Adv. Geom. 1 (2001), 333–344 Advances in Geometry ( de Gruyter 2001 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 A fR;C;H;Og, we define unitals by means of their intersection properties with re-spect to lines, in analogy to properties of classical unitals. Under suitable topological assump-tions, 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 con-sider smooth unitals in smooth projective planes. In this case some results can be improved by using transversality arguments. This paper is inspired by the beautiful paper [3] by Buchanan, Hähl and Löwen on topological ovals. The methods which we will use here, however, are often quite dif-ferent. Unitals were defined originally as sets of absolute points of unitary polarities of Desarguesian finite projective planes. Such unitals have the characteristic propert