Small Triangulations of Simply Connected 4-Manifolds

We present small triangulations of all connected sums of ${\mathbb{C}\mathbb{P}}^2$ 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.