On Kühnel's 9-vertex complex projective plane

Abstract

We present an elementary combinatorial proof of the existence and uniqueness of the 9-vertex triangulation of CP². The original proof of existence, due to Kühnel, as well as the original proof of uniqueness, due to Kühnel and Lassmann, were based on extensive computer search. Recently Arnoux and Marin have used cohomology theory to present a computer-free proof. Our proof has the advantage of displaying a canonical copy of the affine plane over the three-element field inside this complex in terms of which the entire complex has a very neat and short description. This explicates the full automorphism group of the Kühnel complex as a subgroup of the automorphism group of this affine plane. Our method also brings out the rich combinatorial structure inside this complex.