Orientable hyperbolic 4-manifolds over the 120-cell
Jiming Ma Fangting Zheng
math.GT







School of Mathematical Sciences 
Fudan University
Shanghai 200433, China 
majiming@fudan.edu.cn





Department of Mathematical Sciences, Xi'an Jiaotong-Liverpool University, Suzhou 215123, China  
Fangting.Zheng@xjtlu.edu.cn



[2010]32Q45, 51M50, 52B70



Jiming Ma was supported by  NSFC  12171092. Fangting Zheng was supported by NSFC 12101504.













Since there is no hyperbolic Dehn filling theorem for higher dimensions, it is challenging to construct explicit hyperbolic manifolds of small volume in dimension at least four. Here, we build up closed  hyperbolic 4-manifolds of volume 34π^2/3· 16 by using the small cover theory. In particular, we classify all of the orientable four-dimensional small covers over the right-angled 120-cell up to homeomorphism; these are all with even intersection forms.


Orientable hyperbolic 4-manifolds over the 120-cell
    Fangting Zheng
    Jun. 8, 2022
===================================================








§ INTRODUCTION




 §.§ Hyperbolic manifolds

By the Thurston-Perelman's geometrization theorem, the majority of 3-manifolds are known to be hyperbolic 3-manifolds. Moreover,  according to the Jorgensen-Thurston theory, it is straightforward to construct closed hyperbolic 3-manifolds M(ϵ) by generically Dehn filling cusped hyperbolic 3-manifolds M along the tori. However, the Jorgensen-Thurston theory does not hold in ℍ^n when n is greater than or equal to four. More precisely, when conducting a sufficiently large Dehn filling on cusp ends of a higher dimensional cusped hyperbolic manifold, we can obtain a manifold that admits a complete, finite volume Einstein metric; in this case, however, the fundamental group is relatively hyperbolic rather than hyperbolic in general <cit.>. Hence, it is necessary to develop some methods to describe the higher dimensional situations. For example, Martelli and Riolo have studied the four-dimensional hyperbolic Dehn filling to another extent <cit.>. 

Moreover, besides the well-known four color theorem, it is helpful to understand the manifold theory through programming, for example see <cit.>. For the hyperbolic case, few broadly applicable techniques are currently available; a limited number of explicit examples have been reported in the literature. For example, Ratcliffe and Tschantz have determined the structure of the congruence two subgroups  Γ_2^4 of the group Γ^4 of integral, positive, Lorentzian 5× 5 matrices. They have classified all of the corresponding hyperbolic four-manifolds that have a minimum volume <cit.>.  Later, Saratchandran used the Ratcliffe-Tschantz census to study the topology of cusped hyperbolic 4-manifolds <cit.>. 

Theoretically, by initially constructing a hyperbolic orbifold and then applying the Selberg lemma, a hyperbolic manifold can be constructed. However, this approach is often inefficient since there is no universal method to determine the covering degree. For instance, given a Fuchsian group Γ,  the minima of indices of torsion-free subgroups of Γ is bounded above by the twice of the least common multiple (LCM) of orders of the finite subgroups of Γ <cit.>. However, Jones and Reid have shown that the complex version can vary substantially. More precisely, the authors reported a sequence Γ_k of co-compact Kleinian groups for which the ratio of the minimum index to the LCM could be arbitrarily large <cit.>.



 §.§ Small cover

Small covers have been studied by Davis and Januszkiewicz in <cit.>, see also <cit.>.  
A small cover of an n-dimensional simple polytope P is a manifold which arises as an orbifold cover of P of minimal index 2^n.  In other words, an n-dimensional  small covers admits  a locally standard ℤ_2^n-action and the orbit space is an  n-dimensional simple polyhedron.  It turns out that any small cover of P is determined by a map of the (abstract) right-angled Coxeter group Γ associated to P into the group ℤ_2^n. Such map is called a characteristic function or a coloring. For an (abstract) right-angled Coxeter group Γ associated to some three-dimensional simple polytope  P, Γ is guaranteed to have a torsion-free subgroup of index eight according to the four color theorem. In other words, we can always construct a small cover over a three-dimensional simple polytope. In higher dimensions, however, this problem presents significant challenges and has driven the study of the Buchstaber invariant <cit.>.

The algebraic and topological invariants of a small cover can be calculated by the combinatorics of the orbit polytope  and the coloring.  For example, the mod 2 Betti number β_i^(2) of a small cover M over the polytope P agrees with h_i, where h=(h_0, h_1,…, h_n) is the h-vector of the polytope P <cit.>.  According to some recent results of the homology groups and the cohomology ring of small covers <cit.>, a small cover of dimension three or four has at most 2-torsion in its homology group. Moreover, every oriented four-dimensional small cover has  zero signature and even intersection form.  Therefore the isomorphism class of the intersection form is determined by the second Betti numbers. See Section <ref> and <ref> for additional details.

Suppose that the simple polytope P can be realized in hyperbolic n-space as a right-angled polytope. Then any small cover over P is a hyperbolic
manifold.
For example, the first closed hyperbolic 3-manifold that appeared in the literature,
the Löbell manifold, is a small cover <cit.>,  see also <cit.>.
Ratcliffe and Tschantz classified the congruence-2 hyperbolic 4-manifolds of minimal
volume. They provided a list of the resulting 1171 non-compact manifolds, of
which 23 are orientable, together with their homology groups. There is a
straightforward generalization of small covers for the non-compact case. The problem studied by Ratcliffe and Tschantz in <cit.> can be reformulated as the classification of the small covers of a non-compact right-angled hyperbolic 4-polytope discovered by Potyagailo and Vinberg <cit.>. 
The analogous problem in dimension 5 has then been solved similarly by Ratcliffe
and Tschantz in <cit.>.
In this paper, we use the small cover theory to construct a family of closed hyperbolic 4-manifolds, which are obtained  after 16-degree coverings.

In comparison to the results of Ratcliffe and Tschantz, we consider the  minimal index torsion free subgroups of the Coxeter group associated to the right-angled 120-cell. The Coxeter  group used here has many more generators and relations than the group Γ^4 of integral, positive, Lorentzian 5× 5 matrices; this presents additional challenges in the identification of minimal index torsion free subgroups. For other hyperbolic 4-manifolds of small volume. See <cit.>.

Garrison and Scott have considered the simplest ℤ_2^4-coloring case over the right-angled 120-cell <cit.>, where the image of the characteristic functions is {e_1, e_2, e_3, e_4, e_1+e_2+e_3+e_4}. They have found that there is only one, up to homeomorphism, such non-orientable manifold. The manifold and the corresponding characteristic function are denoted as Garrison-Scott manifold and Garrison-Scott coloring, respectively. We obtain some other hyperbolic manifolds from the right-angled 120-cell and prove the following:

 
There are exactly 56 orientable small covers over the right-angled hyperbolic 120-cell up to homeomorphism.


There are exactly two 8-coloring  non-orientable  small covers over the right-angled hyperbolic 120-cell up to homeomorphism.


In addition, one 9-coloring manifold is successfully constructed. See Section <ref> and <ref> for details. Therefore, we obtain a total of 59 new closed hyperbolic 4-manifolds with volume 544π^2/3 through two algorithms. See Table <ref>, <ref> and <ref> for details. The color-recursion algorithm is much more universal, but is too inefficient to solve our problems. The block-pasting algorithm is far more effective as it takes advantage of the symmetry information of the 120-cell; however, it still exhibits an unaffordable running time when considering k-colorings, where k is greater than or equal to nine. We have identified all of the orientable small covers over the 120-cell, which are realized by at most eight colors. See Section 6 for more details. 



 §.§ Intersection form

The Davis manifold <cit.> is the first example of a hyperbolic 4-manifold with  a calculated even type intersection form  ⊕_36[ 0 1; 1 0;   ] <cit.>. Quite recently, Martelli-Riolo-Slavich exhibited the first concrete closed oriented hyperbolic four-manifold with odd intersection form <cit.>. Here, we construct 60 more examples with even intersection forms.
  
	For the 60 manifolds M_1,⋯,M_56,M_57,M_58,M_59,M_60 claimed in Tables <ref>–<ref>, we have their homologies and intersection forms as below:


where, c is the sum of the first reduced Betti numbers of the full-subcomplex K_ω for ω∈ row Λ. The Q(M) means the intersection form of the orientable 4-manifold M. Among these sixty small covers over the 120-cell ℰ, the manifolds M_1,M_2,⋯,M_56 are orientable, whereas M_57,M_58,M_59,M_60  are non-orientable. The manifold M_j is the orientable double cover of M_j, for 57≤ j≤ 60.


Outline of the paper: In Section 2 and Section 3, we present some preliminaries on the algebraic theory  of small covers and the combinatorics of the 120-cell, respectively. In Section 4, we use the color-recursion algorithm, which is broadly applicable but of low efficiency, to address the  case where the total number of colors assigned to the facets of the right-angled 120-cell to construct small covers is 4,5,6 or  7. Moreover, one new small cover constructed by 9 colors is obtained by this approach. Based on the high symmetry of the 120-cell, we then introduce a substantially more efficient block-pasting algorithm in Section 5 and study the more complicated cases, where the total number of colors  assigned to the facets of the right-angled 120-cell to construct small covers is 8 or  9. We classify all of the small covers over the right-angled 120-cell that are obtained in previous sections by homeomorphism in Section 6, thereby proving Theorem  <ref> and Theorem <ref>. Finally, we consider the intersection forms of all of these manifolds and complete the proof of Theorem <ref> in Section 7.

Acknowledgments: The authors appreciate greatly the referees' valuable and constructive  comments on an earlier draft of this paper, which make the paper much more precise and readable. We are also grateful to Leonardo Ferrari who kindly pointed out some typos in figures of adjacent matrices.



 §.§ Small covers

Given an n-dimensional simple polytope P, the manifold corresponding to a 2^n-index torsion-free subgroup Γ of the (abstract) right-angled Coxeter group associated to  P is a small cover over P  <cit.>. There is another equivalent but more practical way to describe a small cover by using the language of coloring <cit.>: 

Let ℱ(P)={F_1,F_2,…,F_m} be  the set of co-dimensional one faces of P.  Let us define the ℤ_2^r-coloring characteristic function, n ≤ r≤ m, to be a function 
    λ:ℱ(P)={F_1,F_2,⋯,F_m}⟶ℤ_2^r,
 that satisfies the non-singularity condition. That is λ(F_i_1),λ(F_i_2),⋯,λ(F_i_n) should generate a  subgroup  of ℤ_2^r, which is isomorphic to ℤ_2^n, when the facets F_i_1, F_i_2, ⋯, F_i_n share a common vertex.  Note that the function λ is not guaranteed to exist. The corresponding characteristic matrix is obtained by placing the vector λ(F_i) column by column. For the sake of brevity, a ℤ_2^r-coloring characteristic function λ is referred to as a coloring.  Every vector in ℤ_2^r is called a color. The vector λ(F_i)∈ℤ_2^r is called a color of F_i. In particular, if r=4, there is a total of 16 colors. Say a coloring is a k-coloring  if exactly k different colors are used for its construction. These k colors form a coloring set {c_1,c_2,⋯,c_k} of λ.
 
If the characteristic function λ is defined successfully, we can recover a smooth n-manifold M(P,λ):=P×ℤ_2^r/∼, which is named real toric manifold over P, by the following equivalence relation:



	(x,g_1)∼ (y,g_2)⟺{[              x=y  and g_1=g_2                            if                     x∈ int P,; x=y    and    g_1^-1g_2 ∈ G_f                            if                       x ∈∂ P, ].


where, f=F_i_1∩⋯∩ F_i_n-l is the unique face of co-dimensional n-l that contains x as an interior point, and G_f is a subgroup generated by λ(F_i_1), λ(F_i_2), ⋯, λ(F_i_n-l). For brevity, we  only refer to the colorings when the polytope is given instead of talking about both colorings and manifolds. If r=n, then the corresponding manifold is called a small cover.  

For example, define a ℤ_2^2-coloring characteristic function λ on the 2-dimensional cube I^2 as shown in Figure <ref>, where (1,0)=e_1 and (0,1)=e_2 are the standard basis of ℤ_2^2. The small cover M(I^2,λ) is the Klein bottle.



Since the dual of the boundary of a simple polytope P is a simplicial complex K <cit.>, the definition of real toric manifolds introduced above has a dual version. By substituting  the facet set ℱ(P) with the vertex set 𝒱 of the simplicial complex K, we can also define the characteristic function λ on K, namely 
    λ:𝒱(K)={v_1,v_2,⋯,v_m}⟶ℤ_2^r
  The non-singularity condition changes as follows: if the convex hull conv{v_i_1, v_i_2, ⋯, v_i_n} of n vertices v_i_1,v_i_2,⋯,v_i_n  is a facet of K, the images λ(v_i_1),λ(v_i_2),…,λ(v_i_n) shall generate a  subgroup isomorphic to ℤ_2^n. 

