Construction of topological affine planes with non-continuous parallelism by bending of lines (Q1208680)

This note comprises the second half of the author's doctoral thesis [Topologische affine Ebenen mit nichtstetigem Parallelismus, Univ. Stuttgart (1989; Zbl 0699.51009)]. The first part was published in Geom. Dedicata 40, No. 3, 297-318 (1991; Zbl 0742.51013)]. There a general method was described how to construct new affine planes from a given plane as the limit of a process of successively bending lines. Passing over to certain algebraic subplanes yields topological affine planes with noncontinuous parallelisms. Consequently, these planes have no topological projective extensions, thus answering a longstanding question in topological geometry in the negative. In the present note the author supplies models of planes for which his construction works. This is achieved by a different process of bending lines at certain curves. Beginning with Moulton planes an infinite family of pairwise non-isomorphic topological affine planes with noncontinuous parallelisms is eventually obtained. These planes are affine subplanes \(E_ A\) of topological affine Salzmann planes. In Abh. Math. Semin. Univ. Hamburg 60, 257-264 (1990; Zbl 0721.51016) the author investigated the general problem of which topological affine subplanes \(E_ A\) of a topological projective plane \(E\) have topological projective extensions. He also constructed topological ternary fields that do not coordinatise topological projective planes. This was achieved by coordinatising the projective extensions of the affine subplanes \(E_ A\) with respect to a different line at infinity. Using a similar process of bending lines at infinitely many vertical lines in J. Geom. 40, No. 1/2, 34-46 (1991; Zbl 0728.51012) the author constructed examples of Cartesian groups such that the parallelisms of the corresponding topological affine planes are continuous but cannot be extended to topological projective planes.