  It is well known that for many semilinear parabolic equations there is a
global attractor which has a cell complex structure with finite dimensional
cells. Additionally, many semilinear parabolic equations have equilibria with
finite dimensional unstable manifolds. In this article, these results are
unified to show that for a specific parabolic equation on an unbounded domain,
the space of heteroclinic orbits has a cell complex structure with finite
dimensional cells. The result depends crucially on the choice of spatial
dimension and the degree of the nonlinearity in the parabolic equation, and
thereby requires some delicate treatment.
