Small Triangulations of Simply Connected 4-Manifolds

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.