Geometrically bounding 3-manifold, volume and Betti number

It is well known that an arbitrary closed orientable $3$-manifold can be realized as the unique boundary of a compact orientable $4$-manifold, that is, any closed orientable $3$-manifold is cobordant to zero. In this paper, we consider the geometric cobordism problem: a hyperbolic $3$-manifold is geometrically bounding if it is the only boundary of a totally geodesic hyperbolic 4-manifold. However, there are very rare geometrically bounding closed hyperbolic 3-manifolds according to the previous research [11,13]. Let $v \approx 4.3062\ldots$ be the volume of the regular right-angled hyperbolic dodecahedron in $\mathbb{H}^{3}$, for each $n \in \mathbb{Z}_{+}$ and each odd integer $k$ in $[1,5n+3]$, we construct a closed hyperbolic 3-manifold $M$ with $\beta^1(M)=k$ and $vol(M)=16nv$ that bounds a totally geodesic hyperbolic 4-manifold. The proof uses small cover theory over a sequence of linearly-glued dodecahedra and some results of Kolpakov-Martelli-Tschantz [9].


Geometrically bounding 3-Manifolds.
It is an open question what kind of closed n-manifolds can bound (n + 1)-manifolds. Farrell and Zdravkovska once conjectured that every almost flat n-manifold bounds a (n + 1)-manifold, for example, see [6,4].
There is a well-known result given by Rohlin in 1951 that Ω 3 = 0, which means every closed orientable 3-manifold M bounds a compact orientable 4-manifold (for example, see Corollary 2.5 of [18]). Farrell and Zdravkovska once conjectured [6] that every flat n-manifold M is the cusp section of a one-cusped hyperbolic (n + 1)-manifold. However, Long-Reid [11] gave a negative answer to this stronger conjecture by showing that if M is the cusp section of a one-cusped hyperbolic 4n-manifold, then the η-invariant of M must be an integer.
Long-Reid also further studied what kind of 3-manifolds bound geometrically [11]. If a hyperbolic n-manifold M is the unique totally geodesic boundary of a hyperbolic (n + 1)manifold N , then we say M bounds geometrically or M is a geometrically bounding hyperbolic n-manifold. See also Ratcliffe-Tschantz [16] for cosmological motivations of studying geometrically bounding hyperbolic 3-manifolds. Geometrically bounding 3-manifolds are difficult to seek since there are very few examples of hyperbolic 4-manifolds. Moreover, Long-Reid showed [11] that if a hyperbolic closed 3-manifold M is geometrically bounding, then the η-invariant η(M ) ∈ Z. By Theorem 1.3 of Meyerhoff-Neumann [13], the set of η-invariants See Ratcliffe-Tschantz [17] for counting questions on the number of totally geodesic hyperbolic 4-manifolds with the same 3-manifold boundary M , and Slavich [19,20] for other topics on geometrically bounding 3-manifolds. See also the recent paper [10] by Kolpakov-Reid-Slavich for geodesically embedding questions about hyperbolic manifolds and the relation between the geodesically embedding and geometrically bounding is subtle.
1.2. Small covers. Small covers, or Coxeter orbifolds, were studied by Davis and Januszkiewicz in [5], see also [24]. They are a class of n-manifolds which admit locally standard Z n 2 -actions, such that the orbit spaces are n-dimensional simple polytopes. The algebraic and topological properties of a small cover are closely related to the combinatorics of the orbit polytope and the coloring on the boundary of that polytope. For example, the mod 2 Betti number β (2) i of a small cover M over the polytope L agrees with h i , where h = (h 0 , h 1 , . . . , h n ) is the h-vector of the polytope L [5].
Those manifolds admitting locally standard Z k 2 -actions would form a class wider than small covers. We will say more about this topic in Section 2, and here we just give the definition for the convenience of stating Theorem 1.5. Definition 1.3. Let L be an n-dimensional simple polytope, F be the set of co-dimensional one faces of L. Such faces are called as facets. A Z k 2 -coloring is a map λ : F −→ Z k 2 satisfying λ(F 1 ), λ(F 2 ), . . . , λ(F n ) generate a subgroup of Z k 2 which is isomorphic to Z n 2 , when the facets F 1 , F 2 , · · · , F n are sharing a common vertex.
