Semiclassical topological flat Laguerre planes obtained by pasting along a circle
A flat Laguerre plane \({\mathcal L}\) has a point set P homeomorphic to a circular cylinder \(S_ 1\times {\mathbb{R}}\). Parallel classes and circles are topologically embedded like the factors \(x\times {\mathbb{R}}\) and \(S_ 1\times 0\), respectively. Thus, there are two natural ways to decompose \({\mathcal L}\) into two ``half planes''. Decompositions of the first kind are obtained by dividing P along a pair of parallel classes. Those of the second kind are formed by dividing along a circle. \({\mathcal L}\) is called a semiclassical plane if \({\mathcal L}\) has a decomposition whose components are separately isomorphic to half planes of the classical real Laguerre plane. According to the two ways of dividing, there are semiclassical planes of the first and second kind. In order to describe such a plane one has to say how the truncated circles in the two classical half planes are to be recombined. For decompositions of the second kind, the author encodes the necessary information in a pair of homeomorphisms \(\Phi\),\(\Psi\) of the real line with the extra condition \(\Phi (-x)=-\Phi (x)\). He shows that every pair \(\Phi\),\(\Psi\) leads to a plane, and he determines all isomorphisms between such planes and, in particular, their automorphism groups \(\Gamma\). For nonclassical planes, \(d=\dim \Gamma\) varies between 4 and 0. The author completely determines how d and k, the dimension of the kernel of \(\Gamma\) on the set of parallel classes, depend on the nature of \(\Phi\) and \(\Psi\). There are 7 possibilities, including the classical case. There are also rigid examples (without any automorphisms). The main method of proof is derivation at a point, leading to semiclassical affine planes (e.g., Moulton planes). This enables the author to use his results about these planes [Arch. Math. 44, 472-480 (1985; Zbl 0564.51006)]. A similar analysis of semiclassical Laguerre planes of the first kind is carried out by the author [Semiclassical flat Laguerre planes obtained by pasting along two parallel classes, J. Geom., to appear].
semiclassical topological flat Laguerre planes
scientific article; zbMATH DE number 4028276