To keep a concise notation, we encode every ℤ_2^*-color by an integer through the binary-to-decimal conversion. For example, in the ℤ_2^3-coloring case, we use the numbers 1,2,⋯ ,7 to represent the following seven colors (1,0,0), (0,1,0),  (1,1,0), (0,0,1) (1,0,1), (0,1,1), and  (1,1,1), respectively. Thus, a characteristic matrix can also be viewed as a characteristic vector.  For example, the characteristic vector C of characteristic matrix


	(λ(ℱ_1),λ(ℱ_2),λ(ℱ_3),λ(ℱ_4),λ(ℱ_5))=
	[ 1 0 0 0 1; 0 1 0 0 1; 0 0 1 0 1; 0 0 0 1 1;   ]


is (1, 2, 4, 8, 15). These integers are also called colors if there is no ambiguity. In the previous example, we may say the color of facet F_5 is 15 or (1,1,1,1). The characteristic function λ, the characteristic matrix Λ, and the characteristic vector C can be inferred from each other easily. The characteristic vector C is the most  concise representational form among them.



 §.§ GL_n(ℤ_2)-equivalence and orientability


Let P be an n-dimensional simple polytope with its first facets F_1,F_2,⋯,F_n sharing a vertex. Suppose λ is a ℤ_2^r-coloring characteristic function over P that satisfies λ(F_i)=e_i, for 1≤ i≤ n, where {e_i}_i=1^n is the standard basis of ℤ_2^n. Colors on other facets are linear combinations of the colors on the first n facets. Another coloring λ is GL_n(ℤ_2)-equivalent to λ if there exists some g∈ GL_n(ℤ_2) such that λ=g∘λ. This relation can be easily checked to determine if it is an equivalence relation. Thus, all the characteristic functions over a given simple polytope can be classified under this equivalence relation. Then, λ is referred to as the GL_n(ℤ_2)-equivalence representative in its equivalence class.  For example, assume P is the 2-dimensional cube I^2 and its facets are ordered as shown in Figure <ref> (1). There are a total of 18 small covers over P as shown in Table <ref>, which can be grouped into three GL_2(ℤ_2)-equivalence classes in total. The GL_2(ℤ_2)-equivalence representatives of these three classes are (e_1, e_2, e_1, e_2), (e_1, e_2, e_1+e_2, e_2) and (e_1, e_2, e_1, e_1+e_2), respectively, as shown in Figure <ref>.





 By the definition of small cover, the GL_n(ℤ_2)-equivalence characteristic functions always give homeomorphic n-dimensional small covers. Besides, small covers from different GL_n(ℤ_2)-equivalence classes may also be  homeomorphic. See Section <ref> for more details. In the following sections, we always first find the GL_n(ℤ_2)-equivalence representatives, and then group them by homeomorphism. For a GL_4(ℤ_2)-equivalence representative k-coloring with coloring set {1,2,4,8,c_5,⋯,c_k}, the (k-4) colors c_5,⋯,c_k are called
 added colors. Especially, there is no added color in the 4-coloring case. In the following, the coloring  with the coloring set C={1, 2, 4, 8, c_5,⋯, ck} is called a (c_5,⋯,c_k)-coloring. The case that considers all of the (c_5,⋯,c_k)-coloring is called the (c_5,⋯,c_k)-coloring case.

H. Nakayama and Y. Nishimura have discussed the orientability of a small cover in <cit.>. Their main theorem is presented below.

  (H. Nakayama-Y. Nishimura <cit.>) For a simple n-dimensional polytope P, and for a basis {e_1,⋯,e_n} of ℤ_2^n, a homomorphism ϵ:ℤ_2^n→ℤ_2={0,1} is defined by ϵ(e_i)=1, for each i=1,⋯, n. A small cover M(L,λ) is orientable if and only if there exists a basis {e_1,...,e_n} of ℤ_2^n such that the image of ϵλ is {1}.


Therefore, a small cover is orientable if and only if there exists a basis of ℤ_2^n such that all the colors used, i.e., the columns of the characteristic matrix, have an odd number of 1 entries. In the space of ℤ_2^4, the vectors (1,0,0,0), (0,1,0,0), (0,0,1.0), (0,0,0,1), (1,1,1,0), (1,0,1,1), (1,1,0,1), and (0,1,1,1), which are the binary form of 1, 2, 4, 8, 7, 11, 13, and 14, are the only eight elements with entry sums of 1 mod 2. These eight vectors are called orientable colors. The seven non-trivial colors left are (1,1,0,0), (1,0,1,0), (0,1,1,0), (1,0,0,1), (0,1,0,1), (0,0,1,1) and (1,1,1,1), which are the binary forms of 3, 5, 6, 9, 10, 12, and 15. An orientable basis in ℤ_2^4 is defined to be a basis in ℤ_2^4 that consists of four linear-independent orientable colors. Especially, the standard basis in ℤ_2^4, i.e., (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1), is an orientable basis. Note that, for an orientable color, the number of entries with value 1 is always odd. In other words, when changing from one orientable basis to another orientable one, we actually add or remove even many 1s from the previous characteristic matrix to form the new one. Hence, parity of the number of 1s in each column is maintained under different orientable basis.  Therefore, we have the following corollary.


	Suppose (1,2,4,8,a_1,⋯,a_m) is a characteristic vector of some GL_4(ℤ_2)-equivalence representative λ over a simple polytope P Then the corresponding small cover M(P,λ) is non-orientable if and only if there is some a_i ∈{3,5,6,9,10,12,15}. .  







§ COMBINATORICS OF THE 120-CELL

In this section, we provide some fundamental combinatorial background information about the 120-cell ℰ with ordered facets. We refer to <cit.> for a nice exposition about some well-known facts on the 120-cell. This part is important in constructing and classifying small covers over the right-angled 120-cell. Most of the content in this section is well-known for experts; we present it here for the convenience of the readers.


 §.§ Dodecahedron


The regular dodecahedron, or pentagonal dodecahedron, is abbreviated as  dodecahedron and denoted as 𝒟 in the following discussion. It has 12 regular pentagonal faces, 20 vertices and 30 edges, see Figure <ref> in Section 5, and is represented by the Schläfli symbol {5, 3}. 

The full symmetry group 𝔸(P^n) of an n-dimensional polytope P^n is the group of all isometries under which the polytope P^n is invariant.
The group operation is the composition. It is the same as the combinatorial automorphism group of the n-dimensional polytope P^n if P^n is regular. The group 𝔸(P^n) includes a subgroup of orientation-preserving isometries 𝔸^+(P^n) and a subset of  orientation-reversing isometries 𝔸^-(P^n). For the symmetry group of the dodecahedron 𝒟, there is a well-known lemma (e.g., <cit.>):


	|𝔸(𝒟)|=120 and |𝔸^+(𝒟)|=60.


The group of orientation-preserving symmetries of the dodecahedron 𝒟 is known as the alternating group on five letters, which is denoted as A_5. The full symmetry group is isomorphic to A_5×ℤ_2, with the second factor being generated by the antipodal map.




 §.§ SO(3) and rigid motion


The special orthogonal group, denoted by SO(n), contains all n× n orthogonal matrices  of determinant 1. This is also the group of rigid motions of ℝ^n fixing the origin. Such a group is also called the rotation group. See <cit.> for detailed discussions in studying the symmetries of 𝒟 from the rigid motion's point of view.



 §.§ Quaternions


Let <1,i,j,k> be the usual basis for ℝ^4. The quaternion space  ℋ=ℝ⊕𝕀=ℝ⊕ iℝ⊕ jℝ⊕ kℝ, where 𝕀=iℝ⊕ jℝ⊕ kℝ is the subspace of the purely imaginary quaternions. The sum of two elements of ℋ is defined to be the sum of them as elements of ℝ^4. Similarly the product of an element of ℋ and a real number is defined to be the same as the scalar multiplication in ℝ^4. The associative quaternion multiplication is defined by first establishing the products of basis elements as follows:


i^2=j^2=k^2=ijk=-1.


Then, the distributive law is applied to define the general situation.

Let q=a+bi+cj+dk be a quaternion. The conjugate of q is the quaternion q^*=a-bi-cj-dk. The conjugate of a product of two quaternions is the product of the conjugates in reverse order. That is, if p and q are quaternions, then (pq)^* = q^*p^*. The square root of the product of a quaternion with its conjugate is called its norm and is denoted by || q||, namely || q||=√(qq^*)=√(q^*q)=√(a^2+b^2+c^2+d^2). The inverse of a quaternion, under quaternion multiplication, is q^*/||q||^2. Besides, using the norm we can define the 3-sphere 𝕊^3 as the set of unit quaternions {q∈ℋ | || q||=1}. The 3-sphere 𝕊^3 forms a group under quaternion multiplication. The point 1+0i+0j+0k serves as the identity; the associativity and inversion are inherited from ℋ. For two points p and q in ℋ, the norm of the difference between them is defined as d(p, q)=|| p-q||. The metric on ℋ naturally induces the round metric d_S(p,q)=arccos(<p,q>) on the 3-sphere. According to <cit.>, the group structure and geometry of 𝕊^3 interact with each other in the following way:


Both left and right actions of 𝕊^3 on ℋ are via orientation-preserving isometries in Isom^+(ℋ^3). The same holds for the 3-sphere's action on itself.



For p∈𝕊^3 and q,r∈ℋ, we have d_ℋ(pq,pr)=| pq-qr|=| p|| q-r|=| q-r|=d_ℋ(q,r). Similarly, d_ℋ(qp,rp)=d_ℋ(q,r). Since 𝕊^3 is connected and the identity 1 acts trivially, the actions are orientation-preserving.



Furthermore, for q∈𝕊^3, we define a twist action ϕ_q:ℋ→ℋ,
p↦ϕ_q(p)=qpq^-1, which is again via isometries. Note that ϕ_q preserves 𝕀⊂ℋ setwise. Thus we can define ψ_q:=ϕ_q|_𝕀:𝕀→𝕀. Recall that SO(3) is the group of the three-by-three matrices with determinant one. Take ⟨ i, j, k⟩ as a basis for 𝕀, the group SO(3) can be identify with Isom_0^+(𝕀), the group of orientation-preserving isometries of 𝕀 fixing the origin. 
Hence the map ψ_q is an element of SO(3) and the reduced map ψ:𝕊^3→ SO(3) is a double cover <cit.>. 

If G⊂ SO(3) is a group, then G^*=ψ^-1(G) is called the binary group corresponding to G. In particular, for the group of orientation-preserving isometries of the dodecahedra 𝒟 𝔸^+(𝒟)⊂ SO(3), we have the binary dodecahedral group 𝒟^*≜ψ^-1(𝔸^+(𝒟))⊂𝕊^3, which has 120 elements in total. More precisely, 𝒟^*, consist of 24 Hurwitz units {±1,± i,± j,± k,1/2 (± 1± i± j ± k )} and 96 quaternions  obtained from 1/2( 0± i±ϕ^-1j ±ϕ k ) by the even permutations of all four coordinates <cit.>. 

 

 §.§ The 120-cell ℰ
 
 We are going to formulate the construction of the 120-cell by means of group 𝒟^*⊂𝕊^3.
 
 	Suppose V is a finite set of points in a metric space X, the Voronoi cell about a point q∈ V is the set Vor(q)={r∈ X | for   all   p∈ V,  d_X(q,r)≤ d_X(r,p)}.
 
Two cells Vor(p) and Vor(q) are said to be adjacent if Vor(p)∩Vor(q) is an (n-2)-cell. Let 𝒯_120 be the tiling of the 3-sphere by the cell {Vor(q) |  q∈𝒟^*} and define 𝒞=Sym(𝒯_120). Here is a useful lemma:
<cit.>
	
		
  * The left action of 𝒟^* on 𝒯_120 is transitive on the 3-dimensional cells.
		
  * The twisted action of 𝒟^* fixes Vor(1) setwise.
		
  * Both above actions give homomorphsims of 𝒟^* to C. 
		
  * Each cell Vor(q) is a regular spherical dodecahedron.
	

 
 
 	The Euclidean 120-cell ℰ is the convex hull, taken in ℋ, of the vertices of 𝒯_120.
 

 
 	The convex hull in ℝ^4 of these 120 points is the regular 4-dimensional polytope 600-cell, which is dual to the 120-cell ℰ. If we forget the group structure of 𝒟^* and regard it as a set, it is the vertex set of the 600-cell.
 
 
 
 	  The right-angled 120-cell ℰ is a 4-dimensional compact hyperbolic polytope of dihedral angle π/2 and volume 34/3π^2 <cit.>. It has the same face lattice with the Euclidean 120-cell.
 
