Triangulated Surfaces and Higher-Dimensional Manifolds - TU Berlin

Triangulated Surfaces and Higher-Dimensional Manifolds
Frank H. Lutz
(TU Berlin)

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Topology: orientable, genus g = 3
Combinatorics:
Geometry:
vertex-minimal, n = 10 vertices,
irreducible
coordinate-minimal,
general position

Combinatorics:
triangulated
2-manifold M :
f-vector f = (n, f 1 , f 2 ).
double counting 3f 2 = 2f 1
Euler characteristic χ(M) = n − f 1 + f 2
=⇒ f = (n, 3n − 3χ(M), 2n − 2χ(M))
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Combinatorics:
triangulated
2-manifold M :
f-vector f = (n, f 1 , f 2 ).
double counting 3f 2 = 2f 1
Euler characteristic χ(M) = n − f 1 + f 2
=⇒ f = (n, 3n − 3χ(M), 2n − 2χ(M))
number of edges: f 1 ≤ ( )
n
2
i.e. n 2 − 7n + 6χ(M) ≥ 0

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PROPOSITION HEAWOOD (1890)
n ≥
⌈
1
2 (7 + √ 49 − 24χ(M))⌉

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PROPOSITION HEAWOOD (1890)
n ≥
⌈
1
2 (7 + √ 49 − 24χ(M))⌉
THEOREM RINGEL (1955)/JUNGERMAN & RINGEL (1980)
For every surface M, with the exception of
M(2, +), M(2, −), and M(3, −),
there is a triangulation with n vertices
iff
⌈
n ≥ 1
2 (7 + √ 49 − 24χ(M))⌉
.
(In the exceptional cases one vertex has to be added.)

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Examples:
χ(S 2 ) = 2: n ≥ 4
χ(RP 2 ) = 1: n ≥ 6
χ(T 2 ) = 0: n ≥ 7
1
1
7 3 4 7
3 4
2
1
6
1
2
4
3
5
2 1
3
6
2 5 2
7 3
4
7

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Enumeration/Realization:
7 3 4 7
[ 1, 2, 3 ], [ 1, 2, 4 ],
[ 1, 3, 7 ], [ 1, 4, 5 ],
[ 1, 5, 6 ], [ 1, 6, 7 ],
[ 2, 3, 6 ], [ 2, 4, 7 ],
[ 2, 5, 6 ], [ 2, 5, 7 ],
[ 3, 4, 5 ], [ 3, 4, 6 ],
[ 3, 5, 7 ], [ 4, 6, 7 ]
1
1
6
2 5 2
7 3
4
7
Möbius’ torus (1861)

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Enumeration/Realization:
7
Coordinates:
[ 1, 2, 3 ], [ 1, 2, 4 ],
[ 1, 3, 7 ], [ 1, 4, 5 ],
[ 1, 5, 6 ], [ 1, 6, 7 ],
[ 2, 3, 6 ], [ 2, 4, 7 ],
[ 2, 5, 6 ], [ 2, 5, 7 ],
[ 3, 4, 5 ], [ 3, 4, 6 ],
[ 3, 5, 7 ], [ 4, 6, 7 ]
1: (3,-3,0)
2: (-3,3,0)
3: (-3,-3,1)
4: (3,3,1)
5: (-1,-2,3)
6: (1,2,3)
7: (0,0,15)
5
6
1
3
4
2
Császár’s torus (1949)

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Enumeration schemes:
1. Generation from irreducible triangulations.
2. Lexicographic enumeration.
3. Strongly connected enumeration.

----------------------------------------------
Enumeration schemes:
1. Generation from irreducible triangulations.
2. Lexicographic enumeration.
3. Strongly connected enumeration.

[Royle, 2001]: n Types
4 1
5 1
6 2
7 5
8 14
9 50
10 233
11 1.249
12 7.595
13 49.566
14 339.722
15 2.406.841
16 17.490.241
17 129.664.753
18 977.526.957
19 7.475.907.149
20 57.896.349.553
21 453.382.272.049
22 3.585.853.662.949
23 28.615.703.421.545
Triangulated
2-spheres
with
4 ≤ n ≤ 23
vertices
----------------------------------------------

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edge contraction ↔ vertex split
A triangulation is irreducible if there is no contractible edge.

