Simple crystallizations of 4-manifolds

TL;DR

The paper constructs minimal crystallizations for all simply connected PL 4-manifolds of standard type, including the K3 surface and certain pairs of homeomorphic but non-PL-homeomorphic manifolds, revealing their combinatorial structures.

Contribution

It provides explicit minimal crystallizations for all standard simply connected PL 4-manifolds, including a new crystallization of the K3 surface and proofs of uniqueness for certain cases.

Findings

Minimal crystallization of the K3 surface is presented.

All standard simply connected PL 4-manifolds have minimal crystallizations.

The minimal 8-vertex crystallization of P^2 is unique.

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.