1. Introduction
One of the central problems in topology is the rigidity question, namely when a weaker equivalence between two spaces implies a stronger equivalence between them. Freedman's work [Reference Freedman11] on the classification of closed oriented simply connected topological 4-manifolds via the intersection form is a good example of this type of question. In toric topology, a similar type of question was posed in [Reference Masuda and Suh15], which is now called the cohomological rigidity problem, which asks if homeomorphism/diffeomorphism classes of quasitoric manifolds can be classified by their integral cohomology rings.
Although the problem looks overambitious, it is a sensible question to ask on the following basis. No counter-example has been found since it was formulated. On the contrary, there is a piece of evidence supporting the cohomological rigidity of quasitoric manifolds. Indeed, the classification result in [Reference Orlik and Raymond16] together with the description of the cohomology ring of a quasitoric manifold [Reference Davis and Januszkiewicz9, theorem 4.14] implies the cohomological rigidity of 4-dimensional quasitoric manifolds. Besides, many affirmative answers have been proved, for instance certain Bott manifolds [Reference Choi5], generalized Bott manifolds [Reference Choi, Masuda and Suh6] and 6-dimensional quasitoric manifolds associated to 3-dimensional Pogorelov polytopes [Reference Bukhshtaber, Erokhovets, Masuda, Panov and Pak1].
Being a generalized notion of quasitoric manifold, a toric orbifold [Reference Davis and Januszkiewicz9] is a 2n-dimensional compact orbifold equipped with a locally standard 
$T^{n}$-action whose orbit space is a simple polytope. It is known that the cohomology rings fail to classify toric orbifolds up to homeomorphism. For instance, there are weighted projective spaces with isomorphic cohomology rings that are not homeomorphic. Therefore, toric orbifolds do not satisfy cohomological rigidity. However, in the above counter-examples, two weighted projective spaces with isomorphic cohomology rings are homotopy equivalent [Reference Bahri, Franz, Notbohm and Ray2]. Hence, we take a step back and ask a homotopical version of the cohomological rigidity:
Question 1.1 Are two toric orbifolds homotopy equivalent if their integral cohomology rings are isomorphic as graded rings?
This paper aims to answer this question for certain 4-dimensional toric orbifolds. We first study certain CW-complexes which model 4-dimensional toric orbifolds and investigate their homotopy theory. In what follows, 
$H^{\ast }(X)$ denotes the cohomology ring with integral coefficients unless otherwise stated, and 
$P^{3}(k)$ denotes the 3-dimensional mod-k Moore space for 
$k>1$. It is known that 
$H^{3}(X)$ is a finite cyclic group for all 4-dimensional toric orbifolds X. We refer to [Reference Fischli10, Reference Jordan13]. Let 
$H^{3}(X)\cong \mathbb {Z}_m$ with 
$m=2^{s}q$ for q odd and 
$s\geq 0$. When 
$q>1$, we show that X decomposes into a wedge of 
$P^{3}(q)$ and a recognizable space.
Theorem 1.2 Let X be a 4-dimensional toric orbifold such that 
$H^{3}(X)\cong \mathbb {Z}_m$. If 
$m=2^{s}q$ for an odd integer 
$q>1$ and 
$s\geq 0$, then X is homotopy equivalent to 
$\hat {X}\vee P^{3}(q)$, where 
$\hat {X}$ is a simply connected 4-dimensional CW-complex with 
$H^{3}(\hat {X})=\mathbb {Z}_{2^{s}}$ and 
$H^{i}(\hat {X})\cong H^{i}(X)$ for 
$i\neq 3$.
If m is odd or equivalently 
$s=0$, then theorem 1.2 implies 
$X\simeq \hat X\vee P^{3}(m)$ where 
$H^{3}(\hat {X})=0$. As an application, we can answer question 1.1 for certain 4-dimensional toric orbifolds in the following theorem.
Theorem 1.3 Let X and 
$X'$ be 4-dimensional toric orbifolds such that 
$H^{3}(X)$ and 
$H^{3}(X')$ have no 2-torsion. Then X is homotopy equivalent to 
$X'$ if and only if there is a ring isomorphism 
$H^{*}(X)\cong H^{*}(X')$.
This paper is organized as follows. In § 2, we review the constructive definition of a 4-dimensional toric orbifold X. In particular, it is important to see that X is the mapping cone of a map from a lens space to a wedge of 2-spheres. This phenomenon is motivated by the study of [Reference Bahri, Sarkar and Song4] and can also be understood in terms of a 
$\mathbf {q}$-CW complex studied in [Reference Bahri, Notbohm, Sarkar and Song3]. In § 3, we define a category 
$\mathscr {C}_{n,m}$ of certain CW-complexes which model 4-dimensional toric orbifolds and study the homotopy theory of 
$\mathscr {C}_{n,m}$. Section 4 aims to give a necessary and sufficient condition for 
$X\in \mathscr {C}_{n,m}$ to decompose into a wedge of 
$P^{3}(q)$ and a space in 
$\mathscr {C}_{n,2^{s}}$. In § 5, we study the p-local version of the discussion of § 4 for some odd prime p and apply this to 4-dimensional toric orbifolds. Combining the equivalent condition (Proposition 4.8) and the p-local decomposition (Proposition 5.3), we finally complete the proofs of theorems 1.2 and 1.3 in § 6.