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THEOREM BARNETTE & EDELSON (1989)
All 2-manifolds have finitely many
irreducible triangulations.

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THEOREM BARNETTE & EDELSON (1989)
All 2-manifolds have finitely many
irreducible triangulations.
THEOREM NAKAMOTO & OTA (1995)
Irreducible triangulations of a closed surface of genus g
have at most O(g) vertices.

----------------------------------------------
Numbers of irreducible triangulations:
2-sphere: 1 [Steinitz, 1922]
2-torus: 21 [Grünbaum, 1970, Lavrenchenko, 1984]
M(2, +): 396.784 [Sulanke, 2005]
M(3, +): ?
RP 2 2 [Barnette, 1982]
Klein bottle: 29 [Sulanke, 2004]
M(3, −): 9.708 [Sulanke, 2005]
M(4, −): 6.297.982 [Sulanke, 2005]
M(5, −): ?

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Enumeration schemes:
1. Generation from irreducible triangulations.
2. Lexicographic enumeration.
3. Strongly connected enumeration.

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Basic property:
I :
Every edge lies in exactly two triangles.

----------------------------------------------
Ex.: n = 5 triangle-edge incidence matrix
123
124
125
134
135
145
234
235
245
345
⎛
⎜
⎝
12 13 14 15 23 24 25 34 35 45
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
⎞
⎟
⎠

Ex.: n = 5 triangle-edge incidence matrix
123
124
125
134
135
145
234
235
245
345
⎛
⎜
⎝
12 13 14 15 23 24 25 34 35 45
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
2 2 2 0 2 2 2 2 2 2
⎞
⎟
⎠
----------------------------------------------

----------------------------------------------
Ex.: n = 5
123
124
125
134
135
145
234
235
245
345
⎛
⎜
⎝
12 13 14 15 23 24 25 34 35 45
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
⎞
⎟
⎠

Ex.: n = 5 backtracking!
123
124
125
134
135
145
234
235
245
345
⎛
⎜
⎝
12 13 14 15 23 24 25 34 35 45
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
⎞
⎟
⎠
1 1 0 0 1 0 0 0 0 0
----------------------------------------------

Ex.: n = 5 backtracking!
123
124
125
134
135
145
234
235
245
345
⎛
⎜
⎝
12 13 14 15 23 24 25 34 35 45
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
⎞
⎟
⎠
2 1 1 0 1 1 0 0 0 0
----------------------------------------------

Ex.: n = 5 backtracking!
123
124
125
134
135
145
234
235
245
345
⎛
⎜
⎝
12 13 14 15 23 24 25 34 35 45
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
⎞
⎟
⎠
3 1 1 1 1 1 1 0 0 0
----------------------------------------------

Ex.: n = 5 backtracking!
123
124
125
134
135
145
234
235
245
345
⎛
⎜
⎝
12 13 14 15 23 24 25 34 35 45
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
⎞
⎟
⎠
2 2 2 0 1 1 0 1 0 0
----------------------------------------------

Ex.: n = 5 backtracking!
123
124
125
134
135
145
234
235
245
345
⎛
⎜
⎝
12 13 14 15 23 24 25 34 35 45
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
⎞
⎟
⎠
lexicographic
enumeration!
2 2 2 0 1 1 0 1 0 0
----------------------------------------------

Pseudomanifold with isolated singularities:
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Pseudomanifold with isolated singularities:
----------------------------------------------

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Pseudomanifold with isolated singularities:
II :
The link of a vertex should be one circle.

Several components:
----------------------------------------------

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Several components:
III :
The surface should be connected.

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Triangulated surfaces with up to 10 vertices [L. 2003]:
n Surface Types n Surface Types n Surface Types
4 S 2 1 8 S 2 14 10 S 2 233
T 2 7 T 2 2109
5 S 2 1 M(2, +) 865
RP 2 16 M(3, +) 20
6 S 2 2 K 2 6
RP 2 1210
RP 2 1 9 S 2 50 K 2 4462
T 2 112 M(3, −) 11784
7 S 2 5 M(4, −) 13657
T 2 1 RP 2 134 M(5, −) 7050
K 2 187 M(6, −) 1022
RP 2 3 M(3, −) 133 M(7, −) 14
M(4, −) 37
M(5, −) 2

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Isomorphism free lexicographic enumeration:
Triangulated surfaces with 11 and 12 vertices
[Sulanke & L., 2006].

