduces braid Floer homology, a new invariant for analyzing the dynamics of area-preserving disc diffeomorphisms through braiding, leading to a Morse-type forcing theory for periodic points.

Contribution

It develops braid Floer homology, proves its invariance, establishes a shift theorem, computes examples, and applies it to forcing results in Hamiltonian disc dynamics.

Findings

Braid Floer homology is topologically invariant.

A shift theorem relates braid twisting to homology shifts.

The theory provides a new Morse-type forcing criterion for periodic points.

Abstract

Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on solid tori, periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a braid Floer homology. This refinement of the 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.