2. Toric orbifolds of dimension 4
We begin with a summary of the constructive definition of a toric orbifold. For our purpose, we focus on the 4-dimensional case. For more details on toric orbifolds see [Reference Davis and Januszkiewicz9, § 7], [Reference Poddar and Sarkar18, § 2] and [Reference Cox, Little and Schenck7, chapters 3, 10].
Let P be an 
$(n+2)$-gon on vertices 
$v_1, \dots , v_{n+2}$ for some 
$n\geq 0$. We denote by 
$E_i$ the edge connecting 
$v_i$ and 
$v_{i+1}$ for 
$i=1, \dots , n+2$, where we take indices modulo 
$n+2$. To each edge 
$E_i$, assign a primitive vector 
$\xi _i=(a_i,b_i)\in \mathbb {Z}^{2}$ such that two adjacent vectors 
$\xi _i$ and 
$\xi _{i+1}$ are linearly independent. We often describe this combinatorial data as in Figure 1.
Figure 1. 
$(n+2)$-gon with primitive vectors on facets.
Identify 
$\mathbb {Z}^{2}$ with 
$\mathrm {Hom}(S^{1}, T^{2})$. Each 
$\xi _i$ defines a one-parameter subgroup of 
$T^{2}$
\[ S^{1}_{\xi_i}= \{(t^{a_i}, t^{b_i})\in T^{2} \mid t\in S^{1}\}. \]Now, define the following identification space
\begin{equation} X= P\times T^{2}/_\sim \end{equation}
where we identify 
$(p, g)$ and 
$(p, h)$ for 
$gh^{-1}\in S^{1}_{\xi _i}$ if p is in the relative interior of 
$E_i$, and for all 
$g, h\in T^{2}$ if p is a vertex of P. Note that there is no identification between 
$(p,g)$ and 
$(q,h)$ unless 
$p=q$. Here, the torus 
$T^{2}$ acts on X by the multiplication on the second factor, which yields the orbit map 
$\pi \colon X \to P$ by the projection onto the first factor.
We roughly describe the orbifold structure on X following the identification (2.1). First, there is a standard presentation of 
$\mathbb {C}^{2}$ given by a homeomorphism 
$\mathbb {R}^{2}_\geq \times T^{2}/_{\sim _{std}}\cong \mathbb {C}^{2}$ that maps 
$[(x,y), (t,s)]$ in 
$\mathbb {R}^{2}_\geq \times T^{2}/_{\sim _{std}}$ to 
$(xt, ys)$ in 
$\mathbb {C}^{2}$. Here, the standard identification 
$\sim _{std}$ is given by 
$((x,y),g)\sim _{std} ((x,y),h)$
(1) for
$gh^{-1}\in 1\times S^{1}$ if $x=0$ and $y\neq 0$;(2) for
$gh^{-1}\in S^{1} \times 1$ if $x\neq 0$ and $y=0$;(3) for all
$g,h\in T^{2}$ if $x=y=0$.
Note that there is no identification between 
$((x_1, y_1), g)$ and 
$((x_2, y_2), h)$ in 
$\mathbb {R}^{2}_\geq \times T^{2}$ unless 
$(x_1, y_1)=(x_2, y_2)$.
Let 
$U_i$ be a neighbourhood of 
$v_i$ in P, which is homeomorphic to 
$\mathbb {R}^{2}_\geq$ as a manifold with corners. Let 
$\psi _i$ be a homeomorphism 
$\mathbb {R}^{2}_{\geq } \cong U_i$ and let 
$\rho _i\colon T^{2} \twoheadrightarrow T^{2}$ be an endomorphism of 
$T^{2}$ given by
\begin{equation} \rho_i\colon T^{2}\to T^{2}, \quad (t_1, t_2)\overset{\rho_i}{\longmapsto}(t_1^{a_i}t_2^{a_{i+1}}, t_1^{b_i}t_2^{b_{i+1}}). \end{equation}
Since 
$\xi _i=(a_i, b_i)$ and 
$\xi _{i+1}=(a_{i+1}, b_{i+1})$ are linearly independent, the kernel 
$K_i =\ker \rho _i$ is a cyclic subgroup of 
$T^{2}$. Then the map 
$\psi _i\times \rho _i$ induces a surjection
\begin{equation} \mathbb{C}^{2} \cong \mathbb{R}^{2}_\geq{\times} T^{2}/_{\sim_{std}} \xrightarrow{\psi_i\times \rho_i} U_i\times T^{2}/_\sim. \end{equation} This shows that 
$U_i\times T^{2}/_\sim$ is homeomorphic to the quotient 
$\mathbb {C}^{2}/K_i$, where 
$K_i$ acts on 
$\mathbb {C}^{2}$ as a subgroup of 
$T^{2}$. Hence, the map (2.3) forms an orbifold chart around the point 
$[v_i, g]\in X$. The gluing maps among these orbifold charts are determined by the underlying polygon.