 
 The 120-cell has 120 dodecahedral facets, 720 pentagonal ridges, 1200 edges, and
 600 vertices. There are four facets meeting at each vertex. Figure <ref> shows a layer structure of the facets of the 120-cell. The first layer is the unbounded dodecahedral facet that contains ∞. The second layer consists of 12 dodecahedral facets. The adjacencies between the second and third layers can be seen in Figure <ref> with little effort. It may be easily checked that the 120-cell is a regular polytope, for example see <cit.>. 
 


 An n-polytope P^n is called simple if every r-face is contained in exactly n-r facets.
 A simple polytope is called a flagtope if every collection of its pairwise intersecting facets has a nonempty intersection. If a simple polytope is a flagtope, all of the adjacency information is included in the adjacency matrix.  The 120-cell ℰ can be verified to be a flagtope with little effort. Moreover, the Euclidean 120-cell defined above is inscribed in the 3-sphere 𝕊^3 and we can define a projection Π:∂ℰ→𝕊^3,x↦x/||x||. We fix the orders of the 120 facets of 120-cell and 120 elements of 𝒟^* by F_1,F_2,⋯ F_120 and q_1,q_2,⋯, q_120 as shown in Tables <ref>–<ref>, where Π(F_i)=Vor(q_i),q_i∈𝒟^*. For brevity, we re-denote Vor(q_i) as Vor(i) and define a function ord:𝒟^*→{1,2,⋯,120},  q_i↦ ord(q_i)=i to obtain the label. The symbols F_i, q_i and ord will be used throughout the rest of the paper with this meaning, unless stated otherwise. Suppose X(ℰ)=(a_ij)_m× m is the adjacency matrix of the 120-cell, then the entries can be defined as
	

		a_ij={[ 1  ; 0   ]..
	



We calculate distances from elements in 𝒟^*, i.e., the vertices dual to the facets of the 120-cell, to the identity 1+0i+0j+0k under the round metric. Then, the facets of ℰ can be divided into nine spherical layers L_i according to the distances as shown in Table <ref>.  A detailed geometric illustration is available in <cit.>.


Note that the distances from the center of the facet F_1 to the centers of those facets adjacent to it, namely the 12 facets in the second layer, are always π/5, under the round metric on 𝕊^3. Due to the symmetry, the distance between the two centers of every adjacent facets should always be π/5. Therefore, we can probe the adjacency matrix X(ℰ) by examining the distance under the round metric d_S(q_i,q_j) for all p,  q∈𝒟^*.  
For a better demonstration of the adjacency matrix X(ℰ), we first depict the adjacency relations among different spherical layers L_i (1≤ i≤ 9). We say two layers L_i and L_j are adjacent if there are two facets F_i_p∈ L_i and F_j_q∈ L_j such that F_i_p and F_i_q are adjacent. Adjacency relations are shown in Figure <ref>, where two layers are connected by edge if they are adjacent.     


All of the adjacency matrices  X_L_i,L_j, abbreviated as X_ij of the adjacent layers, for 1≤ i< j≤ 9, are shown in Figures <ref> – <ref> in Appendix.  For clarity, we represent the matrix by a colored grid; the uncolored entries are all of value 0. The diagonal items are marked specially with dark shading and other positions with values of 1 are filled with light shading. The figures can be obtained from Table <ref>-<ref> in Appendix. The whole symmetric adjacency matrix X(ℰ) for the 120-cell ℰ is


	X_ℰ=[ X_11 X_12    0    0    0    0    0    0    0; X_21 X_22 X_23 X_24    0    0    0    0    0;    0 X_32 X_33 X_34 X_35    0    0    0    0;    0 X_42 X_43 X_44 X_45 X_46    0    0    0;    0    0 X_53 X_54 X_55 X_56 X_57    0    0;    0    0    0 X_64 X_65 X_66 X_67 X_68    0;    0    0    0    0 X_75 X_76 X_77 X_78    0;    0    0    0    0    0 X_86 X_87 X_88 X_89;    0    0    0    0    0    0    0 X_98 X_99;      ],


where X_ji=X_ij^T.

It can be read from the matrix X_12 that the first four facets of the 120-cell ℰ are adjacent to each other. Since the 120-cell is a flagtope, these four facets would share a common vertex. We may always color the facet F_i by e_i, i=1, 2, 3, and 4, where {e_i}_i=1^4 is the standard basis set of ℤ_2^4.



 §.§ Symmetry group of 120-cell ℰ
The symmetry group, i.e., the combinatorial automorphism group, of the 120-cell ℰ, denoted as 𝔸(ℰ), is well known as the Coxeter group H_4. However, that is not enough for our purpose. We want to describe the group 𝔸(E) as the subgroup of the permutation group S_120, writing down precisely each element in 𝔸(E)  as a permutation of 120 facets. Each permutation tells how the facets of the 120-cell are permuted under certain symmetry. The approach we adopt to find all of the permutations is similar to the method used in the proof of <cit.>. The permutation form of 𝔸(E) is of great importance for classifying small covers over the right-angled 120-cell by homeomorphism in Section <ref>. 

Let 𝕏^n∈{𝕊^n,𝔼^n,ℍ^n} be one of the three standard n-dimensional geometric spaces of constant curvature 1, 0 or -1 and P^n be an n-dimensional polytope in the n-space 𝕏^n. A collection of faces Q_0⊂ Q_1⊂⋯⊂ Q_n-1⊂ Q_n=P^n is called a flag of the polytope P^n if Q_i has dimension i. Suppose that 𝒬= {Q_i} is a flag in P^n. The corresponding flag polytope
P^n_𝒬 is defined to be the convex hull of the centres of the Q_i.  
Similarly, we can also define the flag of a tessellation 𝒯 of n-space 𝕏^n. A collection of cells  Q_0⊂ Q_1⊂⋯⊂ Q_n-1⊂ Q_n is called a flag of the tessellation 𝒯 if Q_i is an i-cell in the tiling 𝒯. Suppose that 𝒬= {Q_i} is a flag in 𝒯. The corresponding flag polytope
𝒯_𝒬 is the convex hull in 𝕏^n of the centres of the Q_i.



The orientation preserving
symmetry group of the 120-cell is the quotient of the product of two copies
of the binary icosahedral group by the group generated by (-1,-1). More precisely, 
the flags of the 120-cell ℰ in ℋ are 4-dimensional simplices with one vertex at the origin, which are in one-to-one correspondence with the 14,400 spherical flag tetrahedra of 𝒯_120. Next, we fix a right-handed spherical flag tetrahedron T of Vor(1). By Lemma <ref>, we may use the left action of 𝒟^* to transfer
every tetrahedron T^'∈𝒯_120 into Vor(1). Then, we use the twisted action of 𝒟^* to send T^' to T. We now want to find the distinct 14,400 elements. 

First, we can obtain a symmetry element of 𝔸^+(ℰ) by multiplying one element of the set  𝒟^* from left and one element of the set  𝒟^* from the right (the two factors may be the same) to all of the centers of the Voronoi cells in 𝒯_120, i.e., all of the elements in 𝒟^*. Namely, for p,q∈𝒟^*, define ψ_p,q:𝒟^*→𝒟^*, q_i↦ψ_p,q(q_i)=pq_ip. For every pair (p,q)∈𝒟^*×𝒟^*, we have a permutation [ord(ψ_p,q(q_1)), ord(ψ_p,q(q_2)),⋯,ord(ψ_p,q(q_120))] of an orientation-preserving symmetry element in 𝔸^+(ℰ). Note that the symmetry obtained by left-acting a_1+b_1 i +c_1 j+d_1k and right-acting -a_2+b_2 i +c_2 j+d_2k is the same as the symmetry obtained by left-acting -a_1-b_1 i -c_1 j-d_1k and right-acting a_2-b_2 i -c_2 j-d_2k. Then, among all 120×120=14400 results, there are only 7200 left after removing duplicates. Secondly, the conjugation map a+bi+cj+dk→ a-bi-cj-dk is a product of three reflections, so is orientation reversing in ℋ. It preserves 𝕊^3 and is again orientation reversing there. Since 𝒟^* is a group of quaternions, it is closed under conjugation. Since the tiling 𝒯_120 is metrically defined in terms of 𝒟^*, it is also invariant under conjugation. Hence, if we do the quaternion conjugation to each of the above 7200 results, we can obtain the corresponding orientation-reversing ones. So, there are also 7200 orientation-reversing symmetries of the 120-cell ℰ. Then, we have searched 14400 different elements in the symmetry group 𝔸(ℰ). From the well-known result that |𝔸(ℰ)|=14400, it can be claimed that all of the elements in 𝔸(ℰ) have been depicted clearly by showing how its facets are permuted. 




§ CONSTRUCTING SMALL COVERS WITH THE COLOR-RECURSION ALGORITHM




 §.§ Color-recursion algorithm

The color-recursion algorithm  is broadly applicable. Here, we use it on the 120-cell ℰ to generate small covers.  The 120-cell is dual to the 600-cell. By the dual non-singularity condition introduced in Section 2.1, if the convex hull conv{v_i_1,v_i_2,v_i_3,v_i_4} of four vertices v_i_1,v_i_2,v_i_3,v_i_4 of the 600-cell is a tetrahedral facet of the 600-cell, the images λ(v_i_1),λ(v_i_2),λ(v_i_3),λ(v_i_4) shall generate a subgroup isomorphic to ℤ_2^4. The color of the vertex is the color on the corresponding dual  facet. We now want to exclude the cases that fail to meet the non-singularity condition by systematically “checking simplexes". For an n-dimensional simple polytope, every facet meets exactly n-1 other facets at a vertex.  Thus, we need to conduct the i-simplex criterion checks, where 1≤ i≤ n-1. For the 4-dimensional polytope 120-cell ℰ, we only need to check the cases of 1-simplex, 2-simplex, and 3-simplex.

Given a coloring set C={1,2,4,8,c_1,⋯,c_k} and suppose the first (i-1) facets are colored, we are going  to color the facet F_i. Now, the initial candidate set of admissible colors for F_i, denoted by d^0(F_i), is C. For a color c∈ C, we undertake the following three tests to conclude whether we can color the facet F_i by the color c or not

(a). 1-simplex checking

By the adjacency matrix X_ℰ, we can obtain the set 𝒜^1(F_i) of the facets that are adjacent to F_i and have been colored. If 𝒜^1(F_i)=∅, all of the 2^n-1 colors are qualified. If 𝒜^1(F_i)={F_i_1,F_i_2,...,F_i_k}∅,  there are k 1-simplexes in the 600-cell as shown in Figure <ref>. We still use the notation F_i rather than introducing new notation v_i of the dual vertex because they are one-to-one dual to each other. This convention applies throughout the section. The non-singularity condition requires that these 1-simplexes do not have the same colors on their two vertices.

Let c(F_i_j)∈ℤ^n_2 be the color of colored facet F_i_j∈𝒜^1(F_i). Now the set of admissible colors for F_i is 
    d^1(F_i)=d(F_i)-{c(F_i_1),c(F_i_2),⋯,c(F_i_k)}.

If c∈ d^1(F_i), we move on to the next checking. Otherwise, we check another candidate of color in the coloring set C. 

(b). 2-simplex checking

Next, we find from  𝒜^1(F_i) the set 𝒜^2(F_i) of pairs of adjacent facets in 𝒜^1(F_i). In turn, each pair together with the facet F_i provides us with three vertices in the 600-cell. Then, the complete graph on these vertices give us the 1-skeleton of a 2-simplex as shown in Figure <ref>. The non-singular condition requires that no color of the vertices for every 2-simplex can be represented as the ℤ_2-coefficient linear combination of colors of the other two. If 𝒜^2(F_i)=∅, all of the candidates are qualified. Otherwise, assume 𝒜^2(F_i)={(F_i_1,F_i_2),⋯,(F_i_j,F_i_k)}. Now, the set of admissible colors for F_i is  
    d^2(F_i)=d^1(F_i)-Span{c(F_i_1),c(F_i_2)}-⋯-Span{c(F_i_j),c(F_i_k)}.

If c∈ d^2(F_i), we move on to the next checking. Otherwise, we check another candidate of color in the coloring set C. 



(c). 3-simplex checking

Finally, we find from  𝒜^2(F_i) the set 𝒜^3(F_i) of facet triples such that every two of the three facets are adjacent. In turn, each triple together with the facet F_i provides us with four vertices. Then, the complete graph on these vertices gives us the 1-skeleton of a tetrahedron as shown in Figure <ref>.


If 𝒜^3(F_i)=∅, all of the colors are qualified. Otherwise, assume 𝒜^3(F_i)={(F_i_1,F_i_2,F_i_3),⋯, (F_i_j,F_i_k,F_i_l)}, then

    d^3(F_i)=d^1(F_i)-Span{c(F_i_1),c(F_i_2),c(F_i_3)}-⋯-Span{c(F_i_j),c(F_i_k),c(F_i_l)}.

If c∈ d^3(F_i), we can color F_i by the color c. Otherwise, we turn to check another candidate of color in the coloring set C. 

