  The parameter space $\mathcal{S}_p$ for monic centered cubic polynomial maps
with a marked critical point of period $p$ is a smooth affine algebraic curve
whose genus increases rapidly with $p$. Each $\mathcal{S}_p$ consists of a
compact connectedness locus together with finitely many escape regions, each of
which is biholomorphic to a punctured disk and is characterized by an
essentially unique Puiseux series. This note will describe the topology of
$\mathcal{S}_p$, and of its smooth compactification, in terms of these escape
regions. It concludes with a discussion of the real sub-locus of
$\mathcal{S}_p$.
