- a= number of vertex links homeomorphic to the sphere
- b= number of vertex links homeomorphic to the real projective plane
- c= number of vertex links homeomorphic to the torus
- d= number of vertex links homeomorphic to the Klein bottle
- e= number of vertex links homeomorphic to the genus three nonorientable surface
- f= number of vertex links homeomorphic to the genus four nonorientable surface
- g= number of vertex links homeomorphic to the genus five nonorientable surface

- a - For any vertex v of Δ, g
_{2}(Δ) ≥ g_{2}(link v). - b - If n is the number of singular vertices, then g
_{2}≥ 2 χ - ( n-3 choose 3). If n-3 < 3, then the binomial coefficient is interepreted as zero. - b' - Same theorem, with the minimum g
_{2}realized by a complex with 11 vertices. Hence Γ = minG2 - 1. - c - If Δ has 8 singular vertices and m of them are Klein bottles, then g
_{2}≥ 2 χ - 10 + (m/3) - d - If Δ has 8 singular vertices and any of them are real projective planes, then g
_{2}≥ 2 χ - 7 - d' - Same theorem, with the minimum g
_{2}realized by a complex with 11 vertices. Hence Γ = minG2 - 1.

Except where otherwise noted, the f-vectors are characterized through h- and g-vectors by, h_{0}=1, h_{4}=1-χ, h_{3} - h_{1} = 2 χ, h_{1} ≥ f_{0}-4, and Γ ≤ g_{2} ≤ (g_{1} +1 choose 2), where f_{0} is the minimum number of vertices required for a complex with the given singularities.

- The first entry is the minimum number of vertices possible for the given singularities
- 10 indicates that the possible f-vectors for PL-homeomorphic complexes for
*every*complex in the group are the same and equal all possible f-vectors for that particular group of singularities - 10, # indicates that the possible f-vectors of complexes PL-homeomorphic to complex # equals all possible f-vectors for that group of singularities.
- 10, #, β There is no complex with g-vector (5, Γ) for these singularities.
- 9, # indicates that the possible f-vectors of complexes PL-homeomorphic to complex # at http://www.math.cornell.edu/~takhmejanov/pseudoManifolds.html with the same singularities equals all possible f-vectors for that group of singularities.
- 9, #1, α There is no complex with g-vector (4,6) for these singularities.
- 8, N# indicates that the possible f-vectors of complexes PL-homeomorphic to complex N# in "Three-Dimensional Pseudomanifolds on Eight Vertices", B. Datta and N. Nilakantan, Indian J. of Mathematics and Mathematical Sciences, 2008, equals all possible f-vectors for that group of singularities.
- 7, The one-vertex suspension of the six-vertex triangulation of the real projective plane can be used to prove that the f-vectors of the suspension of the real projective plane has the same f-vectors as all complexes with exactly two singular vertices each with link homeomorphic to the real projective plane.
- 5, f-vectors of three-manifolds equal all possible f-vectors of S
^{3}.