Conversely, through Proposition 1.7 of [5], from a Z k 2 -coloring λ and a principal Z k 2bundle over an n-dimensional simple polytope L, we can get a unique closed n-manifold M . In particular, we can use 2 k copies of L, namely L×Z k 2 , to construct a quotient space M (L, λ) under the following equivalent relation:    p = q and g 1 = g 2 , if p ∈ Int L p = q and g 1 g −1 Here for a face f of the simple polytope L, G f is the subgroup generated by λ(F i 1 ), λ(F i 2 ), . . . , λ(F i k ), where f = F i 1 ∩ F i 2 ∩ . . . ∩ F i k is the unique face that contains p as an interior point and F i j ∈ F. It is easy to see M (L, λ) is a closed n-manifold.
A simple example is that if we color the four co-dimensional one faces of a tetrahedron by e 1 , e 2 , e 3 and e 1 + e 2 + e 3 respectively, where e 1 , e 2 and e 3 are the standard basis vectors of Z 3 2 . Then from the above construction, we can get the closed orientable 3-manifold RP 3 . It should be noted that a tetrahedron admits a unique right-angled spherical structure, and these spherical structures on copies of the tetrahedron are glued together to build up the unique spherical structure on RP 3 . This point of view appeals in this paper.
In the following of this section, we suppose P to be the regular right-angled hyperbolic dodecahedron in H 3 with twelve 2-dimensional facets. We let nP be the linearly-gluing of n copies of P . It is obvious that nP has 12 pentagonal facets and 5n − 5 hexagonal facets. See Section 2.5 for more details. Definition 1.4. From a Z 3 2 -coloring λ on the polytope nP , we obtain a natural Z 4 2 -coloring δ on nP by following manners: Supposing {e 1 , e 2 , e 3 , e 4 } is the standard basis of Z 4 2 . For each facet F of nP , if λ(F ) = Σ 3 i=1 x i e i , x i = 1 or 0, then we take δ(F ) = Σ 4 i=1 x i e i , where x 4 = 1+Σ 3 i=1 x i mod 2. A Z 3 2 -coloring λ is called non-orientable when the 3-manifold M (nP, λ) is non-orientable. Furthermore, if the 3-manifold M (nP, λ) is non-orientable, then its natural Z 4 2 -coloring δ is called an admissible extension of λ or a natural Z 4 2 -coloring associated to λ (say a natural Z 4 2 -extension of λ for short). It can be shown that M (nP, δ) is the orientable double cover of M (nP, λ) when λ is non-orientable.
The following is our main technical theorem. Theorem 1.5. For each n ∈ Z + and each odd integer k ∈ [1, 5n+3], there is a non-orientable Z 3 2 -coloring λ on the polytope nP , such that the first Betti number of the orientable 3-manifold M (nP, δ) is k, where δ is the natural Z 4 2 -extension of λ.
As one orientable 3-manifold M may double cover many non-orientable 3-manifolds, we must show the orientable 3-manifolds under consideration are not homeomorphic in order to prove Theorem 1.2. And here the first Betti number is the classification index we adopt to determine the lower bound. Now on one hand, based on Theorem 1.5, for a given n ∈ Z + and an odd integer k ∈ [1, 5n + 3], we can construct an orientable 3-manifold M (nP, δ) whose first Betti number is exactly k. Moreover, we believe that the inverse side is true as well. That means the first Betti numbers of M (nP, δ) are definitely the odd integers in [1, 5n + 3], where δ is the natural Z 4 2 -extension of λ and λ is among all possible non-orientable Z 3 2 -colorings over the polytope nP . The "only if" part is only checked by programming so far and we haven't proof it precisely yet. So we here just list them together as a question: Question 1.6. Is k ∈ Z the first Betti number of M (nP, δ), where δ is the natural Z 4 2 -coloring associated to a non-orientable Z 3 2 -coloring λ on nP , if and only if k is odd in [1, 5n + 3]?
Proof of Theorem 1.2 For a non-orientable Z 3 2 -coloring λ on the polytope nP , there is a natural Z 4 2 -extension δ on nP . Both M (nP, δ) and M (nP, λ) are 3-manifolds and M (nP, δ) is the orientable double cover of M (nP, λ). See Section 3, in particular, Proposition 3.5 for more details. Now there are two methods to show M (nP, δ) is geometrically bounding: firstly we may use Proposition 2.9 in [9] to extend the Z 4 2 -coloring δ on the 3-dimensional polytope nP to a Z 5 2 -coloring ε on the 4-dimensional polytope nE. Here nE is a 4-dimensional polytope obtained by linearly-gluing n copies of the hyperbolic right-angled 120-cell E. Then M (nE, ε) is an orientable hyperbolic 4-manifold in which M (nP, λ) can be embedded. Secondly since M (nP, δ) is the orientable double cover of M (nP, λ), we can thus apply Corollary 8 of [12] directly because M (nP, δ) admits a fixed-point free orientation-reversing involution. Therefore M (nE, ε) − M (nP, λ) is a totally geodesic hyperbolic 4-manifold with boundary M (nP, δ). Now from Theorem 1.5, Theorem 1.2 follows.
