Extending homomorphisms of dense projective subplanes by continuity (Q1297007)

The author investigates dense subplanes of topological projective planes and their continuous homomorphisms. A homomorphism between projective planes \((P,{\mathcal L})\) and \((P',{\mathcal L}')\) is a map from \(P\) to \(P'\) that takes collinear points to collinear points and whose image contains a quadrangle. The author shows that if \(\Psi=(S,{\mathcal S})\) is a dense subplane of a topological projective plane \(\Pi=(P,{\mathcal L})\) and \(\alpha:S\to P\) is a continuous homomorphism, then \(\alpha\) extends to a continuous homomorphism \(\overline\alpha:P\to P'\). Furthermore, \(\overline\alpha\) is injective if and only if \(\alpha\) is injective. In fact, whether or not a continuous homomorphism can be continuously extended to the entire projective plane can be characterised in terms of the induced map on a single line. It is shown that \(\alpha\) can be extended if and only if there is a line \(Z\in {\mathcal S}\) such that the restriction \(\alpha_{Z\cap S}\) of \(\alpha\) to \(Z\) extends to a continuous map \(\overline\alpha_Z:Z \to P'\). In particular, if \(\alpha\) is a continuous collineation of \(\Psi\) such that \(\overline\alpha_Z\) is surjective onto the image of the line \(Z\) under \(\alpha\) or such that \(\Pi\) is compact, then \(\overline\alpha\) is a continuous collineation of \(\Pi\). This last result has the well-known consequences that a central collineation of \(\Psi\) extends to one of \(\Pi\) and that \(\Pi\) has at least the same Lenz-Barlotti type as \(\Psi\).