A certain cofibration construction of X is studied in [Reference Bahri, Sarkar and Song4] based on the orbifold structure on X. Pick a vertex 
$v_i$ of P and 
$U_i$ is its neighbourhood as above. Consider a line segment 
$\ell _i$ in P connecting two points lying in the relative interior of 
$E_i$ and 
$E_{i+1}$, respectively. The restriction of identification (2.1) to 
$\ell _i$ gives rise to a subspace of X
\[ L_i = \ell_i\times T^{2}/_\sim. \]
By assuming that the homeomorphism 
$\psi _i\colon \mathbb {R}^{2}\to U_i$ sends the arc 
$S^{1}_{\geq }=S^{1}\cap \mathbb {R}^{2}_\geq$ to 
$\ell _i$, the restriction of (2.3) to 
$S^{1}_\geq \times T^{2}/_{\sim _{std}}$ induces a homeomorphism 
$(S^{1}_{\geq }\times T^{2}/_{\sim _{std}})/K_i\cong L_i$. Here, we notice that 
$K_i$ is isomorphic to 
$\mathbb {Z}_{m_{i, i+1}}$, where
\begin{equation} m_{i,j}=|\det \begin{bmatrix} \xi_i^{t} & \xi_{j}^{t} \end{bmatrix}|. \end{equation}
As 
$S^{1}_{\geq }\times T^{2}/_{\sim _{std}}$ is homeomorphic to 
$S^{3}$, we conclude that 
$L_i$ is homeomorphic to 
$S^{3}/\mathbb {Z}_{m_{i, i+1}}$ which is 
$S^{3}$ if 
$m_{i,i+1}=1$ and is a lens space otherwise. This description can be found in [Reference Sarkar and Suh19, Proposition 2.3] including higher dimensional cases.
Moreover, the subspace 
$U_i\times T^{2}/_\sim$ is homeomorphic to a tubular neighbourhood of the cone on 
$L_i$. Let B be the union of all edges 
$E_j$ where 
$j\neq i,i+1$. The subspace 
$B\times T^{2}/_\sim$ is homotopic to a wedge of n copies of 2-spheres and the subspace 
$(P-\{v_i\})\times T^{2}/_\sim$ retracts to 
$B\times T^{2}/_\sim$. As X is a union of 
$(P-\{v_i\})\times T^{2}/_\sim$ and 
$U_i\times T^{2}/_\sim$, it implies a homotopy cofibration
\begin{equation} L_i \xrightarrow{f_i} {\bigvee}^{n}_{j=1} S^{2} \to X \end{equation}
where the map 
$f_i$ is induced by the composition of the inclusion 
$\iota$ and the retraction r
\[ \ell_i\times T^{2}/_\sim\overset{\iota}\hookrightarrow(P-\{v_i\})\times T^{2}/_\sim\xrightarrow{r} B\times T^{2}/_\sim. \]See Figure 2 for a pictorial illustration of (2.5).
Figure 2. 
$X={\bigvee} _{i=1}^{3} S^{2} \cup _{f_i} CL_i$.
Applying the cohomology functor to the cofibre sequence (2.5) and referring to [Reference Bahri, Notbohm, Sarkar and Song3, theorem 1.1], we can compute the free part of 
$H^{\ast }(X)$. The cohomology of X has been discussed using various tools in the studies [Reference Jordan13, theorem 2.5.5], [Reference Fischli10, theorem 2.3] and [Reference Kuwata, Masuda and Zeng14, corollary 5.1] which can be summarized as follows.
Proposition 2.1 Let X be a toric orbifold of dimension 4. Then we have
\begin{equation} \begin{array}{c|c c c c c c} i & 0 & 1 & 2 & 3 & 4 & \geq 5\\\hline H^{i}(X) & \mathbb{Z} & 0 & \mathbb{Z}^{n} & \mathbb{Z}_m & \mathbb{Z} & 0 \end{array} \end{equation}
where m is the greatest common divisor of 
$\{m_{i,j}\mid 1 \leq i< j \leq n+2\}$ for 
$m_{i,j}$'s defined in (2.4). We set 
$\mathbb {Z}_m=0$ if 
$m=1$.
Remark 2.2 The way of realizing X as a cofibre in (2.5) can be understood in a more general framework of a 
$\mathbf {q}$-CW complex. A 
$\mathbf {q}$-CW complex is defined inductively starting from a discrete set 
$X_0$ of points. Then, 
$X_{i}$ is defined by the pushout
where 
$e^{i}$ and 
$S^{i-1}$ are i-dimensional cell and its boundary, respectively, and 
$K_\alpha$ is a finite group acting linearly on 
$e_i$. Every toric orbifold is a 
$\mathbf {q}$-CW complex. We refer to [Reference Bahri, Notbohm, Sarkar and Song3] for more details.
