Topological affine planes composed of two desarguesian halfplanes and projective planes with trivial collineation group (Q1058136)

The author determines all flat affine planes containing a line whose removal produces two copies of the (right) halfplane of the real affine plane. These planes generalize those generalized Moulton planes of \textit{W. A. Pierce} [Can. J. Math. 13, 427--436 (1961; Zbl 0103.13404)] which are constructed over the field \(\mathbb R\) of reals. To each pair \(\varphi\), \(\psi\) of homeomorphisms of \(\mathbb R\) fixing 0 and 1 there corresponds such a plane \({\mathcal A}_{\varphi,\psi}\), and conversely. The isomorphism types of the \({\mathcal A}_{\varphi,\psi}\) are determined. A large class of examples is given where \({\mathcal A}_{\varphi,\psi}\) admits only the identical collineation.