Proof of Grünbaum's Conjecture on the Stretchability of Certain Arrangements of Pseudolines

Any finite set of lines in the real projective plane determines a cell complex; these complexes and their combinatorial properties have been a subject of study at least since 1826 [9]. More recently, Levi [6] considered a topological generalization of this notion, defined as follows: Consider a simple closed curve in RP2 which does not separate RP2; this is called a pseudoline. (It is clear that any two pseudolines must meet, and it is easy to see that if they meet at precisely one point, they must cross there.) If a finite set of pseudolines has the property that any two meet at precisely one point, but that not all pass through a common point, we speak of an urra~~eme~~ of pseudolines; the arrangement is simple if distinct pairs meet at distinct points. An arrangement of pseudolines also determines a cell complex, and Levi and others showed that a number of properties of linear complexes carry over to pseudolinear ones [6,4]. Two arrangements are c&led isomorphic if there is an isomorphism of their associated cell.complexes; it 385 0097.3165/‘80/060385-#6~02.00/0