3. Cohomology of 4-dimensional CW-complexes
3.1 A category of 4-dimensional CW-complexes
Suppose that X is a simply connected CW-complex satisfying (2.6). By [Reference Hatcher12, Proposition 4H.3] it is homotopy equivalent to a CW-complex
\begin{equation} \left(\underset{i=1}{\overset{n}{\bigvee}} S^{2}\vee P^{3}(m)\right)\cup_f e^{4} \end{equation}
where 
$f\colon S^{3}\to {\bigvee} _{i=1}^{n} S^{2}\vee P^{3}(m)$ is the attaching map of the 4-cell. In this section, we study the homotopy theory of CW-complexes in this form.
Define 
$\mathscr {C}_{n,m}$ to be the full subcategory of 
$\text {Top}_*$ consisting of mapping cones as in (3.1). Here, the orientation of the 4-cell 
$e^{4}$ is the induced orientation of the upper hemisphere in 
$S^{5}$. We label the 
$i{\text {th}}$ copy of 2-spheres in 
${\bigvee} _{i=1}^{n} S^{2}$ by 
$S^{2}_i$ for 
$1\leq i\leq n$ and write
\[ Y=\underset{i=1}{\overset{n}{\bigvee}} S^{2}_i\vee P^{3}(m) \]
for short. Let 
$\mu _i,\nu \in H_2(Y)$ be homology classes representing 
$S^{2}_i$ and the 2-cell of 
$P^{3}(m)$ respectively. Then, we have
\begin{equation} H_2(Y)\cong\mathbb{Z}\langle{\mu_1,\ldots,\mu_n}\rangle\oplus\mathbb{Z}_m\langle{\nu}\rangle. \end{equation} Let 
$g\colon Y\to Y$ be a map. Then the induced homology map 
$g_\ast \colon H_2(Y) \to H_2(Y)$ is given by 
$g_*(\mu _i)=\sum ^{n}_{j=1}x_{ij}\mu _j+y_i\nu$ and 
$g_*(\nu )=z\nu$ for some integers 
$x_{ij}$ and mod-m congruence classes 
$y_i$ and 
$z$. Conversely, we have the following lemma.
Lemma 3.1 Given a vector 
$(y_1,\ldots ,y_n,z)\in (\mathbb {Z}_m)^{n+1}$ and an 
$(n\times n)$-integral matrix
\[ \left( \begin{array}{c c c} x_{11} & \ldots & x_{1n}\\ \vdots & \ddots & \vdots\\ x_{n1} & \cdots & x_{nn} \end{array} \right)\in \text{Mat}_n(\mathbb{Z}), \]
there exists a map 
$g\colon Y\to Y$ such that 
$g_*(\mu _i)=\sum ^{n}_{j=1}x_{ij}\mu _j+y_i\nu$ and 
$g_*(\nu )=z\nu$.
Proof. First, consider the string of isomorphisms
\begin{eqnarray*} \bigg[\underset{i=1}{\overset{n}{\bigvee}} S^{2}_i,\underset{j=1}{\overset{n}{\bigvee}} S^{2}_j\vee P^{3}(m)\bigg] &\cong& \bigoplus^{n}_{i=1}\bigg[S^{2}_i, \underset{j=1}{\overset{n}{\bigvee}} S^{2}_j\vee P^{3}(m)\bigg]\\ &\cong&\bigoplus^{n}_{i=1}\pi_2\bigg(\underset{j=1}{\overset{n}{\bigvee}} S^{2}_j\vee P^{3}(m)\bigg)\\ &\cong&\bigoplus^{n}_{i=1}H_2\bigg(\underset{j=1}{\overset{n}{\bigvee}} S^{2}_j\vee P^{3}(m)\bigg)\\ &\cong&\bigoplus^{n}_{i=1}\left(\bigoplus^{n}_{j=1}\mathbb{Z}\oplus \mathbb{Z}_m\right) \end{eqnarray*}
where the third isomorphism is due to the Hurewicz theorem. Under these isomorphisms, take 
$g'\colon {\bigvee} _{i=1}^{n} S^{2}_i\to Y$ to be the map corresponding to
\[ \left( \begin{array}{c c c} x_{11} & \cdots & x_{1n}\\ \vdots & \ddots & \vdots\\ x_{n1} & \cdots & x_{nn} \end{array} \right) \oplus (y_1, \dots, y_n) \in\left(\bigoplus^{n}_{i=1}\bigoplus^{n}_{j=1}\mathbb{Z}\right)\oplus\left(\bigoplus^{n}_{i=1}\mathbb{Z}_m\right). \]
Then 
$g'_*(\mu _i)=\sum ^{n}_{j=1}x_{ij}\mu _j+y_i\nu$.
