Simple connectedness of hyperplane complements in some geometries related to buildings

摘要

For a class of parapolar spaces that includes the geometries E_(6,4,) E_(7,7,) and E_(8,1) and with lines of size at least three, the metasymplectic spaces with lines of size at least four, and the polar line Grassmannians with lines of size at least four except D_(4,2)(3), we show that the subgraph of the point-collinearity graph induced on the complement of a hyperplane is simply connected. We also show that these parapolar spaces have Veldkamp lines.