Realization:
THEOREM STEINITZ (1922)
Every triangulated 2-sphere is realizable
as the boundary of a simplicial 3-polytope.
CONJECTURE DUKE/GRÜNBAUM (1970/1973)
Every triangulated 2-torus is realizable in R 3 .
THEOREM BOKOWSKI & GUEDES DE OLIVEIRA (2000)
There is a non-realizable 12-vertex triangulation
of the orientable surface of genus 6.
----------------------------------------------

Realization:
THEOREM STEINITZ (1922)
Every triangulated 2-sphere is realizable
as the boundary of a simplicial 3-polytope.
CONJECTURE DUKE/GRÜNBAUM (1970/1973)
Every triangulated 2-torus is realizable in R 3 .
THEOREM SCHEWE (2006)
For every surface of genus g ≥ 5
there is a non-realizable triangulation.
----------------------------------------------

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Realization of vertex-minimal triangulations:
g n min Types Realizable?
0 4 1 yes
1 7 1 yes [Császár, 1949]
2 10 865 [L., 2003] yes [Bokowski & L., 2005]
3 10 20 [L., 2003] yes [Hougardy, L., Zelke, 2006]
4 11 821 [L., 2005] yes [Hougardy, L., Zelke, 2006]
5 12 751.593 [Sulanke, 2005] yes/no [Hougardy, L., Zelke, 2006],
[Schewe, 2006]
6 12 59 [Bokowski, 1996] no [Bokowski & Guedes de Oliveira,
2000], [Schewe, 2006]

Realization heuristics:
----------------------------------------------

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Realization heuristics:
random realization and recycling of coordinates: 864 L. (2005)
32768 3

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Realization heuristics:
random realization and recycling of coordinates: 864 L. (2005)
32768 3
geometric intuition: 1 Bokowski (2005)

small coordinates: Hougardy, L., Zelke (2005)
----------------------------------------------

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THEOREM HOUGARDY, L., ZELKE (2005)
All 865 triangulations of the orientable surface
of genus 2 with 10 vertices are realizable in the (4 ×4×4)-cube.

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THEOREM HOUGARDY, L., ZELKE (2006)
At least 17 of the 20 triangulations of the orientable surface
of genus 3 with 10 vertices are realizable in the (5 ×5×5)-cube.

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THEOREM HOUGARDY, L., ZELKE (2006)
At least 17 of the 20 triangulations of the orientable surface
of genus 3 with 10 vertices are realizable in the (5 ×5×5)-cube.
THEOREM HOUGARDY, L., ZELKE (2005)
All 20 triangulations of the orientable surface
of genus 3 with 10 vertices are realizable.

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intersection edge functional: Hougardy, L., Zelke (2005)
E
f
a
c
u
w
b
v
d
e

Example:
----------------------------------------------

----------------------------------------------
g = 4, n = 11:
All 821 examples are realizable [HLZ, 2006].
g = 5, n = 12:
At least 15 of the 751.593 examples are realizable [HLZ, 2006]
and at least 3 are not realizable [Schewe, 2006].

g = 4, n = 11:
All 821 examples are realizable [HLZ, 2006].
g = 5, n = 12:
At least 15 of the 751.593 examples are realizable [HLZ, 2006]
and at least 3 are not realizable [Schewe, 2006].
CONJECTURE
Every triangulated surface of genus 1 ≤ g ≤ 4
is realizable in R 3 .
----------------------------------------------

----------------------------------------------
Enumeration schemes:
1. Generation from irreducible triangulations.
2. Lexicographic enumeration.
3. Strongly connected enumeration.

