Simple connectivity in polar spaces
We settle the simple connectivity of the geometry opposite a chamber in a polar space of rank 3 by completing the job for the non-embeddable polar spaces and some polar spaces with small parameters. Keywords: Non-embeddable polar space, opposition, covering degree, simple connectivity MSC 2010: 51E24 Max Horn: Mathematisches Institut, Justus-Liebig-Universität Gießen, Arndtstraße 35392 Gießen, Germany, e-mail: max.horn@math.uni-giessen.de Reed Nessler: Department of Mathematics, University of Virginia, 401 Kerchof Hall, Virginia VA 22904-4137, USA, e-mail: rcn9f@virginia.edu Hendrik Van Maldeghem: Department of Mathematics, Ghent University, Krijgslaan 281-S2 B-9000 Ghent, Belgium, e-mail: hvm@cage.ugent.be Communicated by: Karl Strambach Introduction In 1996, Peter Abramenko [1] showed that the geometry 0 ( ) induced by the set of chambers opposite a given chamber in "most" rank 3 polar spaces Besides some polar spaces with small parameters, the main rather annoying open case was the family of so-called non-embeddable polar spaces. Due to lack of an e cient, explicit and suitable description, this case remained open ever since. In this paper, we will prove that 0 ( ) is simply connected, for any chamber of a non-embeddable polar space , using the recent coordinate description by De Bruyn and Van Maldeghem [4]. Also, we settle the cases of classical polar spaces with small parameters, which are not covered by Abramenko's result, and we determine the covering degree. For the small cases, we use computer computations. The simple connectivity of 0 ( ) is used in a number of di erent situations and results. All these results excluded, up to now, the non-embeddable polar spaces and some small polar spaces of rank 3. As a corollary to Theorems 3.1 and 3.2 below, all those results now also hold for the non-embeddable polar spaces, and some more classical polar spaces with small parameters. Let us mention three examples....