We keep doing the above procedure until we successfully color F_120 by admissible colors. If so, the resulting characteristic vectors are printed to an assigned file. 
We call it the “color-recursion" algorithm because we use the recursion method to generate the coloring. It automatically turns to color the F_i+1-th facet when the facet F_i  is colored; or it traces back to take an alternative color among the coloring set. In particular, if all of the candidate colors in the coloring set encounter d^1(F_i)=∅, d^2(F_i)=∅ or d^3(F_i)=∅,  we may go back to change the color of the (i-1)-th facet and do a new round of checking. If there are still no other admissible choices for F_i-1, we move one more facet backwards, namely to the (i-2)-th facet, to check. Since the first four mutually adjacent facets have been colored by the standard basis in ℤ_2^4, we start from facet F_5. Therefore, tracing-back ends when we fail in supplying a suitable color for the facet F_5. The pseudocode for the color-recursion algorithm is given in Algorithm 1.


Moreover, by means of the stabilizer of a vertex,  we can actually select out the representatives of the added colors for every k-coloring case to substantially reduce the computational load.

	The stabilizer of every vertex in the symmetry group of the regular polytope 120-cell ℰ is S_4, the symmetry group over four letters.

For example, suppose the first four facets that  intersect at a common vertex v are colored by e_1,e_2,e_3, and e_4. Then one more color e_1+e_3, the combination of the first and the third colors, is added to the coloring set. Namely, the coloring set is {1,2,4,8,3}. If we act upon the first four facets by a symmetry in  S_4 to generate, for example,  a permutation as follows

    [ [ e_1 e_2 e_3 e_4; e_2 e_1 e_3 e_4;     ] ],

then the added color, which is the combination of the first and third colors, changes from e_1+e_3 to e_2+e_3. Namely, the coloring set becomes {1,2,4,8,5}. That is, the manifolds that can be constructed by {1,2,4,8,3} and {1,2,4,8,5} are actually the same up to homeomorphism. 
By the same reason, the coloring sets {1,2,4,8,3}, {1,2,4,8,5}, {1,2,4,8,6}, {1,2,4,8,9}, 


		
		
		1,2input v_1× 120=(0,0,⋯,0) with all the entries valued 0 and run ColorRrecursion(1,v). We obtain all of the characteristic vectors. 

{1,2,4,8,10} and (1,2,4,8,12} are in the same family.  Therefore, through Lemma <ref>, we can classify the k-colorings over the 120-cell ℰ by considering the stabilizer group action. The representatives of the added colors are summarized in Table <ref> in Appendix.  This  means we have the following corollary, which is a standard claim for the 5-coloring case and can be adopted by any k-coloring case based on Table <ref>.

	For every small cover M(ℰ,λ) with | Im(λ)|=5, there exists another characteristic function λ^* with Im(λ^*)∈{{1,2,4,8,3},{1,2,4,8,7},{1,2,4,8,15}} such that M(ℰ,λ^*)≅ M(ℰ,λ).


Garrison-Scott has considered the case of (1,2,4,8,15)-coloring in <cit.>. The added color 15 is actually one of the three representatives when k=5 in Table <ref>. Up to GL_4(ℤ_2)-equivalence, there are only ten characteristic functions in this (1,2,4,8,15)-coloring case. Moreover, they are mutually homeomorphic. We discuss the details in Section <ref>.



 §.§ {4, 5, 6, 7, 8}-coloring small covers over the 120-cell ℰ

Applying the color-recursion algorithm to 4-coloring, 5-coloring, 6-coloring, and 7-coloring situations, abbreviated as {4,5,6,7}-coloring cases, we have the following Proposition <ref>:


	There is only one 5-coloring small cover, i.e., the Garrison-Scott manifold, up to homeomorphism, and no {4,6,7}-coloring small cover over the right-angled 120-cell ℰ.



	
The coloring cases are computed in this work. The machine is equipped with Windows 7 Ultimate. Its processor is an AMD A8-7100 Redeon R5, with eight computing cores 4C+4G (4 CPUs) of a 1.8 GHz clockspeed, and the RAM is 16384 MB. Python is our programming language. It costs 11.069 seconds to recalculate the (15)-coloring case, namely the Garrison-Scott manifold, by this algorithm. We list this characteristic vector in Table <ref>. It took nearly 11 hours to accomplish the whole 7-coloring case. However,  when we move on to address the 8-coloring case, the low efficiency and huge time consumption makes it unaffordable. See Table <ref> for the detailed data.

In the cases of (3,5,9,14)-coloring and (3,5,9,15)-coloring, some new manifolds are constructed. The * here represents the case that can not be calculated within an acceptable amount of time. It is estimated from experimental testing that at least half a year is needed to complete the (7,11,13,14)-coloring case. By Corollary <ref>  and Proposition <ref>, all of the orientable ℤ_2^4-colorings belong to the (7,11,13,14)-coloring case. Thus, we cannot effectively obtain all the orientable small covers over the right-angled 120-cell through this algorithm.


 §.§ A 9-coloring small cover over the 120-cell ℰ

We also try to construct new small covers over the right-angled 120-cell through another approach. Take the unique 5-coloring characteristic vector and color it after freeing the fixed colors from the back to the front. We begin by fixing the first 119 colors of the Garrison-Scott characteristic vector and using the color-recursion algorithm to run over all of the admissible colors on the 120-th facet. Later, we fix the colors of the first 118 facets and color the last two, etc. We conduct this procedure by computer and it is not until we move ahead to free the colors of the facets behind the first 33 facets that we obtain a 9-coloring, which  gives a new small cover. The first 33 facets cover exactly the first three layers of the boundary of the facet-ordered 120-cell, as shown in Table <ref>.

We stop here because these results required approximately 13 days and need to consume much more time to process further. Moreover, the 9-coloring characteristic function we obtain here is in the same homeomorphism class with the 9-coloring results obtained so far by the new block-pasting algorithm that is proposed in Section <ref>. 

In total, we obtain three more new manifolds with the color-recursion algorithm up to homeomorphism. We explain the homeomorphism classification in Section <ref>.



§ CONSTRUCTING SMALL COVERS WITH THE PROPOSED BLOCK-PASTING ALGORITHM

Now we change our strategy and propose a new algorithm. We firstly truncate the set of facets by blocks and produce the truncated characteristic vectors concurrently. Then, we match them up and finally obtain the characteristic vectors.


 §.§ Optimized algorithm method

The new algorithm named block-pasting is designed to address the computational efficiency issue  by making use of the high symmetry of the 120-cell.  For each  facet F_i of an n-dimensional polytope P^n, we define the i-th   block, denoted as b_i, to be the ordered set of facets consisting of F_i and all of the facets adjacent to it with increasing subscripts. We may also use b_i to denote the ordered set of subscripts when b_i is referred to as a set of integers. For example, for a facet-labeled dodecahedron P in Figure <ref>,



the block  b_1={F_1,F_2,F_3,F_4,F_5,F_6} (or {1,2,3,4,5,6}) and the block b_2={F_1,F_2,F_3,F_4,F_7,F_8} (or {1,2,3,4,5,6}) are as shown in Figure <ref>.



Recall that every dodecahedral facet of the 120-cell is adjacent to exactly 12 other dodecahedral facets. So, all of the 120 blocks of the 120-cell ℰ contain 13 dodecahedral facets. 

Each symmetry of the 120-cell clearly induces a permutation of the facets. In Section <ref>, we describe the symmetry group 𝔸(ℰ) of the 120-cell as a subgroup  of the permutation group S_120. For every  1≤ i≤ 120, there is a set made of 120 induced permutations, any of which sends the facet F_1 to the facet F_i. We arbitrarily pick one from each of these 120 sets and  form a representative set 𝒥 with 120 elements. Now, the whole block b_1 may be transferred to another block b_i by some unique element in the set 𝒥, for 1≤ i≤ 120. The set of the selected 120 such permutations can be regarded as a 120× 120 matrix M_𝒥, where every row represents a permutation. 


We are now only concerned with the results of these permutations  on the first block b_1. Hence, we only need the first 120× 13 submatrix M^13_𝒥 of the matrix M_𝒥, where every row represents a result of one permutation on b_1. Parts of the matrix M^13_𝒥 are shown in Table <ref>.



We use X_ℰ(b_i) to represent the 13× 13 submatrix of X_ℰ obtained by selecting all of the k-th rows and the k-th columns, where k ∈ b_i. Given a coloring set C={1,2,4,8,c_1,⋯,c_k}. We use the color-recursion algorithm to construct the truncated characteristic vectors of length 13. We also need to ensure the non-singularity condition by checking simplexes as mentioned in Section <ref>. The only difference is that the set 𝒜^1(F_i) is now read from the truncated adjacency matrix X_ℰ(b_i), rather than the whole adjacency matrix X_ℰ. The value of the q-th entry of a resulting characteristic vector is the color of the q-th facet in  b_i. 

For example, we obtain 
    b_3={F_1,F_2,F_3,F_4,F_8,F_11,F_13,F_16,F_20,F_26,F_28,F_32,F_36}
 from the adjacency matrix X_ℰ by probing those facets that are adjacent to or equal to facet F_3. Then X_ℰ(b_3)=(a_ij)_13× 13, where  a_ij=a_t_i,t_j∈ X_ℰ,  t_i and t_j  are  the  i-th and j-th elements in the ordered set {1,2,3,4,8,11,13,16,20,26,28,32,36}. Now, we input the coloring set  C={1,2,4,8,7,11,13,14}, the vector v_1× 13=(0,0,⋯,0) and the truncated adjacency matrix X_ℰ(b_i) into the color-recursion algorithm and run the function ColorRecursion(1,v). Suppose 
    v=(2,7,1,15,15,4,8,8,11,2,11,7,4)
 is one of the resulting truncated coloring vector. It implies that the facets 
    F_1,F_2,F_3,F_4,F_8,F_11,F_13,F_16,F_20,F_26,F_28,F_32,F_36
 can be colored by 2,7,1,15,15,4,8,8,11,2,11,7,4, respectively.




The set of all truncated coloring vectors with coloring set {1,2,4,8,c_5,⋯,c_k} is called the unfix set of b_i defined on (c_5,c_2,⋯,c_k), where c_5,⋯,c_k are the added colors. We may not emphasize the added colors if there is no ambiguity. Next, we denote the unfix set of b_i by B_i. In particular, when the first four entries are fixed to be e_1, e_2, e_3, and e_4, we obtain  the fixing set of b_i; it is denoted by B_i. By the color-recursion algorithm, we can calculate the sets B_1 and B_1 according to the adjacency information given by X_ℰ(b_1).

The set B_i (or B_i) can be regarded as an l× 13 matrix, where l is the cardinality of B_i (or B_i). In the following, we do not distinguish these two viewpoints and may refer to B_i (or B_i) as either a set or a matrix. From Lemma <ref>, every vertex of the 120-cell ℰ enjoys the same stabilizer. Thus, in a certain k-coloring case, the unfix sets of all the other blocks B_2,B_3,⋯,B_120 can be obtained from B_1 by permuting the columns of B_1. The permutation rule with respect to certain B_i, 2≤ i≤120, can be inferred from the adjacency matrix X_ℰ and the set 𝒥 formed by 120 induced permutations as defined before. 

 For example, the block b_1 and b_2 are 
    (1,2,3,4,5,6,7,8,9,10,11,12)   and   (1,2,3,4,8,10,12,14,18,26,28,30,34),
 respectively. We select the permutation, from the set 𝒥, that transfers 
    (1,2,3,4,5,6,7,8,9,10,11,12,13)  to   (2,34,26,14,12,1,10,28,8,30,4,18,3).
 By matching the positions, the unfix set of b_2 can be obtained by rearranging the column's order of B_1 as illustrated in Table <ref>. The rearranged order now is (6,1,13,11,9,7,5,4,12,3,8,10,2). This means the first column of matrix B_2 corresponds to the 6-th column of matrix B_1, and the second column of B_2 corresponds to the first column of matrix B_1, etc. Similarly, we can obtain all the orders of rearrangements  for B_i, 2≤ i≤ 120.




So far, we have prepared 120 sets B_1, B_2, ..., B_120. Now, every l× 13 matrix is extended to an l× 120 matrix by simply putting each s-th column to the position of the b_i(s)-th column, where b_i(s) means the s-th element in the set b_i, and filling in the value zero in the other positions.
We continue to use the same notation B_i (or B_i) for the extended matrix.

Now, we are going to paste the set B_1 and B_2 based on some assigned columns of them. More precisely, a row from B_1 is matched up with a row of B_2 where every two entries specified by the same index i, where i∈ b_1∩ b_2, are with the same values. The index set b_1∩ b_2 is called a linking key for the pasting. The resulting new row is actually the sum of these two rows. The set of all these new rows is denoted by B_1∪^*B_2.