Outline of the paper: In Section 2, we give some preliminaries on algebraic theory of small covers. In Section 3, we show Lemma 3.12, which is the key of our paper. In Section 4 and Section 5, we prove Theorem 1.5 for n is even and odd respectively.

Preliminaries
2.1. Polyhedral product. Let K be an abstract simplicial complex with ground set [m] := {1, 2, . . . , m}, so we have ∅ ∈ K ⊂ 2 [m] . If σ ∈ K, then for all τ ⊂ σ we have τ ∈ K. We associate m pairs of topological spaces, ( Then the corresponding polyhedral product (X, A) K is defined as That is to say (X, are the same pair (X, A), then (X, A) K is abbreviated as (X, A) K . Specially, (D 1 , S 0 ) K is defined as a real moment-angle complex, denoted by RZ K .
For example, let K to be the 1-skeleton of the 2-simplex, namely an abstract simplicial complex 2 [3] \{1, 2, 3}, then we have (D 1 , S 0 ) By Davis [3] and L. Cai [1], RZ K is a topological n-manifold if and only if K is a generalized homology (n − 1)-sphere K with π 1 (|K|) = 0 when n = 1, 2. Specially, assuming L to be an n-dimensional simple polytope and K is the dual of the boundary of L, then RZ K is definitely to be a topological n-manifold. Then for simple polytope, there is an equivalent but more practical way in describing the moment-angle manifold by using the language of coloring and conducting the re-construction procedure.

2.2.
Re-construction procedure. For an n-dimensional simple polytope L, let F(L) = {F 1 , F 2 , . . . , F m } be the set of co-dimensional one faces of L. Taking {e 1 , e 2 , . . . , e m } to be a basis in Z m 2 . Then we define a Z m 2 -coloring characteristic function by mapping F i to e i . λ is also named as a Z m 2 -coloring for short. Because the images of these facets are the basis elements, it is naturally satisfied that λ(F 1 ), λ(F 2 ), . . . , λ(F n ) generate a subgroup of Z m 2 , which is isomorphic to Z n 2 , when the facets F 1 , F 2 , . . . , F n share a common vertex.
Then we can construct M (L, λ) := L × Z m 2 / ∼ by the following equivalent relation: (x, g 1 ) ∼ (y, g 2 )⇐⇒ x = y and g 1 = g 2 if x ∈ Int L, x = y and g −1 where f = F i 1 ∩ · · · ∩ F i n−k is the unique co-dimensional (n − k)-face that contains x as an interior point, and G f is the subgroup generated by λ(F i 1 ), λ(F i 2 ), . . . , λ(F i n−k ). It is easy to proof that M (L, λ) is exactly the real moment-angle complex RZ K over its dual K = (∂L) * . Hence we also denote the manifold by M (K, λ). We use Example 2.1 to illustrate the homeomorphism between the two spaces according to the two definitions respectively. And by replacing the facet set F(L) with the vertice set of K, the characteristic function λ can also be seen as being defined on the simplicial complex K.  Now we will have eight polytopes, namely 2 × Z 3 2 , as shown in Figure 2.

2.3.
Buchstaber invariant and real toric manifolds. Let L be an n-dimensional simple polytope. From the construction of the real moment-angle complex RZ K , we can easily see that there is a natural Z m 2 -action over RZ K , where m is the cardinal number of the facet set F(L). The maximal rank among subgroups of Z m 2 that acts freely on RZ K is called the Buchstaber invariant and denoted by S R (L). Now we use H r to represent a subgroup of Z m 2 that acts on RZ K freely, where r is the rank of H r satisfying 0 ≤ r ≤ S R (L). Then we can have a smooth closed manifold RZ K /H r by quotient. Those smooth closed manifolds obtained by quotienting free Z r 2 -actions from the real moment-angle manifolds are called real toric manifolds. If S R (L) = m − n, RZ K /H (m−n) is named as a small cover. If L is a 3-dimensional simple polytope, by the Four Color Theorem, S R (L) = m − 3. Namely small covers can always be realized over any 3-dimensional simple polytope.
Then we have a short exact sequence and can define the Z m−r 2 -coloring characteristic function λ H r = q • λ • i −1 d through a commutative diagram as shown below, where q is the quotient map and i d is the identity.
The free action requirement ensures that the non-singularity condition always holds at every vertex. That means for each vertex The re-construction procedure can be applied parallelly to all real toric manifolds. Thus there is a one-to-one correspondence between real toric manifolds {RZ K /H r | H r < Z m 2 and acts freely on RZ K } and the set of pairs of polytopes and characteristic functions {(L, λ H r )}. We always use M (L, λ H r ) to denote the corresponding real toric manifold.