Next, for 
$z\in \mathbb {Z}_m$ let 
$g''\colon P^{3}(m)\to Y$ be the composition
\[ g''\colon P^{3}(m)\xrightarrow{\underline{z}}P^{3}(m)\hookrightarrow Y, \]
where 
$\underline {z}\colon P^{3}(m)\to P^{3}(m)$ is the degree-
$z$ map. Let 
$g:Y\to Y$ be the wedge sum 
$g=g'\vee g''$. Then 
$g_*(\mu _i)=\sum ^{n}_{j=1}x_{ij}\mu _j+y_i\nu$ and 
$g_*(\nu )=z\nu$.□
3.2 Cellular cup product representation
Let 
$C_f\in \mathscr {C}_{n,m}$ be the mapping cone of a map 
$f\colon S^{3}\to Y$. As the inclusion 
$Y\hookrightarrow C_f$ induces an isomorphism 
$H_2(Y)\to H_2(C_f)$, we do not distinguish 
$\mu _i,\nu \in H_2(Y)$ and their images in 
$H_2(C_f)$. Let 
$u_i\in H^{2}(C_f) \text { and } e\in H^{4}(C_f)$ be cohomology classes dual to 
$\mu _i$ and the homology class represented by the 4-cell in 
$C_f$ respectively. Let 
$v\in H^{3}(C_f)$ be the Ext image of 
$\nu$. Then
\begin{equation} \begin{array}{c c c} H^{2}(C_f)\cong\mathbb{Z}\langle{u_1,\ldots,u_n}\rangle, & H^{3}(C_f)\cong\mathbb{Z}_m\langle{v}\rangle, & H^{4}(C_f)\cong\mathbb{Z}\langle{e}\rangle. \end{array} \end{equation}
We call the set 
$\{u_1,\ldots ,u_n,v,e\}$ the cellular basis of 
$H^{*}(C_f)$.
With coefficient 
$\mathbb {Z}_m$, let 
$\bar {u}_i\in H^{2}(C_f;\mathbb {Z}_m)$ and 
$\bar {e}\in H^{4}(C_f;\mathbb {Z}_m)$ be the mod-m images of 
$u_i$ and 
$e$, and let 
$\bar {v}\in H^{2}(C_f;\mathbb {Z}_m)$ be the cohomology class dual to 
$\nu$. Then
\begin{align*} & H^{2}(C_f;\mathbb{Z}_m)\cong\mathbb{Z}_m\langle{\bar{u}_1,\ldots,\bar{u}_n,\bar{v}}\rangle,\quad H^{3}(C_f;\mathbb{Z}_m)\cong\mathbb{Z}_m\langle{\beta(\bar{v})}\rangle, \\ & H^{4}(C_f;\mathbb{Z}_m)\cong\mathbb{Z}_m\langle{\bar{e}}\rangle, \end{align*}
where 
$\beta$ is the Bockstein homomorphism. We call the set 
$\{\bar {u}_1,\ldots ,\bar {u}_n,\bar {v};\bar {e}\}$ the mod-m cellular basis of 
$H^{*}(C_f;\mathbb {Z}_m)$.
Definition 3.2 Let 
$C_f$ be a mapping cone in 
$\mathscr {C}_{n,m}$. Then the cellular cup product representation 
$M_{cup}(C_f)$ of 
$C_f$ is 
$A\in \text {Mat}_n(\mathbb {Z})$ if 
$m=1$, and is a triple 
$(A,\textbf {b},c)\in \text {Mat}_n(\mathbb {Z})\oplus (\mathbb {Z}_m)^{n}\oplus \mathbb {Z}_m$ if 
$m>1$, where
\[ A=\left( \begin{array}{c c c} a_{11} & \cdots & a_{1n}\\ \vdots & \ddots & \vdots\\ a_{n1} & \cdots & a_{nn} \end{array} \right) ~\text{and} ~ \mathbf{b}=(b_1, \dots, b_n) \]
are given by 
$u_i\cup u_j=a_{ij}e, ~\bar {u}_i\cup \bar {v}=b_{i}\bar {e}$ and 
$\bar {v}\cup \bar {v}=c\bar {e}$.
Here, 
$A$ is a symmetric matrix since it is the matrix representation of the bilinear form
\[ (-{\cup}{-})_{\mathbb{Z}}\colon H^{2}(C_f;\mathbb{Z})\otimes H^{2}(C_f;\mathbb{Z})\to H^{4}(C_f;\mathbb{Z}) \]
with respect to the cellular basis 
$\{u_1,\ldots ,u_n;e\}$. Furthermore, universal coefficient theorem implies 
$\bar {u}_i\cup \bar {u}_j=a_{ij}\bar {e}\pmod {m}$. So, 
$(-\cup -)_{\mathbb {Z}_m}\colon H^{2}(C_f;\mathbb {Z}_m)\otimes H^{2}(C_f;\mathbb {Z}_m)\to H^{4}(C_f;\mathbb {Z}_m)$ can be recovered from 
$M_{cup}(C_f)$ as well.
