Absolute Points of Continuous and Smooth Polarities

Abstract

This paper deals with sets of absolute points of continuous or smooth polarities in compact, connected or smooth projective planes, called topological polar unitals or smooth polar unitals, respectively. We will show that topological polar unitals are Z2-homology spheres. In the four-dimensional case, a topological polar unital U is either a topological oval, or any line which intersects U in more than one point intersects in a set homeomorphic to S1. Smooth polar unitals turn out to be smoothly embedded submanifolds of the point space. Moreover, secants intersect such unitals transversally. For these unitals, we will obtain full information on the existence of secants, tangents and exterior lines through given points according to their position with respect to the unital. The main result of this paper states that the possible dimensions of smooth polar unitals coincide with those of sets of absolute points of continuous polarities in the classical projective planes P2F, Fϵ{R,C,H,O}. Finally, we will prove that smooth polar unitals in four-dimensional smooth projective planes are topological ovals or are homeomorphic to S3.