2.4. Algebraic topology of RZ K /H r . In [5], Davis and Januszkiewicz have proven that the Z 2 -coefficient cohomology groups of a small cover depend only on the polytope and its characteristic function. In 2013, Li Cai gave a method to calculate the Z-coefficient cohomology groups of RZ K [1]. Based on the work of Cai and Suciu-Trevisanon's result on rational homology groups of real toric manifolds [21,22], Choi and Park then gave a formula of the cohomology groups of real toric manifolds [2], which can also be viewed as a combinatorial version of Hochster Theorem [8].
Let K be a simplicial complex on [m]. We have a bijective map ϕ : denotes the power set of [m]. Let λ be a Z n 2 -coloring characteristic function, then the binary matrix Λ (n×m) = (λ(F 1 ), λ(F 2 ), . . . , λ(F m )) is called as characteristic matrix. We denote rowΛ to be the Z 2 -space generated by the n rows of Λ, namely the row space of the characteristic matrix Λ. Then we have the following theorem: where K ω is the full sub-complex of K by restricting to ω ⊂ [m]. In particular, Here every full-subcomplex K ω is represented by a vector ϕ −1 (ω), which is actually an element of the row space rowΛ. Such kind of vector is called the representative of K ω .
2.5. Object polytopes nP . In the following, we always assume P to be the regular rightangled hyperbolic dodecahedron in H 3 with twelve 2-dimensional facets. Using nP , specially 1P = P , to denote the linearly-glued n copies of P as shown in Figure 8, and nK, specially 1K = K, is the dual of the boundary of nP . So for each polytope nP , n 2, there are (n+3) layers of facets of nP : both the first and the last layer are pentagons; both the second and the (n + 2)-th layers consist of five pentagons; each layer from the third to the (n + 1)-th consists of five hexagons. There is no hexagonal layer in 1P , and the polytope nP has (5n + 7) facets in total. All the polytopes nP , n ∈ Z + , are right-angled hyperbolic polytopes. Notations nP and nK make sense in the rest of this paper unless other statements. If N is the dual of a simple polytope L, namely N = (∂L) * , there is a one-to-one correspondence between facets of L and vertices of N . Thus we have |F(L)| = |{vertices of N }| = m. Thus the matrix is also called as the adjacent matrix of simple polytope L and can be denoted by either X(L) or X(N ). For nP , n ∈ Z + , is always a flag simple polytope, then all the intersecting information about its facets is included in the adjacent matrix.
In order to get more disciplined adjacent matrices X(nP ) as n increases, we order the facets of the polytope nP by the following manners: the first and the last layer are ordered as 1 and 5n + 7 respectively; the facets between are labeled layer by layer. For an even layer, we start from the middle and then order the rest by doubly siding (left-right). While for an odd layer we adopt a right-left doubly siding.
We illustrate all these descriptions on 5P as shown in Figure 9, where the double sidings of even and odd layers are displayed by the arrow-lines on the second and third layers respectively. Using this ordering manner, we achieve more unified increasing patterns of the adjacent matrices and display some as shown in Figure 10 (the omitted entries are all zeros): Figure 10. Adjacent matrices of the polytopes P , 2P and 3P .
2.6. classification. All the real toric manifolds over the same polytope L are G-manifolds of L [5]. Two G-manifolds M 1 and M 2 are θ-equivariant homeomorphic if there exists a homeomorphism f : M 1 → M 2 and an automorphism θ of G, such that f (gx) = θ(g)f (x) for every g ∈ G and x ∈ M 1 . Those two G-manifolds are then said to be DJ-equivalent over L. For an n-dimensional simple polytope L We fix the colorings of three facets of the polytope nL, which are adjacent to a fixed vertex, to be e 1 , e 2 and e 3 , the standard basis of Z 3 2 . Therefore the general linear group action GL 3 (Z 2 ) have been moduled out. All elements of one class of {M (L, λ)}/GL 3 (Z 2 ) are said to be GL 3 (Z 2 )-equivalent, where {M (L, λ)} is the set of all possible small covers over L. For example, assuming L is a square, and we order the facets as shown in (1) of Figure 11. Then there are totally 18 small covers that recovered from 18 colored L. The image of these characteristic functions on F(L) is arranged in a vector, (λ(F 1 ), λ(F 2 ), λ(F 3 ), λ(F 4 )), as shown in Table 1. Table 1. All the small covers over square.

