Simple crystallizations of 4-manifolds

Abstract

Abstract Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination with known results this yields minimal crystallizations of all simply connected PL 4-manifolds of “standard” type, that is, all connected sums of ℂℙ2, S2 × S2, and the K3 surface. In particular, we obtain minimal crystallizations of a pair of homeomorphic but non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that the minimal 8-vertex crystallization of ℂℙ2 is unique and its associated pseudotriangulation is related to the 9-vertex combinatorial triangulation of ℂℙ2 by the minimum of four edge contractions.