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International Journal of Mathematics and Mathematical Sciences
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International Journal of Mathematics and Mathematical Sciences
/
2008
/
Article
/
Tab 1
/
Research Article
Three-Dimensional Pseudomanifolds on Eight Vertices
Table 1
8-vertex normal 3-pseudomanifolds which are not combinatorial 3-manifolds.
𝑋
𝑓
-vector
(
𝑓
1
,
𝑓
2
,
𝑓
3
)
𝜒
(
𝑋
)
𝑛
𝑠
(
𝑋
)
links of singular vertices
Geometric carriers, Homology
(
𝐻
1
,
𝐻
2
,
𝐻
3
)
𝑁
1
(
2
8
,
5
6
,
2
8
)
8
8
all are
𝑇
|
𝑁
1
|
is simply connected,
𝐻
1
,
𝐻
2
,
𝐻
3
=
0
,
ℤ
8
,
ℤ
𝑁
2
(
2
8
,
4
4
,
2
2
)
2
2
both are
𝑇
|
|
|
𝑁
2
|
|
|
𝑆
=
𝑆
1
×
𝑆
1
𝑁
3
(
2
8
,
4
6
,
2
3
)
3
5
𝑇
,
𝑅
2
,
𝑅
2
,
𝑅
3
,
𝑅
3
𝐻
1
,
𝐻
2
,
𝐻
3
=
0
,
ℤ
2
⊕
ℤ
2
,
0
𝑁
4
(
2
8
,
4
2
,
2
1
)
1
1
𝑇
|
|
|
𝑁
4
|
|
|
=
𝐻
∪
𝐶
(
𝜕
𝐻
)
𝑁
5
(
2
8
,
4
8
,
2
4
)
4
8
all are
𝑅
4
|
|
|
𝑁
5
|
|
|
=
𝐾
3
𝑁
6
,,
,,
,,
all are
𝑅
3
|
|
|
𝑁
6
|
|
|
=
𝐾
3
𝑁
7
(
2
8
,
4
2
,
2
1
)
1
2
both are
𝑅
4
|
|
|
𝑁
7
|
|
|
=
𝑆
ℝ
𝑃
2
𝑁
𝑖
,
8
≤
𝑖
≤
1
5
,,
,,
,,
both are in
𝑅
1
,
…
,
𝑅
4
|
|
|
𝑁
𝑖
|
|
|
=
𝑆
ℝ
𝑃
2
𝑁
𝑖
,
1
6
≤
𝑖
≤
2
4
(
2
7
,
4
0
,
2
0
)
,,
,,
,,
,,
𝑁
𝑖
,
2
5
≤
𝑖
≤
3
1
(
2
6
,
3
8
,
1
9
)
,,
,,
,,
,,
𝑁
𝑖
,
3
2
≤
𝑖
≤
3
5
(
2
5
,
3
6
,
1
8
)
,,
,,
,,
,,
[Here
𝐾
3
is the 3-dimensional Kummer variety,
𝐻
=
𝐷
2
×
𝑆
1
is the solid torus,
𝑆
(
𝑌
)
is the topological suspension of
𝑌
, and
𝑛
𝑠
(
𝑋
)
is the number of singular vertices in
𝑋
.
]