  For any rank $r$ oriented matroid $M$, a construction is given of a
"topological representation" of $M$ by an arrangement of homotopy spheres in a
simplicial complex which is homotopy equivalent to $S^{r-1}$. The construction
is completely explicit and depends only on a choice of maximal flag in $M$. If
$M$ is orientable, then all Folkman-Lawrence representations of all
orientations of $M$ embed in this representation in a homotopically nice way.

Combinatorics