On the division of space by topological hyperplanes

A topological hyperplane   is a subspace of RnRn (or a homeomorph of it) that is topologically equivalent to an ordinary straight hyperplane. An arrangement   of topological hyperplanes in RnRn is a finite set HH such that for any nonvoid intersection YY of topological hyperplanes in HH and any H∈HH∈H that intersects but does not contain YY, the intersection is a topological hyperplane in YY. (We also assume a technical condition on pairwise intersections.) If every two intersecting topological hyperplanes cross each other, the arrangement is said to be transsective. The number of regions formed by an arrangement of topological hyperplanes has the same formula as for arrangements of ordinary affine hyperplanes, provided that every region is a cell. Hoping to explain this geometrically, we ask whether parts of the topological hyperplanes in any arrangement can be reassembled into a transsective arrangement of topological hyperplanes with the same regions. That is always possible if the dimension is two but not in higher dimensions. We also ask whether all transsective topological hyperplane arrangements correspond to oriented matroids; they need not (because parallelism may not be an equivalence relation), but we can characterize those that do if the dimension is two. In higher dimensions this problem is open. Another open question is to characterize the intersection semilattices of topological hyperplane arrangements; a third is to prove that the regions of an arrangement of topological hyperplanes are necessarily cells; a fourth is whether the technical pairwise condition is necessary.