Remark 3.3 When X is an oriented compact smooth 4-manifold, the intersection form 
$I(X)$ is the bilinear form given by cup products of degree 
$2$ cohomology classes modulo torsion
\[ I(X)\colon H^{2}(X)/\text{Tor}\otimes H^{2}(X)/\text{Tor}\to \mathbb{Z},\quad x\otimes y\mapsto\langle{x\cup y, [X]}\rangle, \]
where 
$[X]\in H_{4}(X)$ is the fundamental class. Although defined in a similar fashion, 
$I(X)$ and 
$M_{cup}(X)$ are different. First, 
$I(X)$ only concerns cup products of free elements in 
$H^{2}(X)$ and its matrix representation is a symmetric matrix, while 
$M_{cup}(X)$ concerns cup products of cohomology with integral and 
$\mathbb {Z}_m$-coefficients and is a triple consisting of a matrix, a mod-m vector and a mod-m congruence class that record all data. Second, a matrix representation of 
$I(X)$ depends on the choice of generators of 
$H^{2}(X)$, whereas we define 
$M_{cup}(X)$ using a fixed CW-complex structure of X. In the following section, we will discuss the transformation between cellular map representations of two CW-complex structures of the same X. It is similar to matrix congruence but is slightly more complicated, as cup products of cohomology with 
$\mathbb {Z}_m$ coefficient are involved.
Let 
$g\colon S^{3}\to Y$ be another map and let 
$C_g\in \mathscr {C}_{n,m}$ be its mapping cone. Recall that 
$f+g$ is the composition
\[ f+g\colon S^{3}\xrightarrow{\text{comult}}S^{3}\vee S^{3}\xrightarrow{f\vee g}Y\vee Y\xrightarrow{\text{fold}}Y. \]
Denote its mapping cone by 
$C_{f+g}$.
Lemma 3.4 Let 
$Y$ be (1) 
$S^{2}_1\vee S^{2}_2$ or (2) 
$S^{2}_1\vee P^{3}(m)$ and let 
$f,g:S^{3}\to Y$ be two maps. Then 
$M_{cup}(C_{f+g})=M_{cup}(C_f)+M_{cup}(C_g)$.
Proof. In the following, we only prove case (2). The proof also works for case (1) but is simpler. Let
•
$\{u,v;e\}$, $\{u_1,v_1;e_1\}$ and $\{u_2,v_2;e_2\}$ be the cellular bases of $H^{*}(C_{f+g}), H^{*}(C_f)$ and $H^{*}(C_g)$, respectively;•
$\{\bar {u},\bar {v};\bar {e}\}$, $\{\bar {u}_1,\bar {v}_1;\bar {e}_1\}$ and $\{\bar {u}_2,\bar {v}_2;\bar {e}_2\}$ be the mod-m cellular bases of $H^{*}(C_{f+g};\mathbb {Z}_m)$, $H^{*}(C_f;\mathbb {Z}_m)$ and $H^{*}(C_g;\mathbb {Z}_m)$, respectively;• the cellular cup product representations of
$C_{f+g}, C_f$ and $C_g$ be $M_{cup}(C_{f+g})=(A,\textbf {b},c)$, $M_{cup}(C_{f})=(A_1,\textbf {b}_1,c_1)$ and $M_{cup}(C_{g})=(A_2,\textbf {b}_2,c_2)$, respectively.
Here, 
$A,A_1,A_2$ are integers and 
$\textbf {b},\textbf {b}_1,\textbf {b}_2$ are mod-m congruence classes. We claim that
\[ A=A_1+A_2, ~ \textbf{b}=\textbf{b}_1+\textbf{b}_2 \text{ and } c=c_1+c_2. \]
Consider the mapping cone 
$C'=Y\cup _{f\vee g}(e^{4}_1\vee e^{4}_2)$ of 
$g\vee h\colon S^{3}\vee S^{3}\to Y$. Let
•
$u'\in H^{2}(C')$, $e'_1,e'_2\in H^{4}(C')$ be cohomology classes dual to $S^{2}$, $e^{4}_1$ and $e^{4}_2$;•
$\bar {u}'\in H^{2}(C';\mathbb {Z}_m)$, $\bar {e}'_1,\bar {e}'_2\in H^{4}(C';\mathbb {Z}_m)$ be the mod-m images of $u,e'_1$ and $e'_2$;•
$\bar {v}'\in H^{2}(C';\mathbb {Z}_m)$ be the cohomology class dual to the 2-cell of $P^{3}(m)$.
