Compact connected topological planes and their classification by automorphism groups
by
Hermann Hähl
University of Stuttgart

The talk will first give an introductory survey of the subject of compact connected topological projective planes. The vast research project on this subject initiated by H. Salzmann over forty years ago has since developed into a rich theory, which also has served as a basis and a model for work on other types of topological geometries (stable planes, circle geometries, topological generalized polygons, etc.). The talk will give an impression of fundamental notions and highlights of this theory and explain some of its aims, emphasizing the classification of compact connected planes whose automorphism groups are large (i.e. have large topological dimension, so that the planes present a high degree of freedom). The topological dimension of the point set is known to be 2, 4, 8, 16 or possibly (but improbably) \infty. In the last part of the talk, we shall take a closer look on recent classification results for planes of topological dimension 16, which topologically resemble the classical octonion plane, but geometrically may be quite different. We shall exemplify how one can systematically discover large families of such planes starting out from their groups; the examples will mostly be translation planes (with translation group R¹⁶).

Date received: February 13, 2001