There is a universal topological plane

Jacob E. Goodman; Richard Pollaclc] Rephael Wenger~ and Tudor Zamfirescu~ We show that every arrangement of pseudolines in the plane can be extended to a topological projective plane, a projective geometry whose pcints and topology agree with the real projective p kme and whose lines also have the topology of the projective plane. In this topological projective plane the given arrangement becomes au arrangement of “straight lines”. This makes it possible to realize a “topological sweep of an arrangement” as the familiar sweep by a family of parallel lines. We then use this result to construct a universal topological plane, one in which every arrangement of pseudolines is stretchable. Both results had been conjectured by B. Griinbaum. *City College, City University of New York, ,,New York, NY 10031, U. S.A.. Supported in part by NSA grant MDA904-89-H-2038, PSC-CUNY grant 662330, the Mlttag-Leffler Institute, and the Fulbright Commission. t Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U. S.A.. Supported in part by NSF grant CCR-8901484, NSA grant MDA90489-H-2030, and the Mlttag-Leffler Institute. :Ohio State University, Columbus, OH 43210, U. S.A.. Supported in part by NSA grant MDA904-89-H-2030 and the Mlttag-Leffler Institute. SFachbereich Mathematik, Universitiit Dortnmmi, 4600 Dortmund 50, Germany. Permission to copy without fee atl or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the tide of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.