Observe that 
$C_f$ and 
$C_g$ are subcomplexes of 
$C'$. Let 
$\imath _1\colon C_f\to C'$ and 
$\imath _2\colon C_g\to C'$ be natural inclusions and let 
$q\colon C_{f+g}\to C'$ be the map collapsing the equatorial disk of the 4-cell in 
$C_{f+g}$ to a point. Then
\[ \begin{array}{l l l} q^{*}(u')=u, & q^{*}(\bar{v}')=\bar{v} , & q^{*}(e'_1)=q^{*}(e'_2)=e,\\ [ 5pt] \imath^{*}_j(u')=u_j, & \imath^{*}_j(\bar{v}')=\bar{v}_j, & \imath^{*}_j(e'_k)=\delta_{jk}e_j \end{array} \]
for 
$j,k\in \{1,2\}$, where 
$\delta _{jk}$ is the Kronecker symbol. On the one hand, 
$u'\cup u'=\alpha _1e'_1+\alpha _2e'_2$ for some integers 
$\alpha _1$ and 
$\alpha _2$. Now the naturality of cup products implies
\begin{eqnarray*} \imath^{*}_j(u'\cup u')&=&\imath^{*}_j(\alpha_1e'_1+\alpha_2e'_2)\\ \imath^{*}_j(u')\cup\imath^{*}_j(u')&=&\alpha_1\imath^{*}_j(e'_1)+\alpha_2\imath^{*}_j(e'_2)\\ u_j\cup u_j&=&\alpha_je_j \end{eqnarray*}
for 
$j\in \{1,2\}$. So, 
$\alpha _j=A_j$. On the other hand,
\begin{eqnarray*} u\cup u &=&q^{*}(u')\cup q^{*}(u')\\ &=&q^{*}(u'\cup u')\\ &=&q^{*}(A_1e'_1+A_2e'_2)\\ &=&A_1q^{*}(e'_1)+A_2q^{*}(e'_2)\\ &=&(A_1+A_2)e. \end{eqnarray*}
So, 
$A=A_1+A_2$. Similarly we can show 
$\textbf {b}=\textbf {b}_1+\textbf {b}_2$ and 
$c=c_1+c_2$. Therefore, we have
\[ M_{cup}(C_{f+g})=M_{cup}(C_f)+M_{cup}(C_g).\]□
3.3 Cellular map representations
Let 
$f,f'\colon S^{3}\to Y$ be two maps and 
$C_f,C_{f'}\in \mathscr {C}_{n,m}$ be their mapping cones. Let
•
$\{u_1,\ldots ,u_n,v,e\}$ and $\{u'_1,\ldots ,u'_n,v',e'\}$ be the cellular bases of $H^{*}(C_f)$ and $H^{*}(C_{f'})$,•
$\{\bar {u}_1,\ldots ,\bar {u}_n,\bar {v},\bar {e}\}$ and $\{\bar {u}'_1,\ldots ,\bar {u}'_n,\bar {v}',\bar {e}'\}$ be the mod-m cellular bases of $H^{*}(C_f;\mathbb {Z}_m)$ and $H^{*}(C_{f'};\mathbb {Z}_m)$.
Given a map 
$\psi \colon C_{f'}\to C_{f}$ and a coefficient ring 
$R$, let
\[ \psi^{{\ast}}_R\colon H^{2}(C_f;R)\to H^{2}(C_{f'};R) \]
be the induced morphism on the second cohomology with coefficient 
$R$.
Definition 3.5 Let 
$\psi \colon C_{f'}\to C_f$ be a map. Then the cellular map representation 
$M(\psi )$ of 
$\psi$ is 
$W\in \text {Mat}_n(\mathbb {Z})$ if 
$m=1$, and is the triple 
$(W,\textbf {y},z)\in \text {Mat}_n(\mathbb {Z})\oplus (\mathbb {Z}_m)^{n}\oplus \mathbb {Z}_m$ if 
$m>1$, where
\begin{equation} W=\left(\begin{array}{c c c} x_{11} & \cdots & x_{1n}\\ \vdots & \ddots & \vdots\\ x_{n1} & \cdots & x_{nn} \end{array} \right) ~\text{and}~ \mathbf{y}=(y_1, \dots , y_n ) \end{equation}
are given by 
${\psi ^{\ast }_\mathbb {Z}}(u_j)=\sum ^{n}_{i=1}x_{ij}u'_i$ and 
${\psi ^{\ast }_{\mathbb {Z}_m}}(\bar {v})=\sum ^{n}_{i=1}y_{i}\bar {u}'_i+z\bar {v}'$.
Lemma 3.6 For 
$R=\mathbb {Z}$ or 
$\mathbb {Z}_m$, consider 
$\psi$, 
$\psi ^{\ast }_R$ and 
$M(\psi )$ as above. For 
$1\leq j\leq n$, we have 
$\psi ^{\ast }_{\mathbb {Z}_m}(\bar {u}_j)=\sum ^{n}_{i=1}x_{ij}\bar {u}'_i$. Furthermore, if 
$\psi$ is a homotopy equivalence, then 
$W$ is an invertible matrix and 
$z$ is a unit in 
$\mathbb {Z}_m$.
Proof. Since 
$C_f$ and 
$C_{f'}$ are simply connected, universal coefficient theorem implies that
\begin{align*} & H^{2}(C_f {; R})\cong\text{Hom}(H_2(C_f),R), H^{2}(C_{f'} {; R})\cong\text{Hom}(H_2(C_{f'}),R) \text{ and} \\ &\psi^{{\ast}}_R=\text{Hom}(\psi_\ast, R) \end{align*}
is dual to 
$\psi _\ast \colon {H_2(C_{f'})\to H_2(C_f)}$. So, 
$\psi ^{\ast }_{\mathbb {Z}_m}(\bar {u}_j)$ is the mod-m image of 
$\psi ^{\ast }_{\mathbb {Z}}(u_j)$ and the first part of the lemma follows.