A key lemma
The purpose of this section is to prove Lemma 3.12, which is the key in the proof of Theorem 1.5.
2 -coloring λ on the polytope nP , we can extend it to (2 m −1) many Z 4 2 -colorings on nP by adding a non-zero fourth row to the 3×m characteristic matrix Λ of λ as shown below where m = 5n + 7 and * ∈ {0, 1}. Those characteristic functions are called the extensions of λ and they naturally satisfy the non-singularity condition, 2 -coloring λ on the polytope nP is admissible if there is a Z 4 2 -coloring extension of λ, denoted by δ, satisfying M (nP, λ) is non-orientable and M (nP, δ) is orientable.
H. Nakayama and Y. Nishimura discussed the orientability of a small cover in [15] and below is the main theorem. In particular, a small cover M (L, λ) over a 3-dimensional simple polytope L is orientable if and only if it is colored by e 1 , e 2 , e 3 and e 1 + e 2 + e 3 up to GL 3 (Z 2 )-action. Theorem 3.3 can actually be adjusted to meet all real toric manifolds instead of only small covers by merely parallel generalization. We can also use Theorem 2.2 with rational coefficient to re-check this claim. Because the n-th Betti number of a real toric manifold M (L, δ) is 1 if and only if there is an element in the row space of ∆ with all the entries are 1. Here ∆ is the characteristic matrix of δ, which is a Z 4 2 -coloring extension of λ. When M (L, λ) is non-orientable, then the only possible row of row∆ with all entries are 1 is the row contributed by summing up all four rows of ∆. That is to say the sum of every column of the characteristic matrix ∆ is certainly 1 mod 2.
So by Theorem 3.3 and paragraph above, we can have: Furthermore, we can obtain the following proposition based on previous discussions and some facts about fundamental group of a double cover: Proposition 3.5. For every non-orientable Z 3 2 -coloring λ over the 3-dimensional polytope nP , there is a unique Z 4 2 -coloring extension δ such that the 3-manifold M (nP, δ) is the orientable double cover of the non-orientable 3-manifold M (nP, λ).
Proof. By Theorem 3.3, every non-orientable Z 3 2 -coloring λ over the 3-dimensional simple polytope nP has a unique Z 4 2 -coloring extension δ such that M (nP, δ) is orientable. The characteristic matrix of δ is the only one, among all Z 4 2 -extensions of λ, satisfying that the sum of every column is 1 mod 2.
The Z 4 2 -coloring δ on the polytope nP in Proposition 3.5 is called an admissible extension of λ or a natural Z 4 2 -coloring associated to λ (say a natural Z 4 2 -extension of λ for short). From this proposition, on one hand, the existence and the uniqueness of the admissible extension make sense. On the other hand the orientability of a real toric manifold can be detected easily by the characteristic matrix. In particular, (1, 1, 0), (1, 0, 1) and (0, 1, 1) in Z 3 2 , which are the binary form of 3, 5 and 6, are the only three elements whose item sum are 0 mod 2. So a characteristic vector corresponding to a non-orientable Z 3 2 -coloring, should contain at least one of 3, 5 and 6. Moreover, by Theorem 2.2, the Betti number of the orientable manifold recovered by the admissible extension, called as the natural Z 4 2 -extension δ, can be computed clearly. And we show this routine in Example 3.6 as a demonstration.  (1) is the dodecahedron P whose 12 facets has been ordered, Figure 13 (2) is the simplicial complex K = (∂P ) * with its 12 vertices being ordered correspondingly. For all ω i ∈ RowΛ, we calculated its contribution to the Betti numbers in Table 2. Table 2: β 1 (M (P, δ)).
Therefore the first Betti number β 1 (M (nP, λ)) can be figured out by counting and summing up the numbers of connected components of all those subcomplexes. Here each subcomplex corresponds to a non-zero vector in the row space row∆ as shown in Example 3.6.
We perform a great number of calculations upon such process through the computer. By using CT-tree we are able to seek out a certain coloring family that change regularly and therefore construct a family of geometrically bounding 3-manifolds M (nP, δ). According to the one-to-one corresponding discussed in Section 2.3, constructing such a specific manifold actually means to work out the required characteristic function δ over the polytope nP .
Definition 3.7. A downward pentagon coloring brick (d-brick for short) is a Z 3 2 -colored figure with five pentagons glued as in Figure 14 (1). An upward pentagon coloring brick (u-brick for short) is a Z 3 2 -colored figure with five pentagons glued as in Figure 14 (2). A hexagon coloring brick (h-brick for short) is a Z 3 2 -colored figure with five hexagons glued as in Figure  14 (3). All the bricks can also be represented by an 5-length integer vector through binary transformation. We adopt a uniform format S = (a 1 a 2 a 3 a 4 a 5 ) to denote a coloring brick.
