Floer homology, relative braid classes, and low dimensional dynamics (Q2888972)

The author studies some problems related to knotting and linking of periodic orbits for non-autonomous Hamiltonian vector fields. He assumes that the time-dependence of the vector field is \(1\)-periodic. After defining braid classes in various settings, he presents some braid class invariants and shows that such invariants are computable in some situations. First the author defines the braids on \(2\)-disc, the braid diagrams on the unit interval and discrete braid diagrams. Then he defines an invariant for discrete relative braid classes and presents some results from \textit{R. W. Ghrist}, \textit{J. B. Van der Berg} and \textit{R. C. Vandervorst} [Invent. Math. 152, 369--432 (2003; Zbl 1017.37031)]. Next he deals with braids and dynamics. After presenting some aspects concerning the Cauchy-Riemann equations, the heat flow, discrete parabolic equations and the invariants, the author presents some aspects concerning Floer homology, Morse homology and the Conley index. At the end he states some results from a preprint [``Braid Floer homology'' \url{arXiv:0910.0647}], written with \textit{R. W. Ghrist}, \textit{J. B. Van der Berg} and \textit{W. Wojcik}.NEWLINENEWLINEFor the entire collection see [Zbl 1237.58002].