  An abstract simplicial complex is said to be $d$-representable if it records
the intersection pattern of a collection of convex sets in $\mathbb{R}^d$. In
this paper, we show that $d$-representability of a simplicial complex is
equivalent to the existence of a map with certain properties, from a closely
related simplicial complex into $\mathbb{R}^d$. This equivalence suggests a
framework for proving (and disproving) $d$-representability of simplicial
complexes using topological methods such as applications of the Borsuk-Ulam
theorem, which we begin to explore.
