Triangulations of Normal 3-Pseudomanifolds on 9 Vertices |
| Tair Akhmejanov |
| Boston College |
| Cornell REU 2011 |
The complexes are denoted by the number of vertices that have links homeomorphic to the 2-sphere, real projective plane, 2-torus, and klein bottle, respectively.
Each row corresponds to a distinct homeomorphism class. There are 58 rows.
Notice that there is only one instance where the invariant of quantities of singularity types does not fully characterize the homeomorphism class.
In this case, the group splits into two distinct homoemorphism classes, 3,4,0,2_a and 3,4,0,2_b.
The fundamental group for every space is trivial.
| Complex |
Number of Triangulations |
Minimum g2 |
h3-h1=2 χ (Δ) |
H2 |
H3 |
| 8,0,1,0† |
131 |
6* |
2 |
Ζ |
Ζ |
| 8,0,0,1 |
77 |
6* |
2 |
Ζ2 |
|
| 7,2,0,0† |
4058 |
3* |
2 |
Ζ2 |
|
| 7,0,2,0† |
153 |
6* |
4 |
Ζ2 |
Ζ |
| 7,0,0,2 |
171 |
6* |
4 |
Ζ ⊕ Ζ2 |
|
| 6,2,0,1 |
1003 |
6* |
4 |
Ζ ⊕ Ζ2 |
|
| 6,0,3,0 |
1 |
10 |
6 |
Ζ3 |
Ζ |
| 6,0,1,2 |
1 |
10 |
6 |
Ζ2 ⊕ Ζ2 |
|
| 6,0,0,3 |
9 |
9 |
6 |
Ζ2 ⊕ Ζ2 |
|
| 5,4,0,0 |
277 |
6 |
4 |
Ζ ⊕ Ζ2 |
|
| 5,2,1,1 |
634 |
7 |
6 |
Ζ2 ⊕ Ζ2 |
|
| 5,2,0,2 |
25 |
9 |
6 |
Ζ2 ⊕ Ζ2 |
|
| 5,0,2,2 |
1 |
10 |
8 |
Ζ3 ⊕ Ζ2 |
|
| 5,0,0,4 |
7 |
8* |
8 |
Ζ3 ⊕ Ζ2 |
|
| 4,4,1,0† |
1217 |
6* |
6 |
Ζ2 ⊕ Ζ2 |
|
| 4,4,0,1 |
8 |
9 |
6 |
Ζ2 ⊕ Ζ2 |
|
| 4,2,2,1 |
37 |
9 |
8 |
Ζ3 ⊕ Ζ2 |
|
| 4,2,1,2 |
7 |
10 |
8 |
Ζ3 ⊕ Ζ2 |
|
| 4,2,0,3 |
84 |
8* |
8 |
Ζ3 ⊕ Ζ2 |
|
| 4,0,5,0 |
1 |
10* |
10 |
Ζ5 |
Ζ |
| 4,0,1,4 |
1 |
10* |
10 |
Ζ4 ⊕ Ζ2 |
|
| 3,6,0,0 |
1 |
10 |
6 |
Ζ2 ⊕ Ζ2 |
|
| 3,4,2,0 |
40 |
8 |
8 |
Ζ3 ⊕ Ζ2 |
|
| 3,4,0,2_a |
296 |
7* |
8 |
Ζ3 ⊕ Ζ2 |
|
| 3,4,0,2_b |
25 |
8 |
8 |
Ζ3 ⊕ Ζ2 |
|
| 3,2,2,2 |
23 |
9* |
10 |
Ζ4 ⊕ Ζ2 |
|
| 3,2,1,3 |
8 |
9* |
10 |
Ζ4 ⊕ Ζ2 |
|
| 2,6,0,1 |