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Date: Thu, 3 Feb 2005 17:23:30 +0100
From: John M Sullivan
Subject: triangulations with low edge degree
Dear Frank,
...
Consider triangulated 3-manifolds with every
edge valence at most 5.
Are there really (as I suspect) only a finite number
of such triangulations, and are the manifolds then
all spherical?
?
Do you perhaps already have such a list?
If not, can we modify your programs to produce one?
Best, John

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Date: Thu, 3 Feb 2005 17:23:30 +0100
From: John M Sullivan
Subject: triangulations with low edge degree
Dear Frank,
...
Consider triangulated 3-manifolds with every
edge valence at most 5.
Are there really (as I suspect) only a finite number
of such triangulations, and are the manifolds then
all spherical?
Do you perhaps already have such a list?
If not, can we modify your programs to produce one?
?
No :-(
Best, John

----------------------------------------------
Date: Thu, 3 Feb 2005 17:23:30 +0100
From: John M Sullivan
Subject: triangulations with low edge degree
Dear Frank,
...
Consider triangulated 3-manifolds with every
edge valence at most 5.
Are there really (as I suspect) only a finite number
of such triangulations, and are the manifolds then
all spherical?
Do you perhaps already have such a list?
If not, can we modify your programs to produce one?
Best, John
?
No :-(
Yes :-)

Vertex-links: 2-spheres with face vector f = (n, f 1 , f 2 ).
----------------------------------------------

----------------------------------------------
Vertex-links: 2-spheres with face vector f = (n, f 1 , f 2 ).
By Euler’s equation n − f 1 + f 2 = 2
and double counting 2f 1 = 3f 2
=⇒ f = (n, 3n − 6, 2n − 4)

----------------------------------------------
Vertex-links: 2-spheres with face vector f = (n, f 1 , f 2 ).
By Euler’s equation n − f 1 + f 2 = 2
and double counting 2f 1 = 3f 2
=⇒ f = (n, 3n − 6, 2n − 4)
Vertex-degree ≤ 5: 2f 1 = ∑ deg ≤ 5n.
Thus n ≤ 12

Enumeration (of vertex-links):
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Enumeration (of vertex-links):
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Enumeration (of vertex-links):
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Enumeration (of vertex-links):
----------------------------------------------

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Enumeration (of vertex-links):
. . . . . .

----------------------------------------------
Enumeration (of vertex-links):
. . . . . .

----------------------------------------------
Enumeration (of vertex-links):
. . . . . .
. . . . . .

----------------------------------------------
Enumeration (of vertex-links):
. . . . . .
. . . . . .

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Enumeration (of vertex-links):
. . . . . .
. . . . . .

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Enumeration (of vertex-links):
. . . . . .
strongly connected!
. . . . . .

Vertex-links:
----------------------------------------------

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Combinatorial types of 3-manifolds with edge valence at most 5:
n Types n Types n Types n Types n Types
5 1 19 230 33 115 47 15 62 2
6 2 20 275 34 93 48 10 63 2
7 5 21 240 35 76 49 15 64 3
8 7 22 240 36 72 50 6 65 7
9 16 23 214 37 55 51 9 69 1
10 29 24 250 38 46 52 8 72 1
11 37 25 220 39 37 53 11 73 1
12 61 26 205 40 44 54 18 76 4
13 77 27 191 41 54 55 2 80 2
14 111 28 194 42 45 56 4 87 1
15 134 29 174 43 33 57 3 98 1
16 184 30 156 44 17 58 5 120 1
17 202 31 115 45 23 60 3
18 233 32 122 46 16 61 1

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THEOREM L. & SULLIVAN (2005)
There are exactly 4787 combinatorially distinct
triangulated 3-manifolds with edge valence at most five:
4761 × S 3 ,
22 × RP 3 ,
2 × L(3, 1),
1 × L(4, 1),
1 × S 3 /Q.
THEOREM MATVEEV & SHEVCHISHIN (2005)
Let M be a triangulated 3-manifold such that every
edge has valence at most five.
Then M is spherical and has at most 600 tetrahedra.

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THEOREM L. & SULLIVAN (2006)
There are exactly 41 distinct 3-dimensional combinatorial
pseudomanifolds with edge valence at most five.
They all are of type
susp(RP 2 ).

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THEOREM L. & SULLIVAN (2006)
There are exactly 41 distinct 3-dimensional combinatorial
pseudomanifolds with edge valence at most five.
They all are of type
susp(RP 2 ).
THEOREM L. & SULLIVAN (2005)
Let M be a triangulated d-manifold such that every
codimension-two face has valence at most four.
Then M is the join product of boundaries of simplices.