We use the dodecahedron 𝒟 to explain this process. Figure <ref> is a projective illustration of the dodecahedron. The area circled by the bold line is b_1, the block of F_1; the area circled by the double line is b_2, the  block of F_2. We suppose, for example, that

B_1={x_1,x_2}={(1,2,4,4,2,6,0,0,0,0,0,0), (1,2,4,5,2,6,0,0,0,0,0,0)},

B_2={y_1,y_2,y_3}={(1,2,4,4,0,0,1,7,0,0,0,0),(1,2,4,4,0,0,6,5,0,0,0,0),

(1,2,3,4,0,0,1,7,0,0,0,0)} .


In this example, x_1 has the same color with y_1 and y_2 on the first, second, third and fourth facets. Thus, y_1 and y_2 can paste to x_1, forming the coloring vectors (1,2,4,4,2,6,1,7,0,0,0,0) and (1,2,4,4,2,6,6,5,0,0,0,0), respectively. In contrast, x_2 cannot be pasted to any element of B_2 as there are no vectors of the color 5 on the 4-th position. Therefore, 
    B_1∪^* B_2={(1,2,4,4,2,6,1,7,0,0,0,0),(1,2,4,4,2,6,6,5,0,0,0,0)}.




We then move on to paste the sets B_1 ∪^*B_2 and B_3. We follow the same procedure except that the index set that specifies the matching line, namely the linking key, is b_1∪ b_2∩ b_3. We conduct this procedure until we finish pasting the final set B_120 and obtain the resulting set  S=B_1∪^*B_2⋯∪^*B_120. The set of linking keys of this procedure is 
    {b_1∩ b_2, b_1∪ b_2∩ b_3,⋯,b_1∪ b_2∪⋯∪ b_i-1∩ b_i,⋯ ,b_1∪ b_2∪⋯∪ b_119∩ b_120}.

 
Note that the first set is a fixing set, which is different from other 119 sets. This means after pasting the final facet F_120, we actually obtain results of GL_4(ℤ_2)-equivalence representatives. The pseudocode for the block-pasting algorithm is given in Algorithm 2. 



Compared to the first algorithm, the second one is far more efficient. The color-recursion algorithm is using the method of “series circuit", where the resulting characteristic function is produced one by one. Whereas, the block-pasting algorithm adopts the idea of “parallel circuit", where different parts of a resulting characteristic vector are generated simultaneously and then  pasted together. However, it demands huge internal storage because the step of gluing may cause a surge in data. By experimental testing, at least 2000 GB of internal storage is needed to fulfill this algorithm in the  8-coloring case. Hence, we start the block matching from one single element in B_1 instead of dealing with the whole set at a time to reduce the storage requirements. Moreover, we control the amount of truncated coloring vectors that are being pasted simultaneously with the unfix set B_5 to make the computation more practical. More precisely, we set up an upper threshold of 120,000. This quantity is set in accordance with the device storage. This means the result set with the number of elements larger than 120,000 is automatically pieced into p subsets with cardinalities all less than 120,000. The number p is the smallest satisfied integer that is calculated by the computer automatically. 



 §.§ Verification and results

We have five evidences to validate our computing results:

	
  * Apply the color-recursion algorithm to compute the number of  GL_n(ℤ_2)-equivalence classes of ℤ_2^n-colorings of the n-cube I^n for n=2,3,4,5,6 as shown in Table <ref>. The results coincide with the work of Suyong Choi <cit.>.
	
	
  * Apply the color-recursion algorithm to compute the number of  GL_3(ℤ_2)-equivalence classes of ℤ_2^3-colorings  of the 3-dimensional Löbell polytope ℒ_n for n=5,6,7 as shown in Table <ref>. The results coincide with the work of Xin Fu <cit.>.
	
	
  * Apply the color-recursion algorithm to compute the number of  GL_4(ℤ_2)-equivalence classes of ℤ_2^4-colorings  of the neighborly 4-polytope with eight facets as shown in Table <ref>. The results coincide with the results done by Baralić–Milenković <cit.>
		


  * We cross compare the results of the color-recursion and the block-pasting algorithms for the 120-cell. Both of them produce 10 characteristic vectors in the setting of 5-coloring and no result in the {4,6,7}-coloring cases. Then, by comparing Table <ref> and Table <ref> we can see these two algorithms generate the same resulting characteristic vectors in the cases of (3,5,9,14)-coloring and (3,5,9,15)-coloring.


  * Although we haven't ruled out all of the characteristic vector of 9-coloring setting yet, the two algorithms both calculate a common 9-coloring small cover as reported in Table <ref>.




Through the block-pasting algorithm, we finally solve the three remaining cases as shown in Figure <ref>. There are 480× 128=61,440 orientable GL_4(ℤ_2)-equivalence small covers over the right-angled 120-cell with respect to the (7,11,13,14)-coloring case. The multiplicative equation means that there are 480 truncated coloring vectors in B_1 defined on (7,11,13,14) that can be finally pasted up; each of them equally has 128 resulting vectors.  Moreover, we can now easily compare the efficiency and productivity of the color-recursion and the proposed block-pasting algorithms through Table <ref> and Table <ref>. The latter can  be up to 60 times faster in some cases. The (3,5,9,14)-coloring case costs two hours with the block-pasting algorithm; it takes five days with the color-recursion algorithm. The performance improvement is mainly because after calculating B_1, the time taken to generating B_2, B_3,..., B_120 through permutations in the block-pasting algorithm is negligible.


	



§ SMALL COVERS OVER THE 120-CELL UP TO HOMEOMORPHISM



 §.§ DJ-equivalence rigidity

Two G-manifolds M_1 and M_2 are  DJ-equivalent if there is a homeomorphism  f: M_1 → M_2 and an automorphism θ of G, such that f(gx)=θ (g)f(x),  for all g∈ G and
x ∈ M_1.

For an n-dimensional simple polytope P, let 𝔸=𝔸(P) denote the symmetry group of P and G=GL_n(ℤ_2) denote the general linear group of degree n over ℤ_2. Two small covers M(P,λ) and M(P,μ) are DJ-equivalent if and only if λ=g ∘μ∘ a_*, where g ∈ GL_n(ℤ_2) and a_* is the permutation of facets induced by some some a ∈𝔸 <cit.>.


For example, denote the characteristic functions in (2), (3) and (4) of Figure <ref> by λ_1, λ_2, and λ_3, respectively. The manifolds M(P,λ_2) and M(P,λ_3) can be proved to be DJ-equivalent for λ_3=g∘λ_2∘ a_*, where g is a linear transformation sending the basis {e_1,e_2} to {e_2,e_1+e_2} and a_* is the permutation of facets induced by a  π/2 planar rotation a∈𝔸(P). However M(P,λ_1) is different from M(P,λ_2) and M(P,λ_3) up to DJ-equivalence.

The DJ-equivalence may coincide with the homeomorphism. By the Mostow Rigidity Theorem and the geometric feature of the hyperbolic right-angled 120-cell ℰ, Garrison and Scott have proven the following DJ-equivalence rigidity theorem.

 (Garrison-Scott <cit.>)
	Two small covers over the right-angled 120-cell are homeomorphic if and only if they are DJ-equivalent.



 
	Two hyperbolic small covers M(ℰ,λ_i) and M(ℰ,λ_j) over the right-angled 120-cell are homeomorphic if and only if (λ_i·𝔸(ℰ))⋂ (GL_4(ℤ_2)·λ_j)∅. Here λ_i·𝔸(ℰ) and GL_4(ℤ_2)·λ_j are the orbit spaces of λ_i and λ_j under the group actions of  𝔸(ℰ) and GL_4(ℤ_2), respectively.
	

Although the proof of Lemma <ref> is quite trivial, this lemma is of great importance in designing a practical grouping algorithm.
The problem of the classification up to homeomorphism has been transferred into detecting the intersections of some orbit spaces of  different characteristic functions.
We aim at finding the non-empty intersection of two small sets by Lemma <ref> instead of checking the belonging of a coloring to a big set by Theorem <ref> in order to reduce the amounts of both storage and computation. Namely we use (λ_i·𝔸(ℰ))⋂ (GL_4(ℤ_2)·λ_j)∅ rather than λ_1∈𝔸(ℰ)·λ_2· GL_4(ℤ_2) as the basic rule in designing the algorithm. For an arbitrary characteristic function λ_i, the cardinalities of 𝔸(ℰ)·λ_i and GL_4(ℤ_2)·λ_j are 14,400 and 20,160 respectively, which are much smaller than that of the large-scale multiplicative case with  290,340,000 items before duplication. This reduced  level of computational effort is much more feasible and efficient in actual experiments.



 §.§ DJ-equivalence classes of small covers over the n-cubes I^n
We firstly use the grouping algorithm to calculate the number  of DJ-equivalence classes over the n-cube as an experimental test before addressing the 120-cell case as shown in Table <ref>.

By the color-recursion algorithm, we can calculate that the number of GL_n(ℤ_2)-equivalence classes of small covers over the n-cubes I^n are 25, 543, and 29281 where, n=3, 4, and 5, respectively. These results coincide with the number of acyclic digraphs with n labeled nodes, denoted by R_n, which are counted by Robinson and Stanley in <cit.>. Based on the work of Mckay et al. <cit.>, the number of GL_n(ℤ_2)-equivalence classes over I^n equals R_n; see Section 2.1 of <cit.> for more detail. 
The diffeomorphism classes of small covers over an n-cube is given by Choi, Masuda, and Oum <cit.>. There is no direct previous result about the number DJ-equivalence classes. However, It is at least reasonable that our result on the number of the DJ-equivalence classes is bounded by the cardinalities of the results presented in <cit.>.





 §.§ Homeomorphism classes of 8-coloring small covers over the 120-cell ℰ



By Lemma <ref>, we can calculate the homeomorphism classes of the GL_4(ℤ_2)-equivalence representatives that are obtained by the proposed  block-pasting algorithm. We prove here the following two theorems; the characteristic vectors are reported afterwards.

Theorem <ref>. There are exactly 56 orientable small covers over the right-angled hyperbolic 120-cell up to homeomorphism.




We apply the Grouping algorithm to classify the 61,440 GL_4(ℤ_2)-equivalence classes of the (7,11,13,14)-coloring by DJ-equivalence and obtain 56 final results. They are all orientable according to Corollary <ref>. The characteristic vectors are all of size of 1× 120 and the 2-digits denoting forms of the colors are replaced by letters successively, namely 10, 11, 12, 13, 14, and 15 are denoted by a,  b,  c,  d,  e, and f, respectively. For example, the first result in Table <ref> actually means the characteristic vector below:



	(1,2,4,8,2,8,4,11,14,14,7,7,13,13,1,14,14,13,4,7,2,7,2,8,4,11,13,1,11,4,8,2,11,7,7,13,4,1,11,2,
		
		13,11,1,1,1,13,2,8,14,8,7,2,8,14,4,8,1,13,8,13,11,4,14,4,1,14,8,14,2,8,14,11,14,11,7,2,11,1,1,11,
		
		4,13,7,7,7,7,13,4,1,2,11,8,8,7,11,2,14,4,1,13,7,11,13,13,4,14,2,14,2,4,4,8,13,2,14,11,1,1,8,7)。



All of the 56 characteristic vectors are reported in Table <ref>. The readers can obtain the closed orientable hyperbolic 4-manifolds from the characteristic vectors  and the combinatorics of the 120-cell given in Tables <ref>–<ref>.

cccc
		
		font=normalsize
        All of the orientable small covers over the right-angled hyperbolic 120-cell ℰ up to homeomorphism.

