Polar unitals in compact eight-dimensional planes

Abstract

We determine centralizers and unitals for the polarities of eight-dimensional compact planes with at least 17-dimensional group of automorphisms, and discuss transitivity properties. Introduction.Let P = (P , L, ∈) be a compact projective plane (cf. [6]): point set P and line set L are compact spaces, and intersection and joining are continuous operations. We say that P is d-dimensional if P has topological (covering) dimension d.A polarity of P = (P , L, ∈) is a continuous involution π from P ∪ L onto L ∪ P such that p ∈ ⇐⇒ π ∈ p π holds for all p ∈ P and all ∈ L. The absolute points of the polarity π are those points p with p ∈ p π . The polar unital U π = (U π , B π ) for π is the set U π of absolute points, together with the set B π of traces (also called blocks) of lines that meet U π in more than one point. In many of the cases that we have to deal with, there is a distinguished line (at infinity) fixed by all automorphisms of the plane, and the polar unital contains exactly one point p ∞ at infinity. In these cases, we consider the affine part A π := U π \{p ∞ }.A unital (U, B) is called of spherical type (n, s) if U is homeomorphic to an n-sphere and each block b ∈ B is homeomorphic to an s-sphere. It has been conjectured that each nonempty polar unital of any 8-dimensional compact projective plane is of spherical type (7, 3) or (5, 1). Under additional hypotheses, this has been proved, cf. [3, 3.2, 3.3, 4.5].All 8-dimensional compact projective planes with at least 17-dimensional group of automorphisms have been determined, see [5], cf. [6, 84.28]. Apart from the Hughes planes discussed in Section 6, every such plane is a translation plane. A translation plane can only be self-dual if it has Lenz type V. There are three (closely related) families of compact eight-dimensional planes of Lenz type V with 18-dimensional group of automorphisms, but only one of these families contains self-dual planes (cf. [11, Thm. 6.1]): these are the Spin planes discussed in Section 3. Every eight-dimensional plane of Lenz type V with 17-dimensional group of automorphisms is coordinatized by a mutation of H, or by a Rees

Baer involutions and polarities in Moufang planes of characteristic two

“…Polarities with this absolute flag and at least one more absolute point can be treated as in [9], [10]. In particular, we have the following source of examples (see [2], cf.…”

Unitals over composition algebras

“…Polarities with this absolute flag can be treated as in [34,36]. Many examples can be constructed using an involution of the semifield.…”

Polarities of Schellhammer planes

“…Polarities of compact connected planes have been investigated by Salzmann [18, p. 260], [19], Bedürftig [1], Polster [16], Immervoll [9] and the second author, see [26], [25]. If the plane is topological, we will tacitly assume that the polarity is continuous.…”