Discrete d-pseudomanifolds with at most 2d+6 vertices receive a complete classification; flag normal d-pseudomanifolds in this range are simplicial d-spheres, while at 2d+7 vertices for d>=3 they are either spheres or suspensions of RP^2.
Knill: Coloring discrete manifolds, arXiv preprint arXiv:2106.14374, (2021)
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A Complete Classification of Discrete $d$-Pseudomanifolds with at Most $2d+7$ Vertices
Discrete d-pseudomanifolds with at most 2d+6 vertices receive a complete classification; flag normal d-pseudomanifolds in this range are simplicial d-spheres, while at 2d+7 vertices for d>=3 they are either spheres or suspensions of RP^2.