Here a 1 , a 2 , a 3 , a 4 and a 5 are the five colors in Z 3 2 . A colored nP consists of one d-brick, (n − 1) many h-bricks and one u-brick successively. In particular, in the case of a Z 3 2 -colored dodecahedron P , there are only one downward pentagon coloring brick, one downward pentagon coloring brick and no hexagon coloring bricks. Moreover, we have shown before that there are 2155 GL 3 (Z 2 )-equivalent classes of Z 3 2coloring on P that can recovered as non-orientable manifolds. Then by adding the information of 120 permutation vectors that encode all the 120 elements information of dodecahedron P 's symmetry group A, we further obtain DJ-equivalent classes through computer programming. We obtain exactly 24 non-orientable DJ-equivalent classes and one orientable DJ-equivalent class in all the small covers over the dodecahedron P . Such result also coincides with Corollary 3.4 of Garrison-Scott's paper [7]. is a coloring vector of Figure 15 (1), whose characteristic vector C is (1, 2, 4, 4, 2, 6, 1, 7, 7, 1, 3, 2, 4, 4, 2, 6, 1). Figure 15 (2) is the order required in Section 2.5. We can get the characteristic vector C from a coloring vector simply by adjusting elements' positions according to the assigned order. For example, the 5-th facet is colored by (0, 1, 0), which is the binary form of 2, so the 5-th item of C is 2. Figure 15. The left is a coloring over the polytope 2P and the right is the facet order of 2P .
Definition 3.9. Two Z 3 2 -coloring bricks are said to be compatible if the non-singularity condition is satisfied at all ten intersecting vertices. They form an compatible pair. See Figure 16 (1). Definition 3.11. The affix set of a downward/upward pentagon Z 3 2 -coloring brick S, denoted by A(S), is the set of colorings in Z 3 2 that are compatible with S. Namely the non-singularity condition holds at all five vertices as shown in (2) and (3)   Now we can prove the key lemma of this section: Lemma 3.12. Let n ∈ Z + be an even number, there is a non-orientable Z 3 2 -coloring λ over the polytope nP , such that for its natural associated Z 4 2 -coloring extension δ, we have β 1 (M (nP, δ)) = n + 1.
Proof. We start with the case n = 2. The polytope 2P has five layers of facets: both the first and the fifth layer are pentagons; the second and the fourth layers consist of five pentagons respectively; and the third layer contains five hexagons. These three coloring bricks appeal in all Lemma 3.12, Lemma 4.1 and Lemma 4.4. We still use S 1 , S 2 , S 3 to denote the coloring bricks even after being rotated. For example we keep S 2 S 3 to represent (65372 57163) although the second brick (57163) of this pair is actually what we rotate S 3 = (35716) by angle 2π 5 from left. But we will claim the elements clearly before adopting the abbreviation sign to avoid misunderstanding. The unified format of our claim is (65372 57163)=S 2 S 3 A. Next we select out a = 1, b = 1 and c = 4 as corresponding affix elements from the affix set of S 1 , S 2 and S 3 respectively.
Next, we can use these coloring bricks and their affix elements to build up 3 1 · 2 1 · 2 1 · · · 2 1 n+1 = 3 · 2 n many Z 3 2 -coloring functions over the polytope nP . The entries of coloring bricks may be adjusted by rotating in order to fit the facet structure of nP as well as obey the adjacent relations required by the three compatible pairs. Denoting Now by [aS 1 A 1 a] we mean the color polytope 2P as shown in Figure 17. Following the ordering manner required in Section 2.5, the adjacent matrix of (∂(2P )) * is: Because the labels of facets and the adjacent relations go with the rotation, parts of corresponding coloring vectors are the same if they yield to the same brick. And here the characteristic vector C of the colored polytope 2P is (1, 2, 4, 4, 2, 7, 7, 1, 5, 6, 3, 2, 4, 4, 2, 7, 1).
In order to make things more concise, we use the language of CT-Tree to illustrate our proof. For every i-th row ω i = (w i1 , ..., w ij , ..., w im ) of the row space row∆, where m = 5n + 7 is the number of facets of nP and 1 ≤ i ≤ 2 4 − 1. We define the following indices: For the adjacent matrix X(nP ) of nP , we use X(nP, ω i ) to denote the l i ×l i complement minor after excluding all q-th rows and q-th columns from X(nP ) where q ∈ ω 0 i . We call b(i) := {j|j > i and a ij = 1, where a ij ∈ X(nP, ω i )} as i-band or band of i, and b(i) is a subset of [m].