		
		   1248284bee77dd1eed4727284bd1b482b77d41b2       	1248284bee77dd1eed4727284bd1b482b77d41b2

		
		1   db111d28e8728e481d8db4e41e8e28ebeb72b11b    2   	db111d2e8e72e8481d8db48418ee2e8beb72b11b

		
		   4d7777d412b887b2e41d7bdd4e2e2448d2eb1187      	4d7777d412b887b2e41d7bdd4e2e2448d2eb1187


		
		
		   1248284bee77dd1eed4727284bb1d482b77d41d2       	1248284bee77dd1eed4727284bb1d482b77d41d2

		
		3   bb111b28e8728e481d8dd4e41e8e28ebeb72b11d    4   	bb111b2e8e72e8481d8dd48418ee2e8beb72b11d

		
		   4b7777d412b887b2e41b7ddd4e2e2448d2eb1187      	4b7777d412b887b2e41b7ddd4e2e2448d2eb1187

		
		

		   1248284bee77dd1eed4747282bd1b482b77d21b4       	1248284bee77dd1eed4747282bd1b482b77d21b4

		
		5   db111d28e8748e481d8db4e21e8e28ebeb74b11b    6   	db111d2e8e74e8481d8db48218ee2e8beb74b11b

		
		   2d7777d212b887b4e41d7bdd4e2e2448d2eb1187      	2d7777d212b887b4e41d7bdd4e2e2448d2eb1187

		
		
		
		
		   1248284bee77dd1eed4747282bd1b482b77d21b4       	1248284bee77dd1eed4747282bd1b482b77d21b4

		
		7   db111d28e8748e481d8db4e21e8e28ebeb74b11b    8   	db111d2e8e74e8481d8db48218ee2e8beb74b11b

		
		   2d7777d212b887b4e41d7bdd4e2e2448d2eb1187      	2d7777d212b887b4e41d7bdd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd1eed4e17287bd8b482be7d71b1       	1248284bee77dd1eed4e17287bb8d482be7d71d1

		
		9   db118d24247142488d8db427124e242bebe1b18b    10   	bb118b22427124488d8dd447142e224bebe1b18d

		
		   7d7e77d782b887b1e41debdd4e2e2448d2eb1187      	7b7e77d782b887b1e41beddd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd1eed4e17287bb8d482be7d71d1       	1248284bee77dd1eed4e77281bd8b482be7d11b7

		
		11   bb118b24247142488d8dd427124e242bebe1b18d    12   	db118d22427724488d8db441142e224bebe7b18b

		
		   7b7e77d782b887b1e41beddd4e2e2448d2eb1187      	1d7e77d182b887b7e41debdd4e2e2448d2eb1187

		
		
		   1248284bee77dd1eed4e77281bd8b482be7d11b7       	1248284bee77dd1eed4e77281bb8d482be7d11d7

		
		13   db118d24247742488d8db421124e242bebe7b18b    14   	bb118b22427724488d8dd441142e224bebe7b18d

		
		   1d7e77d182b887b7e41debdd4e2e2448d2eb1187      	1b7e77d182b887b7e41beddd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd1eed4127284bd7b482b17d41b2       	1248284bee77dd1eed4127284bb7d482b17d41d2

		
		15   db117d2e8e72e8487d8db48418ee2e8beb12b17b    16   	bb117b28e8728e487d8dd4e41e8e28ebeb12b17d

		
		   4d7177d472b887b2e41d1bdd4e2e2448d2eb1187      	4b7177d472b887b2e41b1ddd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd1eed4127284bb7d482b17d41d2       	1248284bee77dd1eed4147282bd7b482b17d21b4

		
		17   bb117b2e8e72e8487d8dd48418ee2e8beb12b17d    18   	db117d28e8748e487d8db4e21e8e28ebeb14b17b

		
		   4b7177d472b887b2e41b1ddd4e2e2448d2eb1187      	2d7177d272b887b4e41d1bdd4e2e2448d2eb1187

		
		
		   1248284bee77dd1eed4147282bd7b482b17d21b4   
	    	1248284bee77dd1eed4147282bb7d482b17d21d4

		
		19   db117d2e8e74e8487d8db48218ee2e8beb14b17b    20   	bb117b28e8748e487d8dd4e21e8e28ebeb14b17d

		
		   2d7177d272b887b4e41d1bdd4e2e2448d2eb1187      	2b7177d272b887b4e41b1ddd4e2e2448d2eb1187

		
		
		   1248284bee77dd1eed4147282bb7d482b17d21d4   
	   	1248284bee77dd1eed4817287bdeb482b87d71b1

		
		21   bb117b2e8e74e8487d8dd48218ee2e8beb14b17d    22   	db11ed2242712448ed8db447142e224beb81b1eb

		
		   2b7177d272b887b4e41b1ddd4e2e2448d2eb1187      	7d7877d7e2b887b1e41d8bdd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd1eed4817287bbed482b87d71d1       	1248284bee77dd1eed4817287bbed482b87d71d1

		
		23   bb11eb2242712448ed8dd447142e224beb81b1ed    24   	bb11eb2424714248ed8dd427124e242beb81b1ed

		
		   7b7877d7e2b887b1e41b8ddd4e2e2448d2eb1187      	7b7877d7e2b887b1e41b8ddd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd1eed4877281bdeb482b87d11b7       	1248284bee77dd1eed4877281bbed482b87d11d7

		
		25   db11ed2242772448ed8db441142e224beb87b1eb    26   	bb11eb2242772448ed8dd441142e224beb87b1ed

		
		   1d7877d1e2b887b7e41d8bdd4e2e2448d2eb1187      	1b7877d1e2b887b7e41b8ddd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd7eed4721284bd1b482b71d47b2      	1248284bee77dd7eed4721284bd1b482b71d47b2

		
		27   db111d28e8128e481d8db4e47e8e28ebeb72b71b    28   	db111d2e8e12e8481d8db48478ee2e8beb72b71b

		
		   4d1777d412b881b2e47d7bdd4e2e2448d2eb1187      	4d1777d412b881b2e47d7bdd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd7eed4721284bb1d482b71d47d2      	1248284bee77dd7eed4e11287bd8b482be1d77b1

		
		29   bb111b2e8e12e8481d8dd48478ee2e8beb72b71d    30   	db118d22421124488d8db447742e224bebe1b78b

		
		   4b1777d412b881b2e47b7ddd4e2e2448d2eb1187      	7d1e77d782b881b1e47debdd4e2e2448d2eb1187


		
		
		   1248284bee77dd7eed4e11287bd8b482be1d77b1      	1248284bee77dd7eed4e11287bb8d482be1d77d1

		
		31   db118d24241142488d8db427724e242bebe1b78b    32   	bb118b22421124488d8dd447742e224bebe1b78d

		
		   7d1e77d782b881b1e47debdd4e2e2448d2eb1187      	7b1e77d782b881b1e47beddd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd7eed4e71281bb8d482be1d17d7      	1248284bee77dd7eed4121284bd7b482b11d47b2

		
		33   bb118b22421724488d8dd441742e224bebe7b78d    34   	db117d2e8e12e8487d8db48478ee2e8beb12b77b

		
		   1b1e77d182b881b7e47beddd4e2e2448d2eb1187      	4d1177d472b881b2e47d1bdd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd7eed4121284bb7d482b11d47d2      	1248284bee77dd7eed4141282bd7b482b11d27b4

		
		35   bb117b2e8e12e8487d8dd48478ee2e8beb12b77d    36   	db117d28e8148e487d8db4e27e8e28ebeb14b77b

		
		   4b1177d472b881b2e47b1ddd4e2e2448d2eb1187      	2d1177d272b881b4e47d1bdd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd7eed4141282bd7b482b11d27b4      	1248284bee77dd7eed4141282bb7d482b11d27d4

		
		37   db117d2e8e14e8487d8db48278ee2e8beb14b77b    38   	bb117b28e8148e487d8dd4e27e8e28ebeb14b77d

		
    	   2d1177d272b881b4e47d1bdd4e2e2448d2eb1187      	2b1177d272b881b4e47b1ddd4e2e2448d2eb1187

    	
		
		
		   1248284bee77dd7eed4141282bb7d482b11d27d4      	1248284bee77dd7eed4811287bbed482b81d77d1

		
		39   bb117b2e8e14e8487d8dd48278ee2e8beb14b77d    40   	bb11eb2424114248ed8dd427724e242beb81b7ed

		
		   2b1177d272b881b4e47b1ddd4e2e2448d2eb1187      	7b1877d7e2b881b1e47b8ddd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd7eed4871281bdeb482b81d17b7      	1248284bee77dd7eed4871281bdeb482b81d17b7

		
		41   db11ed2242172448ed8db441742e224beb87b7eb    42   	db11ed2424174248ed8db421724e242beb87b7eb

		
		   1d1877d1e2b881b7e47d8bdd4e2e2448d2eb1187      	1d1877d1e2b881b7e47d8bdd4e2e2448d2eb1187

		
		
		
		   1248284bee77dd7eed4871281bbed482b81d17d7      	1248284bee77db1eed4727284dd1b482b77b41b2

		
		43   bb11eb2242172448ed8dd441742e224beb87b7ed    44   	dd111d2e8e72e8481b8db48418ee2e8bed72d11b

		
		   1b1877d1e2b881b7e47b8ddd4e2e2448d2eb1187      	4d7777b412b887d2e41d7bbd4e2e2448d2eb1187

		
		
		
		   1248284bee77db1eed4e17287dd8b482be7b71b1      	1248284bee77db1eed4e17287dd8b482be7b71b1

		
		45   dd118d22427124488b8db447142e224bede1d18b    46   	dd118d24247142488b8db427124e242bede1d18b

		
		   7d7e77b782b887d1e41debbd4e2e2448d2eb1187      	7d7e77b782b887d1e41debbd4e2e2448d2eb1187

		
		
		
		   1248284bee77db1eed4e17287db8d482be7b71d1      	1248284bee77db1eed4127284dd7b482b17b41b2

		
		47   bd118b22427124488b8dd447142e224bede1d18d    48   	dd117d2e8e72e8487b8db48418ee2e8bed12d17b

		
		   7b7e77b782b887d1e41bedbd4e2e2448d2eb1187      	4d7177b472b887d2e41d1bbd4e2e2448d2eb1187

		
		
		
		   1248284bee77db1eed4817287dbed482b87b71d1      	1248284bee77db1eed4877281dbed482b87b11d7

		
		49   bd11eb2242712448eb8dd447142e224bed81d1ed    50   	bd11eb2424774248eb8dd421124e242bed87d1ed

		
		   7b7877b7e2b887d1e41b8dbd4e2e2448d2eb1187      	1b7877b1e2b887d7e41b8dbd4e2e2448d2eb1187

		
		
		
		   1248284bee77db7eed4721284db1d482b71b47d2      	1248284bee77db7eed4741282db1d482b71b27d4

		
		51   bd111b2e8e12e8481b8dd48478ee2e8bed72d71d    52   	bd111b28e8148e481b8dd4e27e8e28ebed74d71d

		
		   4b1777b412b881d2e47b7dbd4e2e2448d2eb1187      	2b1777b212b881d4e47b7dbd4e2e2448d2eb1187

		
		
		
		   1248284bee77db7eed4e11287dd8b482be1b77b1      	1248284bee77db7eed4121284db7d482b11b47d2

		
		53   dd118d24241142488b8db427724e242bede1d78b    54   	bd117b2e8e12e8487b8dd48478ee2e8bed12d77d

		
		   7d1e77b782b881d1e47debbd4e2e2448d2eb1187      	4b1177b472b881d2e47b1dbd4e2e2448d2eb1187

		
		
		
		   124828eed4bb7bde4872b1248d771e8d21b8d7bd      	12482d48be77e1bebe4d2d2d471b74828dd14b72

		
		55   27111d2e4e21e448d17e784b84e2ee44db7721b8    56   	1711b128e8d28e4db18e74e4be8b28e8e7d27bb7

		
		   71dbbbd18d2e4712e8bbdd787242e887442d11eb      	41dd7714b28d8d72b4b1d71e4e2b244de2e81187
	
		


Then, we apply the Grouping algorithm to classify the other 80 GL_4(ℤ_2)-equivalence classes of the small covers over the right-angled 120-cell. They are all non-orientable by Theorem <ref>.


Theorem <ref>. There are exactly two  8-coloring  non-orientable  small covers over the right-angled hyperbolic 120-cell up to homeomorphism.


The resulting characteristic vectors are reported in Table <ref>, and we adopt the same denotation manner.

clcc
				
				font=normalsize
				8-coloring non-orientable small covers over the 120-cell ℰ.

				
		
		   1248285e4ee9351324e1e21181e911481e9e5e82      	124827b8fff4217d2f8f1d11bf11b811b4f7fbfd

		
		 57     384245e395511e9e33195e31559e9e191335e9e3      58     	d47278fd8417f1fb2d418fd7148bf1f14d2f7f4d

		
		   28e4248e1e42315811219e1184184224e5839ee1      	dfb72741f8f2d71b11db11fb1187422f8b824ff1

		
				

	



We also list the unique, up to homeomorphism, 5-coloring characteristic vector in Table <ref>; it was found by Garrison-Scott <cit.>.





 §.§ 9-coloring small covers over 120-cell ℰ


When considering the 9-coloring case, the computation far exceeds the capabilities of today's laptops. We summarize the details in Table <ref>.

	
	
So far, we can only report that, the 9-coloring characteristic functions exist within the following three cases of added colors:

    (3, 5, 9, 14, 15), (3,7,11,13,14),  (7,11,13,14,15).

The three 9-coloring results we have found so far are all homeomorphic to the coloring in Table <ref>, which is exactly what we obtain in Section <ref> by the color-recursion algorithm. It has the same colors with the 5-coloring resulting vectors in Table <ref> in the first 33 facets.



Therefore, we find a total of 59 new characteristic functions of small covers over the 120-cell. The notation M(ℰ,λ_i), abbreviated as M_i, for 1≤ i ≤ 60, is used to denote the corresponding 60 known small covers over the 120-cell ℰ. This notation appears in Theorem <ref>.



