Braid Floer homology
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.
Cite
@article{arxiv.0910.0647,
title = {Braid Floer homology},
author = {J. -B. van den Berg and R. Ghrist and R. Vandervorst and W. Wojcik},
journal= {arXiv preprint arXiv:0910.0647},
year = {2009}
}
Comments
This is a "verbose" version of an article submitted to a special issue on "Future Directions in Dynamical Systems"