SIMPLICIAL 2-SPHERES OBTAINED FROM NON-SINGULARCOMPLETE FANSYUSUKE SUYAMAAbstract. We prove that a simplicial 2-sphere satisfying a certain conditionis the underlying simplicial complex of a 3-dimensional non-singular complete
fan. In particular, this implies that any simplicial 2-sphere with ≤ 18 vertices
is the underlying simplicial complex of such a fan.

1. Introduction
A rational strongly convex polyhedral cone in Rn is a cone σ spanned by finitely
many vectors in Zn which does not contain any non-zero linear subspace of Rn .
A fan in Rn is a non-empty collection ∆ of such cones satisfying the following
conditions:(1) If σ ∈ ∆, then each face of σ is in ∆;(2) if σ, τ ∈ ∆, then σ ∩ τ is a face of each.
A fan ∆ is non-singular
if any cone in ∆ is spanned by a part of a basis of Zn , and∪complete if σ∈∆ σ = Rn .
A toric variety of complex dimension n is a normal algebraic variety X over C
containing (C∗ )n as an open dense subset, such that the natural action of (C∗ )n on
itself extends to an action on X. The category of toric varieties is equivalent to thecategory of fans (see [3]). A toric variety is smooth if and only if the corresponding
fan is non-singular, and compact if and only if the fan is complete.
Given a non-singular fan ∆ with m edges spanned by v1 , . . . , vm ∈ Zn , we define
its underlying simplicial complex as{I ⊂ {1, . . . , m} | {vi | i ∈ I} spans a cone in ∆}.
The underlying simplicial complex of an n-dimensional complete fan is a simplicial(n − 1)-sphere, that is, a triangulation of the (n − 1)-sphere.
For n ≥ 4, a simplicial (n − 1)-sphere is not always the underlying simplicial
complex of an n-dimensional non-singular complete fan (see [2, Corollary 1.23]).
On the other hand, successive equivariant blow-ups of CP 2 produce non-singularcomplete fans whose underlying simplicial complexes are all simplicial 1-spheres.
We consider the following problem:
Problem 1. Is any simplicial 2-sphere the underlying simplicial complex of a 3dimensional non-singular complete fan?
Date: September 12, 2014.
2010 Mathematics Subject Classification. Primary 52B05; Secondary 14M25.
Key words and phrases. triangulation, fan, toric topology.
1

2YUSUKE SUYAMA

No counterexamples to Problem 1 are currently known. In this paper we givea partial aﬃrmative answer to Problem 1. The degree of a vertex of a simplicial
2-sphere is the number of incident edges.
Theorem 2. Let K be a simplicial 2-sphere with mK vertices. We denote thenumber of vertices of K with degree k by pK (k). If pK (3) + pK (4) + 18 ≥ mK , then
K is the underlying simplicial complex of a 3-dimensional non-singular complete
fan. In particular, if mK ≤ 18, then K is the underlying simplicial complex of sucha fan.
The proof is done by reducing a given simplicial 2-sphere to another one in acollection of certain simplicial 2-spheres with minimum degree 5. For each such
simplicial 2-sphere, we use a computer to find a non-singular complete fan whose
underlying simplicial complex is the simplicial 2-sphere.
The structure of the paper is as follows: In Section 2, we give a complete list of
the simplicial 2-spheres with minimum degree 5 up to 18 vertices. In Section 3, weprove Theorem 2.
2. The simplicial 2-spheres with minimum degree 5 up to 18 vertices
G. Brinkmann and B. D. McKay calculated the number of combinatorially different simplicial 2-spheres with minimum degree 5 [1]:vertices
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

simplicial 2-spheres simplicial 2-spheres with min. deg. 5
10
10
2
0
50
14
0
50
0
233
0
1,249
07,595
149,566
0339,722
12,406,841
117,490,241
3
129,664,753
4977,526,957
12Table 1. The number of simplicial 2-spheres.