§ INTERSECTION FORMS OF CLOSED HYPERBOLIC 4-MANIFOLDS

 Davis and Januszkiewicz formulated how to obtain the ℤ_2-coefficient cohomology groups of a small cover by the orbit polytope and its characteristic function <cit.>. In 2013, Li Cai gave  a method to calculate the ℤ-coefficient cohomology groups of the real moment-angle manifold over a simplicial complex K <cit.>. Based on the work of Cai and Suciu-Trevisan's result on rational homology groups of real toric manifolds  <cit.>, Choi and Park gave a formula of the cohomology groups of real toric manifolds <cit.>, which can also be viewed as a combinatorial version of the Hochster Theorem <cit.>.


Suppose that  P is a simple polytope and K is the simplicial complex dual to the boundary of P. Recall that we can also define the characteristic function λ on K by substituting  the facet set ℱ(P) with the vertex set 𝒱 of the simplicial complex K. See Section 2.1 for more details. We denote the linear space ℤ_2^|𝒱| by ℤ_2^𝒱. In addition, we can identify ℤ_2^𝒱 with the power set 2^𝒱 in the canonical way, where ∅ corresponds to the identity element and multiplication to the symmetric difference. Namely, we have a map φ:ℤ_2^𝒱⟶ 2^𝒱. 
Accordingly, every full-subcomplex K_ω of K, ω⊆𝒱, is identified with an element of 2^𝒱, i.e. K_ω is the full sub-complex of K=(∂ L)^* by restricting to ω⊆𝒱. The notations φ, K and K_ω will be used throughout this section with this meaning.



Let λ be a ℤ_2^r-coloring characteristic function. Denote by rowΛ the row space of the characteristic matrix Λ. The following Choi-Park theorem shows that the cohomology group of a real toric manifold M(P,λ) is the direct sum of the cohomology groups of some full sub-complexes of the dual polytope K=(∂ L)^*. The sub-complexes are determined by the characteristic function.

   (Choi-Park <cit.>) Assume G is the coefficient ring ℚ or ℤ_q for a positive odd integer q. There is an additive isomorphism 
	
    H^p(M(P,λ);G)≅φ^-1(ω) ∈ row Λ⊕H^p-1(K_ω;G).



0.
 
	For a simple polytope P,  
    β^1(M(P,λ);ℚ)=φ^-1(ω) ∈ row Λ∑β^0(K_ω;ℚ),

	where K_ω is the full sub-complex of K=(∂ P)^* by restricting to ω⊆𝒱.


By means of Corollary <ref>, we may calculate the first Betti number of a real toric manifold using the combinatorial information of the colored polytope and the row space of its characteristic matrix.

Besides, Cai and Choi discussed the cohomology groups of real toric manifolds in <cit.>. Let K=(∂ P)^* and denote by b,c and d the sum of the zeroth, first and second reduced Betti numbers of K_ω for ω∈rowΓ. They showed the following table in their paper. 



From Table <ref>, Theorem 3.1 in <cit.> and the fact that the Euler characteristic is independent of the choice of the coefficient ring, one sees that the integral cohomology group H^*(M(P,λ);ℤ) is completely determined by the reduced cohomology group of K_ω for ω∈ row Λ and the h-vector of K. As corollaries, small covers satisfy some topological restrictions. For instance, in dimension n=4, the only (if any) torsion in homology is 2-torsion. 


Over the right-angled 120-cell, there is a canonical way to construct a natural  ℤ_2^5-extension δ from a non-orientable ℤ_2^4-coloring λ such that the real toric manifold M(ℰ,δ) is the orientable double cover of the non-orientable small cover M(ℰ,λ). See <cit.> for more details. Thus, we have four natural ℤ_2^5-extensions δ_57, δ_58, δ_59, and δ_60 of the non-orientable ℤ_2^4-colorings λ_57, λ_58, λ_59, and λ_60 respectively. We use M(ℰ,δ_i), abbreviated as M_i, to denote the corresponding real toric manifold, where 57≤ i≤ 60.  These notations appear in Theorem <ref>.

Proof of Theorem <ref> (homology group part): By Theorem <ref>, the first Betti numbers of the known 60 small covers M(ℰ, λ_i) can be computed by the row spaces of their characteristic matrices. By calculation, each subcomplex K_ω_i of every small cover, φ^-1(ω_i) is a row in rowΛ and 1≤ i≤ 15 , has at most one connected component. Thus, the first Betti numbers are all zero.

Next, we use Theorem <ref> to calculate the first Betti numbers of the four orientable real toric manifolds M_57, M_58, M_59 and M_60. The results are summarized in Table <ref>.

 
Here, β^0(j) equals the 0-th Betti number of the j-th full sub-complex K_ω_j, β^1 is the first Betti number of the  manifold M(ℰ,δ_i), and i represents the serial numbers of the characteristic vectors as claimed in Tables <ref>–<ref>. For example, considering K_ω_5 of λ_60, the first full sub-complex with a non-trivial reduced zeroth homology group, the corresponding row vector is:



	(
		0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,0,0,1,0,1,1,1,0,
		
		1,0,0,0,1,1,1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,
	
	1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,1,0).



The orders of facets being selected by the non-trivial entries are:


	(9,10,11,18,20,26,35,37,39,46,48,49,50,52,56,57,58,59,61,63,64,66,67,68,73,74,75,76,78,82,89,91,103,112,118,119).



According to the adjacency matrix M_ℰ, there are a total of eight connected components. This means the reduced zeroth Betti number of K_ω_5 is seven as shown in Table <ref>. The eight groups of facets with respect to  the eight connected components are shown in Table <ref>.



With Table <ref>, Theorem 3.1 of <cit.> and some simple calculations, we complete the homology part of the proof of Theorem <ref>.   





Suppose M is a compact, oriented, topological 4-manifold. We have the concept of intersection form Q(M) as follows <cit.>.


	The symmetric bilinear form Q(M): H^2(M, ∂ M; ℤ) × H^2(M, ∂ M; ℤ) →ℤ
	defined by Q(M)(a, b)=⟨ a∪ b, [M]⟩ = a · b∈ℤ is called the intersection form
of M, where [M]∈ H^4(M, ∂ M;ℤ) is the fundamental class. 

Note that  Q(M)(a, b)=0 if a or b is a torsion element. Hence Q(M)
descends to a pairing on homology mod torsion. By choosing a basis of
the free part of H^2(M,∂ M; ℤ), we can represent Q(M) by a matrix. 

By the Hirzebruch
 signature formula, every closed oriented hyperbolic four-manifold has zero signature. Therefore, the intersection form of a closed oriented hyperbolic 4-manifold M is either
 

 	⊕_k[ 0 1; 1 0;   ]  or ⊕_k(1)⊕_k(-1).
 

The Davis manifold M^* constructed by M. Davis in <cit.> is a hyperbolic 4-manifold obtained by the 2π/5-angled 120-cell. From the characteristic number χ(M^*)=26, the volume of M^* can be obtained immediately by the Gauss-Bonnet formula as well.
Moreover, Ratcliffe and Tschantz calculated the homology of the Davis manifold and it is the first closed hyperbolic 4-manifold example for which the signature and intersection form have been determined <cit.>.



For real toric manifolds, Choi and Park have studied the  multiplication structure of the cohomology ring in <cit.>. Suppose M(P,λ) is a real toric manifold and Λ is the characteristic matrix of λ. We have 

   (Choi-Park <cit.>) Suppose ℛ is a commutative ring with unity in which 2 is a unit.There is a ring isomorphism

    H^*(M(P,λ);ℛ)≅ω∈ row Λ⊕H^*-1(K_ω;ℛ),

where the product structure on ω∈ row Λ⊕H^*-1(K_ω;ℛ) is given by the
canonical maps

    H^k-1(K_ω_1;ℛ)⊗H^l-1(K_ω_2;ℛ)→H^k+l-1(K_ω_1+ω_2;ℛ),

which are induced by simplicial maps K_ω_1+ω_2→ K_
ω_1⋆ K_
ω_2, where ⋆ denotes
the simplicial join and K_ω_1+ω_2 is the full sub-complex of K=(∂ P)^* by restricting to φ(φ^-1(ω_1)+φ^-1(ω_2)) ⊆𝒱.


By Theorem <ref>, we have the following theorem.
 (The intersection part of Theorem <ref>)
	Let M be a closed, oriented, hyperbolic, real toric 4-manifold with second Betti number β_2. Then the intersection form of M is equivalent over 
	ℤ to ⊕_β_2/2[ 0 1; 1 0;   ]. 



	By the Hirzebruch signature formula, every closed oriented hyperbolic 4-manifold has zero signature (see <cit.> for instance). Thus M has zero signature. In addition, one of the referees also point out one more straightforward approach specific to the real toric manifolds as follow: an orientable 4-dimensional real toric manifold, not necessarily hyperbolic, always admits an orientation-reversing involution, obtained by reflecting M along a hypersurface corresponding to a facet of the orbit polytope. Thus the signature of M vanishes.
	 
	Suppose a basis of ω_i∈ row Λ⊕H^1(K_ω_i;ℚ) is e_1,e_2,⋯, e_l. From the product structure on  ω_i,ω_j∈ row Λ⊕H^3(K_ω_i+ω_j;ℚ), it is obvious that e_i⊗ e_j and e_j⊗ e_i map to the same element in ω_i,ω_j∈ row Λ⊕H^3(K_ω_i+ω_j;ℚ). In addition, H^1(K_ω_i;ℚ)⊗H^1(K_ω_i;ℚ)→H^3(K_ω_i+ω_i;ℚ) is always a zero mapping for K_ω_i+ω_i=∅. Then, it can be inferred that the diagonals of the matrix of the intersection form of M are all zero. We then complete the proof by <cit.>.









10

Anderson:2006	
M. Anderson.
Dehn filling and Einstein metrics in higher dimensions.
J. Diff. Geom. 73 (2006), 219–261.

Baralic:2017	
D. Baralić and Lazar Milenković.
Surprising examples of manifolds in toric topology!
ArXiv:math:AT/1704.05932.

BP
V. M. Buchstaber and T. E. Panov.
Toric topology.
Mathematical Surveys and Monographs, vol.204, American Mathematical Society, Providence, RI, 2015.

2licai:2013
L. Cai.
On products in a real moment-angle manifold.
Journal of the Mathematical Society of Japan, Vol. 69 (2017), no. 2, 503–528.

caichoi:2017
L. Cai and S. Choi.
Integral cohomology groups of real toric manifolds and small covers.
ArXiv:math:AT/1604.06988.

suyong:08
S. Choi.
The number of small covers over cubes.
Algebraic & Geometric Topology, 8 (2008), 2391–2399.

choi_masuda:2017
S. Choi, M. Masuda and S. Oum.
Classification of real Bott manifolds and acyclic digraphs.
Trans. Amer. Math. Soc. 369 (2017), 2987–3011

ChoiP:2013
S. Choi and H. Park.
On the cohomology and their torsion of real toric objects.
Form. Math. 29 (2017), no. 3, 543–553.

choi_park:2017
S. Choi and H. Park.
Multiplication structure of the cohomology ring of real toric spaces.
ArXiv:math:AT/1711.04983v1.

conder_dobcsanyi:2005
M. Conder and P. Dobcsányi.
Applications and adaptations of the low index subgroups procedure.
Mathematics of  Computation 74 (2005), no.249, 485–497.

conder_maclachlan:2005
M. Conder and C. Maclachlan.
Compact hyperbolic 4-manifolds of small volume.
Proc. Amer. Math. Soc. 133 (2005), no.8, 2469–2476.

coxeter:1969
H. S. M. Coxeter.
Introduction to Geometry.
2nd ed. New York: Wiley, 1969.


davis:1985
M. Davis.
A hyperbolic 4-manifold.
Proc. Amer. Math. Soc. 93 (1985), 325–328.

dj:1991
M. Davis and T. Januszkiewicz.
Convex polytopes, Coxeter orbifolds and torus actions.
Duke Math. J. 62 (1991), 417–451.

EEK:1982
A. L. Edmonds, J. H. Ewing and R. S. Kulkarni.
Torsion free subgroups of Fuchsian
groups and tessellations of surfaces.
Invent. Math. 69 (1982), 331–346.

Everitt_Ratcliffe_Tschantz:2011
B. Everitt, J. G. Ratcliffe and S. T. Tschantz.
Right-angled Coxeter polytopes, hyperbolic
six-manifolds, and a problem of Siegel.
Math. Ann. 354 (2012), 871–905.

Xin:2017
X. Fu.
Small cover over Löbell polytopes.
Chin. Ann. Math. Ser. B. 38 (2017), no.2, 187–202.

gabai_meryerhoff_milley:2009
D. Gabai, R. Meyerhoff and P. Milley.
Minimun cusped hyperbolic three-manifolds.
Jour. Amer. Math. Soc. 22 (2009), no.4, 1157–1215.

gabai_meyerhoff_thurston:2003
D. Gabai, G. R. Meyerhoff and N. Thurston.
Homotopy hyperbolic 3-manifolds
are hyperbolic.
Ann. Math. 157 (2003), 335–431.

