  We prove that the rank of the cohomology of a closed symplectic manifold with
coefficients in a field of characteristic $p$ is smaller than the number of
periodic orbits of any non-degenerate Hamiltonian flow. Following Floer, the
proof relies on constructing a homology group associated to each such flow, and
comparing it with the homology of the ambient symplectic manifold. The proof
does not proceed by constructing a version of Floer's complex with
characteristic $p$ coefficients, but uses instead the canonical (stable)
complex orientations of moduli spaces of Floer trajectories to construct a
version of Floer homology with coefficients in Morava's $K$-theories, and can
thus be seen as an implementation of Cohen, Jones, and Segal's vision for a
Floer homotopy theory. The key feature of Morava K-theory that allows the
construction to be carried out is the fact that the corresponding homology and
cohomology groups of classifying spaces of finite groups satisfy Poincar\'e
duality.
