Geometriae Dedicata 50: 1-13, 1994. 1 © 1994 KluwerAcademic Publishers. Printed in the Netherlands. On Ktihnel's 9-Vertex Complex Projective Plane BHASKAR BAGCHI 1 and BASUDEB DATTA 2 1Stat-Math Unit, Indian Statistical Institute, R. V. College Post, Bangalore-560059, India 2Mathematics Department, Indian Institute of Science, Bangalore 560012, India (Received: December 1991;in final form: June 1992) Abstract. We present an elementary combinatorial proof of the existence and uniqueness of the 9-vertex triangulation of C p2. The original proof of existence, due to Ktthnel, as well as the original proof of uniqueness, due to Ktthnel and Lassmann, werebasedon extensivecomputersearch. Recently Arnoux and Matin have used cohomologytheory to present a computer-freeproof. 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 K0hnel complexas a subgroupof the automorphismgroup of this affine plane. Our method also brings out the rich combinatorial structure inside this complex. Mathematics Subject Classification (1991). Primary 57Q15, 51E20. 1. Introduction Recall that a simplicialcomplex IC is a collection of nonempty sets (sets of vertices) such that all nonempty subsets of a member of the collection are again members. A member of/C with i + 1 vertices is called an i-face (or simplex of dimension i). For a E /C, Lk(a) (=link of a):= {7 C /C: 7 f) a = ¢,7 U a E /C}. A simplicial complex may be thought of as a prescription for the construction of a topological space by pasting together geometric simplexes. The topological space thus obtained from a simplicial complex/C is called a polyhedron and is denoted by I /C ]. Let E1 and/C2 be two simplicial complexes• A map f :1 E1 1--+[ )U2 ] • . , I I is called PL if there are subdivisions/(;1 and ~2 of E1 and/C2 respectively such I ! that f • /(;1 --+ /~2 is simplicial. We write I E1 I -I/c2 I if l /(;1 [ and [/~2 [ are PL homeomorphic. A simplicial complex/C (respectively a polyhedron I E 1) is called a combinatorial d-manifold (respectively PL d-manifold) if for every vertex v in/C [ Lk(v) I~ S d-t, the standard (d - 1)-sphere. In 1962, Eells and Kuiper [7] proved that a PL manifold M a with PL Morse number #(M 't) = 3 has dimension d = 0, 2, 4, 8 or 16. If d = 0, M a consists of three points. If d = 2, M d is the real projective plane. For d = 4, 8 or 16, M a is a simply connected cohomology projective plane over complex numbers, quaternions or Cayley numbers, respectively. Each of the manifolds of above type is called a manifold like a projective plane. This classification turned up in the 1987 paper of Brehm and Ktihnel [4] on combinatorial manifolds with few vertices. Specifically, they proved that if M is a combinatorial d-manifold on n vertices with n < 3[d/2] + 3 then I M 1~ Sa; and ifn = 3(a/2) + 3 and [ M I~ ,ca then