Remark 3. An n-dimensional small cover of a simple n-polytope is a closed nmanifold M with a locally standard (Z2 )n -action such that the orbit space M/(Z2 )nis the simple polytope. It follows from Steinitz’s theorem that any simplicial 2sphere is the boundary of a simplicial 3-polytope. The dual of the simplicial 3polytope is a simple 3-polytope P . It follows from the four color theorem that Pis the orbit space of a 3-dimensional small cover. A 3-dimensional small cover ofP admits a hyperbolic structure if and only if P has no triangles or squares as

SIMPLICIAL 2-SPHERES OBTAINED FROM NON-SINGULAR COMPLETE FANS

3facets, that is, the original simplicial 2-sphere has no vertices with degree 3 or 4 [2].
Table 1 shows that “most” 3-dimensional small covers do not admit any hyperbolicstructure.
We give a complete list of such
∏ simplicial 2-spheres up to 18 vertices (see Tables
2 and 3). They are labeled as k≥5 k p(k) . If there are more than one simplicial
2-spheres with the same label, then we add (i), (ii), ... to the label. Letters and ?
on vertices in Tables 2 and 3 are used in Section 3.
For each simplicial 2-sphere, we consider the subcomplex consisting of the vertices with degree greater than or equal to 6 and the edges whose both endpoints
have degree greater than or equal to 6 (red vertices and edges in Tables 2 and 3).
These show that all simplicial 2-spheres in Tables 2 and 3 are distinct except 512 66(ii) and 512 66 (iii) (they have the same subcomplex).
Since the subcomplexes of 512 66 (ii) and 512 66 (iii) are cycles, each cycle determines two subcomplexes surrounded by the cycle (see Figures 1 and 2). These areclearly distinct.

Figure 1. Subcomplexes of 512 66 (ii).

Figure 2. Subcomplexes of 512 66 (iii).
So all simplicial 2-spheres in Tables 2 and 3 are distinct.
For m ≤ 18, the number of the simplicial 2-spheres with m vertices in Tables 2and 3 agrees with the number in Table 1. So this is a complete list of the simplicial
2-spheres with minimum degree 5 up to 18 vertices.
3. Proof of the Theorem 2Let K be a simplicial 2-sphere with mK vertices.
Lemma 4. If K is the underlying simplicial complex of a non-singular completefan, then a simplicial 2-sphere obtained from K by an operation (i), (ii) or Ck (k ≥5) is also the underlying simplicial complex of such a fan (see Figure 3).

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YUSUKE SUYAMA

512

512 62512 63514 72512 64 (i)512 64 (ii)

513 63 71

512 65 (i)512 65 (ii)

512 65 (iii)
Table 2. The simplicial 2-spheres with minimum degree 5 up to17 vertices.

SIMPLICIAL 2-SPHERES OBTAINED FROM NON-SINGULAR COMPLETE FANS

516 82514 62 72 (i)514 62 72 (ii)

514 62 72 (iii)513 64 71 (i)513 64 71 (ii)

512 66 (i)512 66 (ii)

512 66 (iii)

512 66 (iv)
512 66 (v)
512 66 (vi)Table 3. The simplicial 2-spheres with minimum degree 5 and 18 vertices.

5

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YUSUKE SUYAMA

(i)

(ii)

Ck−→−→−→For the operation Ck , the degree of the vertex in the center of the diagram is k.

Figure 3. Operations (i), (ii) and Ck .
Proof. Suppose that the three vertices of a 2-face of K correspond to edge vectors
v1 , v2 , v3 ∈ Z3 . Then we have det(v1 , v2 , v3 ) = 1. We assign v1 + v2 + v3 to thenew vertex made by the operation (i). The corresponding fan is non-singular andcomplete since det(v1 , v2 , v1 + v2 + v3 ) = det(v2 , v3 , v1 + v2 + v3 ) = det(v3 , v1 , v1 +v2 + v3 ) = 1. Thus the lemma holds for an operation (i) (see Figure 4).

(i)−→Figure 4. An operation (i).
Suppose that K contains a subcomplex in Figure 5 and the vertices correspondto edge vectors v1 , v2 , v3 , v4 ∈ Z3 as in Figure 5. Then we have det(v1 , v2 , v3 ) =det(v4 , v3 , v2 ) = 1. We assign v2 + v3 to the new vertex made by the operation(ii). The corresponding fan is non-singular and complete since det(v1 , v2 , v2 + v3 ) =det(v3 , v1 , v2 + v3 ) = det(v2 , v4 , v2 + v3 ) = det(v4 , v3 , v2 + v3 ) = 1. Thus the lemma
holds for an operation (ii).