1-st branch
Conducting the same operation to every i-th row of X(nP, ω i ), we generate the 2ndbranch of the CT-tree as shown in Table 4, which actually means placing the b(i j ) row by row where i j ∈ {i 11 , i 12 , ..., i 1l(i 1 ) }.
branch structure 3 7, 9 1st-branch 7 9, 13 2nd-branch 9 13 The second branch is obtained by branching a row. There is only one single band b(i 1 ) = {i 11 , i 12 , ..., i 1l(i 1 ) } in the first branch. And the band part of the second branch consists of all the bands of elements of that row, that is (b(i 11 ), ... , b(i 1l(i 1 ) )). We use band as a unit to depict the shelf structure within a branch. There are two bands in the 2nd-branch of ω 1 as shown in Table 5. Moreover, the operation of u-branching a band means cancelling the repeating elements when branching a band and leaving only the one in the top-left position. Then we can define the operation of u-branching a branch, which resulting from u-branching every band of a branch and later place them row by row into the table. Thus we can get the (n + 1)-th branch (n > 2) by u-branching the n-th branch. We generate the CT-tree by u-branching until we reach a branch whose band part is empty after removing the elements that are identical with those of the corresponding lead part. Such branch is called a cadence branch. Denoting the last lead as i q . Then we have a lead vector of CT-tree ω 1 and applying the above CT-tree procedure to ω 1 (1) , we then find the second connected component ω 1 and applying the above CT-tree procedure to ω 1 (2) . Continuing this procedure until we reach a t ∈ Z ≥1 , such that ω 1 (t) is empty. Then we can conclude that there are exactly t connected components in the subcomplex K ω 1 .
By this method, we can calculate out that β 1 (M (2P, δ))=3 as shown in Table 8 and thus finish the proof of Lemma 3.12 in the case n = 2. Table 8. The adjacent matrix of (∂(nP )) * changes regularly as n increases. Here "regularly" results from the same adjacency changing pattern upon the same parity. Three ordered polytopes 2P , 4P and 6P are shown in Figure 18. The adjacent manner of the 17th facet of the polytope 2P is the only one that is different from the same position of the polytope 4P , while the facets from 17th to 27th of the polytope 4P follow the same order manner of facets from 7th to 17th of the polytope 2P by simply plus 10. The same changing pattern makes sense when we compare the adjacency patterns between the polytopes 4P and 6P . Which means the adjacent manner of the 27th facet of the polytope 4P is the only one that is different from the same position of the polytope 6P , while the facets from 27th to 37th of the polytope 6P follow the same order manner of facets from 17th to 27th of the polytope 4P by simply plus 10. Generally speaking, the adjacent manner of the (5n + 7)th facet of the polytope nP is the only one that different from the same position of the polytope (n + 2)P while facets from (5n + 7)th to (5(n + 2) + 7)th of the polytope (n + 2)P are following the order manner of facets from (5(n − 2) + 7)th to (5n + 7)th of the polytope nP by simply plus 10. From the procedure of the Betti number calculation, we can see the first Betti number of M (nP, δ) increases steadily if β 0 of each full-subcomplex increases in a certain way. The number of connected components of each full-subcomplex is determined by the selected vertices and their adjacent patterns, which are set by characteristic vector and adjacent matrix respectively. From the previous paragraph we can see the adjacent matrices change regularly from nP to (n + 2)P . Then if the last two coloring bricks of (n + 2)P are actually the copy of the last two bricks of nP , the 0-skeleton is deemed to increase by constant when changes between successive steps remain the same, like that from 2P to 4P and from 4P to 6P . Such statement can be easily proved by CT-trees.
By the previous methodology we can calculate out that β 1 (M (4P, δ 1 ))=5 as shown in Table 9. And the β 0 of the first subcomplex is 2 by CT-tree process as displayed in Table 10 and Table 11. Table 9.
where δ (t) represents a Z 4 2 -coloring over the polytope (n + 2t)P . The coloring vector of δ (t) is obtained by duplicating the last two bricks of δ for t times.

Proof of Theorem 1.2 for n is even
In this section, we prove Theorem 1.2 for an even n ∈ Z + . It is similar to the proof of Lemma 3.12.

