Arrangements of pseudocircles on surfaces.

Abstract

A pseudocircle is a simple closed curve on some surface. Arrangements of pseudocircles were introduced by Grunbaum, who defined them as collections of pseudocircles that pairwise intersect in exactly two points, at which they cross. There are several variations on this notion in the literature, one of which requires that no three pseudocircles have a point in common. Working under this definition, Ortner proved that an arrangement of pseudocircles is embeddable into the sphere if and only if all of its subarrangements of size at most $4$ are embeddable into the sphere. Ortner asked if an analogous result held for embeddability into a compact orientable surface $\Sigma_g$ of genus $g>0$. In this paper we answer this question, under an even more general definition of an arrangement, in which the pseudocircles in the collection are not required to intersect each other, or that the intersections are crossings: it suffices to have one pseudocircle that intersects all other pseudocircles in the collection. We show that under this more general notion, an arrangement of pseudocircles is embeddable into $\Sigma_g$ if and only if all of its subarrangements of size at most $4g+5$ are embeddable into $\Sigma_g$, and that this can be improved to $4g+4$ under the concept of an arrangement used by Ortner. Our framework also allows us to generalize this result to arrangements of other objects, such as arcs.