(ii)−→Figure 5. An operation (ii).
Suppose that K contains a subcomplex in Figure 6 and the vertices correspondto edge vectors v, v1 , . . . , vk ∈ Z3 as in Figure 6. Then we have det(v, vi , vi+1 ) = 1
for any i = 1, . . . , k, where vk+1 = v1 . For each i = 1, . . . , k, we assign v + vi
to the new vertex between v and vi , which is made by the operation Ck . The
corresponding fan is non-singular and complete since det(v, v + vi , v + vi+1 ) =det(vi , v + vi+1 , v + vi ) = det(vi , vi+1 , v + vi+1 ) = 1 for any i = 1, . . . , k. Thus thelemma holds for an operation Ck . This completes the proof.
SIMPLICIAL 2-SPHERES OBTAINED FROM NON-SINGULAR COMPLETE FANS

7Ck−→Figure 6. An operation Ck .
Now we prove Theorem 2 by induction on mK . The tetrahedron is the only
simplicial 2-sphere with 4 vertices, which is the underlying simplicial complex of
the fan of CP 3 . Assume that mK ≥ 5.
(1) The case where there exists a vertex with degree 3. All adjacent vertices
have degree greater than or equal to 4, since, if two vertices with degree 3 areadjacent, then K must be the tetrahedron, which contradicts mK ≥ 5. Thus wecan perform an inverse operation of (i) and we get a simplicial 2-sphere K 0 . We
see that pK 0 (3) + pK 0 (4) ≥ pK (3) + pK (4) − 1. So we have pK 0 (3) + pK 0 (4) + 18 ≥pK (3) + pK (4) + 18 − 1 ≥ mK − 1 = mK 0 . K 0 is the underlying simplicial complexof a non-singular complete fan by the induction hypothesis. Hence K is also theunderlying simplicial complex of such a fan by Lemma 4.
(2) The case where there does not exist a vertex with degree 3 and there exists avertex with degree 4. Since all adjacent vertices have degree greater than or equal
to 4, we can perform an inverse operation of (ii) and we get a simplicial 2-sphereK 0 . We see that pK 0 (3) + pK 0 (4) ≥ pK (3) + pK (4) − 1. The same argument as (1)implies that K is the underlying simplicial complex of a non-singular complete fan.
(3) The case where∑
there does not exist a vertex with degree 3 or 4. The Euler
relation implies that k≥3 (6 − k)pK (k) = 12 (see [3, p.190]). This shows that
K must have a vertex with degree 5. Since mK ≤ pK (3) + pK (4) + 18 = 18 byassumption, K falls into 22 types in Tables 2 and 3.
Suppose that K has a vertex v with degree k ≥ 5 such that any vertex adjacent
to v has degree 5, and any vertex adjacent to a vertex adjacent to v has degree
greater than or equal to 5. Then we can perform an inverse operation of Ck and we
get a simplicial 2-sphere K 0 . Since mK 0 = mK −k < 18 ≤ pK 0 (3)+pK 0 (4)+18, K 0 is
the underlying simplicial complex of a non-singular complete fan by the induction
hypothesis. Hence K is also the underlying simplicial complex of such a fan by
Lemma 4.
Each of 512 , 512 65 (i) and 514 62 72 (i) has such a vertex for k = 5; each of 512 62 ,12 3
5 6 , 512 64 (i), 512 65 (iii) and 513 64 71 (ii) has such a vertex for k = 6; each of
514 72 , 513 63 71 and 514 62 72 (ii) has such a vertex for k = 7; 516 82 has such a vertex
for k = 8 (these vertices are indicated by ? in Tables 2 and 3). So they are theunderlying simplicial complexes of non-singular complete fans.
We show that the rest of simplicial 2-spheres 512 64 (ii), 512 65 (ii), 514 62 72 (iii),
13 4 15 6 7 (i) and 512 66 (i)–(vi) are the underlying simplicial complexes of non-singularcomplete fans with a computer aid. We assign vectors to the vertices as in Table 4.

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YUSUKE SUYAMA

They determine complete fans and it can be checked that all fans are non-singularby calculation.
vertexabcdefghijkl
mnopqrvertexabcdefghijkl
mnopqr

512 64 (ii)
(1, 0, 0)(0, 1, 0)(0, 0, 1)(−1, 2, −1)(0, −1, −1)(1, 0, −1)(1, −1, 0)(1, −1, 1)(−1, 0, 1)(−1, 1, 0)(−1, 1, −1)(0, −2, −1)(1, −1, −1)(0, −1, 1)(0, −1, 0)(0, −2, 1)

