Small Triangulations of Simply Connected 4-Manifolds

TL;DR

This paper constructs minimal triangulations of certain simply connected 4-manifolds, specifically connected sums of complex projective planes and products of spheres, with the smallest known number of 4-simplices.

Contribution

It provides explicit small triangulations for all such manifolds with minimal pentachora count, confirming a conjecture about their optimality.

Findings

Triangulations have 2β₂+2 pentachora, matching the conjectured minimum.

Triangulations are of the standard PL structure.

Results apply to all connected sums of  and S^2 imes S^2.

Abstract

We present small triangulations of all connected sums of $\mathbb{CP}^2$ and $S^2 \times S^2$ with the standard piecewise linear structure. Our triangulations have $2\beta_2+2$ pentachora, where $\beta_2$ is the second Betti number of the manifold. By a conjecture of the authors and, independently, Burke, these triangulations have the smallest possible number of pentachora for their respective topological types.