Topological and smooth unitals (Q2761647)

A unital in a finite projective plane is a set \(U\) of points such that every line intersects \(U\) in \(0\), \(1\) or \(n\) points, \(n\geq 2\) fixed, and for every point \(p\) of \(U\) there is precisely one line (the tangent through \(p\)) which intersects \(U\) only in \(p\). NEWLINENEWLINENEWLINEThe author shows that there are no such objects besides ovals (\(n=2\)) in compact connected projective planes, i.e. projective planes whose point and line spaces are compact connected Hausdorff spaces such that the geometric operations \(\vee\) of joining points and \(\wedge\) of intersecting lines are continuous. In particular, the set of absolute points of a continuous polarity of the projective planes over \(\mathbb R\), \(\mathbb C\), \(\mathbb H\) (quaternions) and \(\mathbb O\) (octonions) does not match the definition above. For this reason, the author introduces another definition of unitals: A topological unital is a set \(U\) of points of a compact connected projective plane such that there exists precisely one tangent \(T_p\) through every point \(p \in U\) and every secant intersects \(U\) in a topological \(k\)-sphere, where \(k \geq 0\) is a fixed number. If, moreover, \(U\) is homeomorphic to a \((k+l)\)-sphere (where \(l\) denotes the dimension of a line), then \(U\) is called a spherical unital. NEWLINENEWLINENEWLINEThe main result in the paper under review deals with spherical unitals \(U\) of codimension at least \(2\). Suppose that there exists a point \(p \not\in U\) such that the following two conditions are satisfied: NEWLINENEWLINENEWLINE(R1) Every line through \(p\) intersects \(U\) and the set \({\mathcal T}_p\) of tangents through \(p\) is a sphere of dimension \(k\). NEWLINENEWLINENEWLINE(R2) \(U \setminus \{ U \cap T |T \in {\mathcal T}_p \}\) is a locally trivial fibration over \({\mathcal L}_p \setminus {\mathcal T}_p\) with projection \(x \mapsto x\vee p\), where \({\mathcal L}_p\) denotes the line pencil through \(p\). NEWLINENEWLINENEWLINEThen \(l>1\) and the dimension of \(U\) equals \((3/2)l-1\), which is the dimension of the set of absolute points of the so-called planar polarity of classical plane over \(\mathbb C\), \(\mathbb H\), and \(\mathbb O\), respectively. NEWLINENEWLINENEWLINEIn the last section, the author investigates unitals in smooth projective planes, i.e. in projective planes whose point and line spaces are smooth manifolds such that \(\vee\) and \(\wedge\) are differentiable maps. In this situation a spherical unital \(U\) is called smooth if it is a smooth submanifold of the point space such that every secant intersects \(U\) transversally. Then (R2) follows from (R1) and, hence, a smooth unital satisfying (R1) always has the same dimension as one of the classical unitals.