512 65 (ii)
(1, 0, 0)(1, 0, 1)(2, −1, 1)(3, 0, −1)(2, 1, −1)(1, 1, 0)(1, −1, 1)(2, 0, −1)(1, 1, −1)(0, 1, 0)(0, 0, 1)(0, −1, 1)(2, −1, 0)(1, 0, −1)(0, 1, −1)(−1, 1, 0)(−1, 0, 0)

514 62 72 (iii), 513 64 71 (i), 512 66 (i)(0, −1, 0)(1, −1, 0)(0, −1, 1)(−1, −1, 1)(−1, −1, 0)(−1, −1, −1)(0, −1, −1)(1, 0, 0)(0, 0, 1)(−1, 0, 1)(−1, 0, −1)(0, 0, −1)(0, 1, −1)(1, 1, 0)(0, 1, 1)(−1, 0, 0)(−1, 1, −1)(0, 1, 0)
12 6
12 6
12 6
12 6
5 6 (ii) 5 6 (iii) 5 6 (iv) 5 6 (v)
512 66 (vi)
(1, 0, 0)(1, 0, 0)(1, 0, 0)(0, −1, 0)(0, −1, 0)(3, 0, −1) (3, 0, −1) (3, 0, −1) (−1, 1, −1) (−1, 0, −1)(2, 1, −1) (2, 1, −1) (2, 1, −1) (0, −2, −1) (0, −2, −1)(1, 1, 0)(1, 1, 0)(1, 1, 0)(1, −1, −1) (1, −1, −1)(3, 0, 1)(1, 0, 1)(1, 0, 1)(0, −1, 1)(0, −1, 1)(3, −1, 1) (3, −1, 1) (2, −1, 1)(−1, 0, 1)(−1, 0, 1)(2, 0, −1) (2, 0, −1) (2, 0, −1)(−1, 1, 0)(−1, 1, 0)(1, 1, −1) (1, 1, −1) (1, 1, −1) (0, −1, −1) (0, −1, −1)(0, 1, 0)(0, 1, 0)(0, 1, 0)(1, 0, −1)(1, 0, −1)(1, 0, 1)(0, 0, 1)(0, 0, 1)(1, −1, 0)(1, −1, 0)(1, −1, 1) (1, −1, 1) (1, −1, 1)(1, −1, 1)(1, −1, 1)(2, −1, 1) (2, −1, 1) (3, −1, 0)(0, 0, 1)(0, 0, 1)(1, 0, −1) (1, 0, −1) (1, 0, −1)(−1, 2, 0)(−1, 2, 2)(−1, 1, 0) (0, 1, −1) (0, 1, −1) (−1, 2, −1) (−2, 2, −1)(0, 0, 1)(−1, 1, 0) (−1, 1, 0)(0, 1, 2)(0, 1, 2)(0, −1, 1) (0, −1, 1) (0, −1, 1)(0, 1, 1)(0, 1, 1)(2, −1, 0) (2, −1, 0) (2, −1, 0) (−1, 2, −2) (−1, 1, −1)(−1, 0, 0) (−1, 0, 0) (−1, 0, 0)(0, 1, 0)(0, 1, 0)
Table 4. Assigning vectors to the vertices.

For example, we show that 514 62 72 (iii) is the underlying simplicial complex of anon-singular complete fan. Vectors in Table 4 determine a 3-dimensional completefan. Its underlying simplicial complex is illustrated in Figure 7, which confirms that

SIMPLICIAL 2-SPHERES OBTAINED FROM NON-SINGULAR COMPLETE FANS

9

there are no overlaps among the 3-dimensional cones. Calculating determinants,
say det(a, b, c) = 1, we see that every cone is non-singular.

Figure 7. 514 62 72 (iii).
Acknowledgement. The author wishes to thank Professor Mikiya Masuda for hisvaluable advice and continuing support, and Professors Hiroshi Sato, Tadao Oda
and Masanori Ishida for useful comments on Problem 1 in the introduction.
References[1] G. Brinkmann and B. D. McKay, Construction of planar triangulations with minimum degree
5, Discrete Math., 301 (2005), 147–163.
[2] M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions,
Duke Math. J., 62 (1991), 417–451.
[3] T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric
Varieties, Ergeb. Math. Grenzgeb. (3), 15, Springer-Verlag, Berlin, 1988.
Department of Mathematics, Graduate School of Science, Osaka City University,
3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585 JAPANE-mail address: m13saU0r13@st.osaka-cu.ac.jp