Topological and smooth unitals

Abstract

We first show that under natural topological assumptions in compact, connected projective planes there are no objects besides ovals which have incidence geometric properties analogous to those of unitals in finite projective planes. After giving a brief account of `classical unitals', i.e.\ sets of absolute points of continuous polarities in the classical projective planes 𝒫2𝔽, 𝔽 ∈ {ℝ, ℂ, ℍ, 𝕆}, we define unitals by means of their intersection properties with respect to lines, in analogy to properties of classical unitals. Under suitable topological assumptions, unitals turn out to be homeomorphic to spheres. The existence of exterior lines is related to the codimension of the unital in the point space. Our main result is that unitals satisfying an additional regularity condition have the same dimensions as classical unitals. Finally, we consider smooth unitals in smooth projective planes. In this case some results can be improved by using transversality arguments.