Proof. Using the notations of {S
and the affix elements a, b to represent elements in Table 19. Firstly we find out a non-orientable Z 3 2 -coloring λ over the polytope 2P , whose coloring vector is [aS 1 S 3 S 2 b] A. Its natural Z 4 2 -extension is denoted by δ. Using λ (t 1 ,t 2 ) to represent the Z 3 2 -coloring characteristic function of coloring vector , which is also defined on the polytope 2(t 1 + t 2 + 1)P . Specially we have λ (0,0) = λ. The coloring vectors of λ (1,0) and respectively. It is necessary to ensure the compatibility of S 2 and S 4 if we want to glue A 2 after A 1 . And here everything has checked to be fine. Then we calculate out the Betti numbers and arrange them in Table 20.   β 1 (M (2(t 1 + t 2 + 1)P, δ (t 1 ,t 2 ) )) = β 1 (M (2(t 1 + t 2 + 2)P, δ (t 1 +1,t 2 ) )).
By (4.4) and (4.5) we obtain where n is even, and 0 t n 2 − 1. Thus we finish the proof of Lemma 4.3.
Lemma 4.4. For any even integer n ∈ Z + and any odd integer k ∈ [n + 3, 5n − 5], there is a non-orientable Z 3 2 -coloring λ over the polytope nP , such that for its natural associated  Table 21. Firstly we seek out two non-orientable Z 3 2 -colorings λ and λ on the polytope 2P , whose colorings are [aS 1 S 2 S 3 c] and [aS 1 S 2 S 1 a]. Their natural Z 4 2 -extensions are denoted as δ and δ. By calculation we have over the polytope 2(t 1 + t 2 + 2)P , here * is the corresponding affix element, which means * = c, b, a, c for i = 1, 2, 3, 4 respectively. Specially the coloring vector of λ Then for i = 1, 2, 3, 4, we have By (4.8) and (4.9), we have (4.10) for each n is even, and 0 t n 2 − 2.
By (4.7), (4.10) and (4.11) we finish the proof of Lemma 4.4. And we arrange all the Betti numbers of Lemma 4.4 together in Table 22.

Proof of Theorem 1.2 for n is odd
In this section, we prove Theorem 1.2 for an odd n ∈ Z ≥1 , which are similar to the arguments in Section 4.
And then we repeat the last two bricks for t times to construct the expected coloring vector over the polytope (3 + 2t)P . The corresponding characteristic function is denoted by λ t . By Theorem 3.3 and Proposition 3.5 we can get its unique admissible extension δ t , a Z 4 2 -coloring characteristic function. The sum of every column of characteristic matrix of δ t is 1 mod 2. That is, M ((3 + 2t)P, δ t ) is the orientable double cover of the non-orientable manifold M ((3 + 2t)P, λ t ). Finally we figure out the increasing law of β 1 (M ((3 + 2t)P, δ t )) as shown in Table 23. Table 23. β 1 (M (nP, δ t )) for n = 3 + 2t in Lemma 5.1. n = 3 n = 5 n = 7 · · · n = 2a + 1, a ∈ Z >0 1 1 By now we finally fulfill the proof of Lemma 5.1.
Proof. Similar to Lemma 4.2, we start at n = 3 and construct six suitable characteristic vectors whose corresponding manifolds' Betti numbers would add up by 10t when repeating the last pair of coloring bricks for t times.
First of all we claim some bricks and affixes in Table 24. Those bricks are useful in constructing coloring vectors of the expected Z 3 2 -coloring characteristic function λ t i . For every i = 1, 2, 3, let λ 0 i , λ 1 i and λ 2 i be the three Z 3 2 -coloring characteristic functions of the three colorings over the polytopes 3P , 5P and 7P respectively as shown in Table 25.
Here t represents how many times the last compatible pair of λ 0 i has been repeated.
Let δ t i be the natural Z 4 2 -extensions of λ t i , for i = 1, 2, 3. Then by calculation we have the first Betti number of the recovered manifold of M ((3 + 2t)P, δ t i ) is 5 + 2i + 10t, for t = 0, 1 and 2. Thus Again, we construct some bricks and affixes as shown in Table 26. Those items are useful in building up the coloring vectors of the expected characteristic function λ t i . For every i = 1, 2, 3, let λ 0 i , λ 1 i and λ 2 i be the three Z 3 2 -coloring characteristic functions of the three colorings over the polytopes 3P , 5P and 7P respectively as shown in Table 27. Here t represents how many times the last compatible pair of λ 0 i has been repeated. Let δ t i be the natural Z 4 2 -extensions of λ t i , for i = 1, 2, 3. Then by calculation we have the first Betti number of the recovered manifold of δ t i is 11 + 2i + 10t, for t = 0, 1 and 2. Lemma 5.3. For any odd integer n ∈ Z >1 and any odd integer k ∈ [1, n − 1], there is a nonorientable Z 3 2 -coloring λ over the polytope nP , such that for its natural associated Z 4 2 -coloring δ, we have β 1 (M (nP, δ)) = k.