The classical Arnol'd Conjecture concerns the number of 1-periodic orbits of 1-periodic Hamiltonian dynamics on a symplectic manifold. The resolution of this conjecture was the impetus for and first triumph of Floer homology. The present talk considers the problem of periodic orbits of higher periods. In the case (trivial for the Arnol'd Conjecture) of a 2-dimensional disc, these orbits are braids.

This talk describes a relative Floer homology that is a topological invariant of (pairs of) braids. This can be used as a forcing theorem for implying the existence of periodic orbits in 1-periodic Hamiltonian dynamics on a disc.

This represents joint with with J.B. van den Berg, R. Vandervorst, and W. Wojcik.