A short proof of the uniqueness of Kühnel's 9-vertex complex projective plane (Q2726407)

The following theorem: Up to simplicial isomorphism there is a unique 4-manifold \((CP^2)\) with 9 vertices which satisfies complementarity (in any decomposition of the vertices into two subsets, exactly 1 of the subsets spans a simplex), has been proved by \textit{W. Kühnel} and \textit{G. Lassmann} [J. Comb. Theory, Ser. A 35, 173-184 (1983; Zbl 0526.52008)], \textit{P. Arnoux} and \textit{A. Marin} [Mem. Fac. Sci., Kyushu Univ., Ser. A 45, No. 2, 167-244 (1991; Zbl 0753.52002)] and others (including the present authors in an earlier paper). In this paper another (combinatorial) proof of the theorem is presented.