  We examine the five-dimensional super-de Rham complex with $N = 1$
supersymmetry. The elements of this complex are presented explicitly and
related to those of the six-dimensional complex in $N = (1, 0)$ superspace
through a specific notion of dimensional reduction. This reduction also gives
rise to a second source of five-dimensional super-cocycles that is based on the
relative cohomology of the two superspaces. In the process of investigating
these complices, we discover various new features including branching and
fusion (loops) in the super-de Rham complex, a natural interpretation of "Weil
triviality", $p$-cocycles that are not supersymmetric versions of closed
bosonic $p$-forms, and the opening of a "gap" in the complex for $D > 4$ in
which we find a multiplet of superconformal gauge parameters.
