Three-Dimensional Pseudomanifolds on Eight Vertices

Table 1

8-vertex normal 3-pseudomanifolds which are not combinatorial 3-manifolds.

𝑋	𝑓-vector (𝑓1,𝑓2,𝑓3)	𝜒(𝑋)	𝑛𝑠(𝑋)	links of singular vertices	Geometric carriers, Homology (𝐻1,𝐻2,𝐻3)
𝑁1	(28,56,28)	8	8	all are 𝑇	|𝑁1| is simply connected, 𝐻^1,𝐻2,𝐻3=0,ℤ^8,ℤ
𝑁2	(28,44,22)	2	2	both are 𝑇	|||𝑁2|||=𝑆𝑆1×𝑆^1
𝑁3	(28,46,23)	3	5	𝑇,𝑅2,𝑅2,𝑅3,𝑅3	𝐻^1,𝐻2,𝐻3=0,ℤ^2⊕ℤ2,0
𝑁4	(28,42,21)	1	1	𝑇	|||𝑁4|||=𝐻∪𝐶(𝜕𝐻)
𝑁5	(28,48,24)	4	8	all are 𝑅4	|||𝑁5|||=𝐾3
𝑁6	,,	,,	,,	all are 𝑅3	|||𝑁6|||=𝐾3
𝑁7	(28,42,21)	1	2	both are 𝑅4	|||𝑁7|||=𝑆ℝ𝑃2
𝑁𝑖, 8≤𝑖≤15	,,	,,	,,	both are in 𝑅^1,…,𝑅4	|||𝑁𝑖|||=𝑆ℝ𝑃2
𝑁𝑖,16≤𝑖≤24	(27,40,20)	,,	,,	,,	,,
𝑁𝑖,25≤𝑖≤31	(26,38,19)	,,	,,	,,	,,
𝑁𝑖,32≤𝑖≤35	(25,36,18)	,,	,,	,,	,,

[Here 𝐾³ is the 3-dimensional Kummer variety, 𝐻=𝐷²×𝑆¹ is the solid torus, 𝑆(𝑌) is the topological suspension of 𝑌, and 𝑛_𝑠(𝑋) is the number of singular vertices in 𝑋.]