290 |
7 |
8 |
Ζ3 ⊕ Ζ2 |
|
| 2,4,2,1 |
14 |
9 |
10 |
Ζ4 ⊕ Ζ2 |
|
| 2,4,1,2 |
40 |
8 |
10 |
Ζ4 ⊕ Ζ2 |
|
| 2,2,2,3 |
3 |
9 |
12 |
Ζ5 ⊕ Ζ2 |
|
| 2,2,1,4 |
2 |
10 |
12 |
Ζ5 ⊕ Ζ2 |
|
| 2,2,0,5 |
1 |
10 |
12 |
Ζ5 ⊕ Ζ2 |
|
| 1,8,0,0† |
119 |
6 |
8 |
Ζ3 ⊕ Ζ2 |
|
| 1,6,1,1 |
5 |
9 |
10 |
Ζ4 ⊕ Ζ2 |
|
| 1,4,4,0 |
4 |
10 |
12 |
Ζ5 ⊕ Ζ2 |
|
| 1,4,2,2 |
14 |
9 |
12 |
Ζ5 ⊕ Ζ2 |
|
| 1,4,1,3 |
10 |
9 |
12 |
Ζ5 ⊕ Ζ2 |
|
| 1,4,0,4 |
5 |
10 |
12 |
Ζ5 ⊕ Ζ2 |
|
| 1,2,4,2 |
1 |
10 |
14 |
Ζ6 ⊕ Ζ2 |
|
| 1,2,3,3 |
1 |
10 |
14 |
Ζ6 ⊕ Ζ2 |
|
| 1,2,2,4 |
2 |
10 |
14 |
Ζ6 ⊕ Ζ2 |
|
| 1,2,0,6 |
1 |
10 |
14 |
Ζ6 ⊕ Ζ2 |
|
| 1,0,8,0† |
12 |
6* |
16 |
Ζ8 |
Ζ |
| 1,0,4,4 |
1 |
10 |
16 |
Ζ7 ⊕ Ζ2 |
|
| 0,8,1,0 |
1 |
10 |
10 |
Ζ4 ⊕ Ζ2 |
|
| 0,6,0,3 |
4 |
10 |
12 |
Ζ5 ⊕ Ζ2 |
|
| 0,4,4,1 |
1 |
10 |
14 |
Ζ6 ⊕ Ζ2 |
|
| 0,4,3,2 |
2 |
10 |
14 |
Ζ6 ⊕ Ζ2 |
|
| 0,4,0,5 |
1 |
10 |
14 |
Ζ6 ⊕ Ζ2 |
|
| 0,2,4,3 |
2 |
10 |
16 |
Ζ7 ⊕ Ζ2 |
|
| 0,2,3,4 |
1 |
10 |
16 |
Ζ7 ⊕ Ζ2 |
|
| 0,2,2,5 |
3 |
10 |
16 |
Ζ7 ⊕ Ζ2 |
|
| 0,0,9,0 |
2 |
10 |
18 |
Ζ9 |
Ζ |
| 0,0,5,4 |
2 |
10 |
18 |
Ζ8 ⊕ Ζ2 |
|
| 0,0,3,6 |
1 |
10 |
18 |
Ζ8 ⊕ Ζ2 |
|
| 0,0,1,8 |
1 |
10 |
18 |
Ζ8 ⊕ Ζ2 |
|
| 0,0,0,9 |
1 |
10 |
18 |
Ζ8 ⊕ Ζ2 |
|
*full characterization of h-vectors.
† denotes spaces previously found by B. Datta and N. Nilakantan in Three dimensional pseudomanifolds on eight vertices. For these spaces, * denotes previously known characterization of h-vectors.
The h-vectors are of the following form.
\(h_4=-\tilde{\chi}\)
\(h_3-h_1=2\chi(\Delta)\) (fourth column above)
\(h_2\leq\binom{h_1+1}{2}\)
\(h_1\geq 4\) (if this is a space from Datta, Nilakantan paper), \(h_1 \geq 5\) (if this is a new space)
\(g_2\geq\)(third column above)
Edited C program
The original program can be found at The Manifold Page.