The homeomorphism type of unital-like submanifolds in smooth projective planes

Abstract.

We prove that a compact, connected submanifold of the point space of a smooth projective plane is homeomorphic to a sphere provided that certain intersection properties with lines are satisfied. As an application, we show that the set of absolute points of a smooth polarity in a smooth projective plane of dimension 2l is empty or homeomorphic to a sphere of dimension 2l - 1 or $\frac {3}{2}\, l - 1$ .