Scott:02
A. Garrison and R. Scott.
Small covers of the dodecahedron and the 120-cell.
Proc. Amer. Math.Soc. 131 (2002), no.2, 963–971.

kirby_calculus
R. E. Gompf and  A. I. Stipsicz.
4-manifolds and Kirby calculus.
Graduate Studies in Mathematics, 20. American Mathematical Society, Providence, RI, 1999. ISBN: 0-8218-0994-6.

Hochster:1977
M. Hochster.
Cohen-Macaulay rings, combinatorics, and simplicial complexes.
Ring Theory 2 (Proc. second Oklahoma conference), B. R. MacDonald and R. Morris, eds. Dekker, New York, 1977,  171–223.

Izmestiev:01
I. Izmestiev.
Three dimensional manifolds defined by coloring a simple polytope.
Math. Notes,  69 (2001), 340–346.

jtcube
M. Jacquemet and S. T. Tschantz.
All hyperbolic Coxeter n-cubes.
ArXiv:math:GT/1803.10462.

JR:1998
K. N. Jones and A. W. Reid.
Minimal index torsion-free subgroups of Kleinian groups.
Math. Ann. 310 (1998), 235–250.

Linowitz_Voight:2015
B. Linowitz, M. Stover and J. Voight.
Commensurability classes of fake quadrics.
Sel. Math. New Ser. (2019) 25: 56. https://doi.org/10.1007/s00029-019-0502-y

DLiu:2018
D. Liu. 
Cellular Chain Complex of Small Covers with Integer Coefficients and Its Application.
Acta. Math. Sin. English Ser. 34 (2018), no. 11, 1742–1754.

long:2008
C. Long.
Small volume closed hyperbolic 4-manifolds.
Bull. London Math. Soc. 40 (2008), 913–916.

longR:2000
D. D. Long and A. W.  Reid.
On the geometric boundaries of hyperbolic 4-manifolds.
Geom. Topol. 4 (2000), 171–178.

mz1
J. Ma and  F. Zheng. 
Torsions of 3-dimensional small covers. 
Chin. Ann. Math. Ser. B 38 (2017), 1311–1320 .

mz
J. Ma and  F. Zheng.
Geometrically bounding 3-manifold, volume and Betti number.
ArXiv:math:GT/1704.02889.

marshall_martin:2012
T. H. Marshall and G. J. Martin.
Minimal co-volume hyperbolic lattices, II:
Simple torsion in a Kleinian group.
Ann. Math. 176 (2012), 261–301.

Martelli:2018
B. Martelli.
Hyperbolic four-manifolds.   Handbook of group actions. Vol. III, 37–58,
Adv. Lect. Math. (ALM), 40, Int. Press, Somerville, MA, 2018.

MRS:2019	
B. Martelli, S. Riolo and L. Slavich.
Compact hyperbolic manifolds without spin structures
ArXiv:math:GT/1904.12720.

martelliRiolo:2017	
B. Martelli and S. Riolo.
Hyperbolic Dehn filling in dimension four.
Geom. Topol. 22 (2018), no.3, 1647–1716.

mckay:04
B. McKay, F. Oggier, G. Royle, N. Sloane, I. Wanless and H. Wilf.
Acyclic digraphs and eigenvalues of (0,1)-matrices.
J. Integer Seq 7 (2004), Artivle 04.3.3.

NakayamaN:05
H. Nakayama and  Y. Nishimura.
The orientability of small covers and coloring simple polytopes.
Osaka J. Math. 42 (2005), no.1, 243–256.

NV:15
S. Nugment and J. Voight.
On the arithmetic dimension of triangle groups.
Math. Comp. 86 (2017), 1979-2004.

pv:2005
L. Potyagailo and E. Vinberg.
On right-angled reflection groups in hyperbolic spaces.
Comment. Math. Helv. 80 (2005), 63–73

rt:2000	
J. Ratcliffe and S. Tschantz.
The volume spectrum of hyperbolic 4-manifolds.
Expe. Math. 9 (2000), no.1, 101–125.

rt:2001	
J. Ratcliffe and S. Tschantz.
On the Davis hyperbolic 4-manifold.
Topol. Appl. 111 (2001), no.3, 327–342.

rt:2004	
J. Ratcliffe and S. Tschantz.
Integral congruence two hyperbolic 5-manifolds. Geom. Dedicata 107 (2004), 187–209.

Robinson:70
R. Robinson.
Enumeration of acyclic digraphs.
From:“Proc. second Chapel Hill conf. on combinatorial mathematics and its applications", Univ. North Carolina, Chapel Hill (1970), 391–399.

Sara1
H. Saratchandran.
Kirby diagrams and the Ratcliffe-Tschantz hyperbolic 4-manifolds.
Topol. Appl. 202 (2016), 301–317.

Sara2
H. Saratchandran.
A four dimensional hyperbolic link complement in a standard S^4.
ArXiv:math:GT/1503.07778.

Sara3
H. Saratchandran.
A four dimensional hyperbolic link complement in a standard S^2× S^2.
ArXiv:math:GT/1504.01366.

Sara4
H. Saratchandran.
Complements of tori in #_2kS^2× S^2 that admit a hyperbolic structure.
ArXiv:math:GT/1504.06703.

S.Schleimer H.Segerman:15
S. Schleimer and H. Segerman.
Puzzling the 120-cell.
Notices Amer. Math. Soc. 62 (2015), no.11, 1309–1316.

Stabley:73
R. Stanley.
Acyclic orientations of graphs.
Discrete Math. 5 (1973), 171–178.

Suciu:13
A. I. Suciu.
The rational homology of real toric manifolds.
arXiv:AT/1302.2342v1(2013).

SuciuT:12
A. I. Suciu and  Alvise Trevisan.
Real toric varieties and abelian covers of generalized Davis-Januszkiewicz spaces.
Preprint, 2012.

trevisan
Alvise Trevisan.
Generalized Davis-Januszkiewicz spaces and their applications in algebra and topology.
Ph.D. Thesis, Vrije University Amsterdam, 2012.

Vesnin:1987
A. Vesnin.
Three-dimensional hyperbolic manifolds of Löbell type.
Sibirsk. Mat. Zh. 28 (1987), no.5, 50–53.

Vesnin:1991
A. Vesnin.
Three-dimensional hyperbolic manifolds with a common fundamental polyhedron. 
Math. Notes, 49 (1991), 575–577.

Vesnin:17
A. Vesnin.
Right-angled polyhedra and hyperbolic 3-manifolds.
Russ. Math. Surv., 72 (2017), no.2, 335–374.









	



 












	
	
	
	
	
	
	
		


|c|c|p11.3cm|
		
		font=normalsize
		Classification of k-coloring and representatives

		
	k     # representative      representatives of added color

	5     3     (3), (7), (15)

	6     7     (3,5), (3,7), (3,12), (3,13), (3,15), (7,11), (7,15)

	        (3,5,6), (3,5,7), (3,5,9), (3,5,10), (3,5,11), (3,5,14), 

	
	7    16    (3,5,15), (3,7,11), (3,7,12), (3,7,13), (3,7,15)

	
	        (3,12,15), (3,13,14), (3,13,15), (7,11,13), (7,11,15)

	        (3,5,6,7), (3,5,6,9), (3,5,6,11), (3,5,6,15), (3,5,7,9),

	
	        (3,5,7,10), (3,5,7,11), (3,5,7,14), (3,5,7,15), (3,5,9,14), 

	
	         (3,5,9,15), (3,5,10,12), (3,5,10,13), (3,5,10,15), (3,5,11,13), 

	
	8   28     (3,5,11,14), (3,5,11,15), (3,5,14,15), (3,7,11,12), (3,7,11,13),

	
	        (3,7,11,15), (3,7,12,13), (3,7,12,15), (3,7,13,14), (3,7,13,15), 

	
	        (3,13,14,15), (7,11,13,14), (7,11,13,15)

	        (3,5,6,7,9), (3,5,6,7,11), (3,5,6,7,15), (3,5,6,9,10), (3,5,6,9,11),

	
	        (3,5,6,9,14), (3,5,6,9,15), (3,5,6,11,13), (3,5,6,11,15), 

	
	        (3,5,7,9,11), (3,5,7,9,14), (3,5,7,9,15), (3,5,7,10,11), (3,5,7,10,12), 

	
	          (3,5,7,10,13), (3,5,7,10,14), (3,5,7,10,15), (3,5,7,11,13),  

	
	9    35    (3,5,7,11,14), (3,5,7,11,15), (3,5,7,14,15), (3,5,9,14,15),   

	
	       (3,5,10,12,15), (3,5,10,13,14), (3,5,10,13,15), (3,5,11,13,14), 

	
	       (3,5,11,13,15), (3,5,11,14,15), (3,7,11,12,13), (3,7,11,12,15),

	
	      (3,7,11,13,14), (3,7,11,13,15), (3,7,12,13,15), (3,7,13,14,15),

	
	       (7,11,13,14,15)

	        (3,5,6,7,9,10), (3,5,6,7,9,11), (3,5,6,7,9,14), (3,5,6,7,9,15), 

	
	        (3,5,6,7,11,13), (3,5,6,7,11,15), (3,5,6,9,10,12), (3,5,6,9,10,13),

	
	       (3,5,6,9,10,15), (3,5,6,9,11,13), (3,5,6,9,11,14), (3,5,6,9,11,15),  

	
	         	(3,5,6,9,14,15), (3,5,6,11,13,14), (3,5,6,11,13,15), (3,5,7,9,11,13),  

	
	10    35    	(3,5,7,9,11,14), (3,5,7,9,11,15), (3,5,7,9,14,15), (3,5,7,10,11,12), 

	
	       (3,5,7,10,11,13), (3,5,7,10,11,15), (3,5,7,10,12,14),  

	
	        (3,5,7,10,12,15), (3,5,7,10,13,14), (3,5,7,10,13,15),  

	
	       (3,5,7,10,14,15), (3,5,7,11,13,14), (3,5,7,11,13,15),

	
	        (3,5,7,11,14,15), (3,5,10,13,14,15), (3,5,11,13,14,15), 

	
	       (3,7,11,12,13,14), (3,7,11,12,13,15), (3,7,11,13,14,15)

	      (3,5,6,7,9,10,11), (3,5,6,7,9,10,12), (3,5,6,7,9,10,13), 

	
	      (3,5,6,7,9,10,15),  (3,5,6,7,9,11,13), (3,5,6,7,9,11,14),  

	
	      (3,5,6,7,9,11,15), (3,5,6,7,9,14,15), (3,5,6,7,11,13,14),

	
	       (3,5,6,7,11,13,15), (3,5,6,9,10,12,15), (3,5,6,9,10,13,14),

	
	11   28   (3,5,6,9,10,13,15), (3,5,6,9,11,13,14), (3,5,6,9,11,13,15),  

	
	      (3,5,6,9,11,14,15), (3,5,6,11,13,14,15), (3,5,7,9,11,13,14), 

	
	      (3,5,7,9,11,13,15), (3,5,7,9,11,14,15), (3,5,7,10,11,12,13),

	
	       (3,5,7,10,11,12,15), (3,5,7,10,11,13,14), (3,5,7,10,11,13,15),

	
	      (3,5,7,10,12,14,15), (3,5,7,10,13,14,15), (3,5,7,11,13,14,15), 

	      (3,7,11,12,13,14,15), (3,5,6,7,9,10,11,12),  

	
	      (3,5,6,7,9,10,11,13), (3,5,6,7,9,10,12,15), (3,5,6,7,9,10,13,14),

	
	       (3,5,6,7,9,10,13,15), (3,5,6,7,9,11,13,14), (3,5,6,7,9,11,13,15),  

	
	12   16   (3,5,6,7,9,11,14,15), (3,5,6,7,11,13,14,15), (3,5,6,9,10,13,14,15), 

	
	      (3,5,6,9,11,13,14,15), (3,5,7,9,11,13,14,15), (3,5,7,10,11,12,13,14), 

	
	      (3,5,7,10,11,12,13,15), (3,5,7,10,11,13,14,15)

	      (3,5,6,7,9,10,11,12,13), (3,5,6,7,9,10,11,12,15),
       (3,5,6,7,9,10,11,13,14), (3,5,6,7,9,10,11,13,15),
 13   7   (3,5,6,7,9,10,13,14,15), (3,5,6,7,9,11,13,14,15), 

	      (3,5,7,10,11,12,13,14,15)

	      (3,5,6,7,9,10,11,12,13,14), (3,5,6,7,9,10,11,12,13,15),

	14   3   (3,5,6,7,9,10,11,13,14,15)

	15   1   (3,5,6,7,9,10,11,12,13,14,15)

	











