Simple crystallizations of 4-manifolds

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 ${\mathbb{CP}}^2$, $S^2\times S^2$, 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 ${\mathbb{CP}}^2$ is unique and its associated pseudotriangulation is related to the 9-vertex combinatorial triangulation of ${\mathbb{CP}}^2$ by the minimum of four edge contractions.

23 pages, 7 figures. Minor update, replacement of Figure 7. To appear in Advances in Geometry