On the division of space by topological hyperplanes

Abstract

A topological hyperplane is a subspace of R^n (or a homeomorph of it) that is topologically equivalent to an ordinary straight hyperplane. An arrangement of topological hyperplanes in R^n is a finite set H such that k topological hyperplanes in H, if their intersection is nonempty, meet in a subspace that is a topological hyperplane in the intersection of any k-1 of them; but two topological hyperplanes that do intersect need not cross each other. If every intersecting pair does cross, the arrangement is affine. The number of regions formed by an arrangement of topological hyperplanes has the same formula as for arrangements of affine hyperplanes. Hoping to explain this geometrically, we ask whether parts of the topological hyperplanes in any arrangement can be reassembled into an arrangement of affine 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 affine topological hyperplane arrangements correspond to oriented matroids; they need not, 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.