On a vertex-minimal triangulation of RP

We give three constructions of a vertex-minimal triangulation of 4-dimensional real projective space RP4. The first construction describes a 4-dimensional sphere on 32 vertices, which is a double cover of a triangulated RP4 and has a large amount of symmetry. The second and third constructions illustrate approaches to improving the known number of vertices needed to triangulate n-dimensional real projective space. All three constructions deliver the same combinatorial manifold, which is also the same as the only known 16-vertex triangulation of RP4. We also give a short, simple construction of the 22-point Witt design, which is closely related to the complex we construct.