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Recognition:
Algorithms for the recognition of the 3-sphere:
[Rubinstein, 1992], [Thompson, 1994], [S. Matveev, 1995], . . .

----------------------------------------------
Recognition:
Algorithms for the recognition of the 3-sphere:
[Rubinstein, 1992], [Thompson, 1994], [S. Matveev, 1995], . . .
. . . are exponential, difficult to implement.

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Recognition:
Algorithms for the recognition of the 3-sphere:
[Rubinstein, 1992], [Thompson, 1994], [S. Matveev, 1995], . . .
. . . are exponential, difficult to implement.
Heuristics for the recognition of the 3-sphere:
[Björner & L., 1997]
. . . simulated annealing with bistellar flips.

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Bistellar Flips/Pachner moves!
2d:
3d:

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THEOREM CSORBA & L. (2005)
Hom(C 6 , K 5 ) ∼ = (S 2 × S 2 ) #29 .
f = (1920, 30780, 104520, 126000,50400)
=⇒ f = (33, 379, 1786, 2300, 920)
THEOREM CSORBA & L. (2005)
Hom(C 5 , K 5 ) ∼ = V 4,2
∼ = S 2 × S 3 .
f = (1020, 25770, 143900, 307950,283200,94400)
=⇒ f = (12, 66, 220, 390, 336, 112)

BUT. . .
. . . there is a 3-sphere with no flips (16 vertices)
[Dougherty, Faber, and Murphy, 2004]
. . . there are non-shellable/non-constructible 3-spheres
with n ≥ 13 vertices [L., 2004]
. . . there are worse examples [King, 2004]
. . . there is no algorithm to recognize 4-manifolds [Markov, 1958]
. . . there is no algorithm to recognize d-spheres for d ≥ 5
[Novikov, 1962]
. . . there are non-PL spheres [Edwards, 1975], [Cannon, 1979]
. . . there are non-PL d-spheres with d + 13 vertices for d ≥ 5
[Björner & L., 2000]
----------------------------------------------

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THEOREM KING (2004)
From any triangulation of S 3 , one can obtain
an edge contractible triangulation of S 3
by a sequence of at most 2 401f2 3
successive expansions.

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THEOREM KING (2004)
From any triangulation of S 3 , one can obtain
an edge contractible triangulation of S 3
by a sequence of at most 2 401f2 3
successive expansions.
THEOREM LICKORISH (1991), ZIEGLER (1996), KING (2004)
Every 3-manifold has infinitely many
irreducible triangulations.

Why don’t we run into bad examples?
----------------------------------------------

----------------------------------------------
Why don’t we run into bad examples?
Where do we get our examples from?

----------------------------------------------
Why don’t we run into bad examples?
Where do we get our examples from?
1. Enumeration: bad examples need more space.
2. Combinatorial constructions: mostly harmless.
3. Topological constructions: we usually know the outcome in advance.

Construction of a non-realizable 3-ball with 16 vertices:
----------------------------------------------

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Construction of a non-realizable 3-ball with 16 vertices:
15
1
3
7
12
13
10
4 6
5
9
8
11
14
2
non-realizable [non-constructible/non-shellable] 3-ball: (16, 75, 106, 46)
[non-constructible/non-shellable 3-sphere: (17, 91, 148, 74)]

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Construction of a non-realizable 3-ball with 12 vertices:
1
3
10
7
4 6
5
8
11
12
9
2
non-realizable [non-constructible/non-shellable] 3-ball: (12, 58, 84, 37)
[non-constructible/non-shellable 3-sphere: (13, 69, 112, 56)]

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COROLLARY L. (2003)
There are non-realizable [non-constructible]
d-balls with d + 9 vertices and 37 facets for d ≥ 3.

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COROLLARY L. (2003)
There are non-realizable [non-constructible]
d-balls with d + 9 vertices and 37 facets for d ≥ 3.
COROLLARY L. (2003)
There are non-constructible/non-shellable
d-spheres with d + 10 vertices for d ≥ 3.