If 
$\psi$ is a homotopy equivalence, then 
$W\in \text {Mat}_{n}(\mathbb {Z})$ and
\[ \left(\begin{array}{c c c c} \bar{x}_{11} & \cdots & \bar{x}_{1n} & y_1\\ \vdots & \ddots & \vdots & \vdots\\ \bar{x}_{n1} & \cdots & \bar{x}_{nn} & y_n\\ 0 & \cdots & 0 & z \end{array} \right)\in\text{Mat}_{n+1}(\mathbb{Z}_m) ,\quad\text{where } \bar{x}_{ij}\equiv x_{ij}\pmod{m} \]are invertible matrices. So, the second part follows.□
The cellular map representation records the data of 
$\psi _\mathbb {Z}^{\ast }$ and 
$\psi _{\mathbb {Z}_m}^{\ast }$. The square matrix 
$W$ in (3.4) is the map representation of 
$\psi ^{\ast }_{\mathbb {Z}}$ with respect to bases 
$\{u_1,\ldots ,u_n\}$ and 
$\{u'_1,\ldots ,u'_n\}$. Lemma 3.6 implies that
\[ \left(\begin{matrix} \overline{W} & \textbf{y}^{t}\\ \textbf{0} & z \end{matrix} \right)\in\text{Mat}_{n+1}(\mathbb{Z}_m), \]
is the matrix representation of 
$\psi ^{\ast }_{\mathbb {Z}_m}$ with respect to bases 
$\{\bar {u}_1,\ldots ,\bar {u}_n,\bar {v}\}$ and 
$\{\bar {u}'_1,\ldots ,\bar {u}'_n,\bar {v}'\}$, where 
$\overline {W}$ is the mod-m image of 
$W$ and 
$\textbf {0}=(0,\ldots ,0)$.
Recall that in linear algebra, matrix representations of a bilinear form 
$V\otimes V\to \mathbb {Z}$ with respect to different bases of 
$V$ are congruent to each other. So, we have the following lemma.
Lemma 3.7 For 
$C_f,C_{f'}\in \mathscr {C}_{n,m}$, let 
$M_{cup}(C_f)=(A,\textbf {b},c)$ and 
$M_{cup}(C_{f'})=(A',\textbf {b}',c')$ be their cellular cup product representations. If there is a homotopy equivalence 
$\psi \colon C_{f'}\to C_f$ with 
$M(\psi )=(W,\textbf {y},z)\in \text {GL}_n(\mathbb {Z})\oplus (\mathbb {Z}_m)^{n}\oplus \mathbb {Z}^{*}_m$, then
\[ \begin{array}{c c c} A'=W^{t}AW, & \textbf{b}'=\textbf{y}AW+z\textbf{b}W, & c'=\textbf{y}A\textbf{y}^{t}+2z\textbf{y}\textbf{b}^{t}+z^{2}c. \end{array} \] In particular, if two maps 
$f$ and 
$f'\colon S^{3}\to Y$ are homotopic, then the matrix cup product representations of their mapping cones are the same.
Lemma 3.8 If 
$f$ is homotopic to 
$f'$, then 
$M_{cup}(C_f)=M_{cup}(C_{f'})$.
Proof. Take a homotopy 
$\phi \colon S^{3}\times I\to Y$ between 
$f$ and 
$f'$. It induces a homotopy equivalence 
$\Phi \colon C_f\to C_{f'}$ such that its restriction to 
$Y$ is the identity map. So, 
$M(\Phi )=(I_n,\textbf {0},1)$. Then the lemma follows from lemma 3.7. □
Lemma 3.9 Let 
$C_f\in \mathscr {C}_{n,m}$ and let 
$(W,\textbf {y},z)$ be a triple in 
$\text {GL}_n(\mathbb {Z})\oplus (\mathbb {Z}_m)^{n}\oplus \mathbb {Z}_m^{*}$. Then there exist a CW-complex 
$C_{f'}\in \mathscr {C}_{n,m}$ and a homotopy equivalence 
$\psi \colon C_f\to C_{f'}$ such that the cellular map representation 
$M(\psi )$ is 
$(W,\textbf {y},z)$.
Proof. Let
\[ W=\left(\begin{array}{c c c} x_{11} & \cdots & x_{1n}\\ \vdots & \ddots & \vdots\\ x_{n1} & \cdots & x_{nn} \end{array} \right) ~\text{and}~ \mathbf{y}=(y_1, \dots, y_n). \]
By lemma 3.1, there exists a map 
$\tilde {\psi }\colon Y\to Y$ such that 
$\tilde {\psi }_\ast (\mu _i)=\sum ^{n}_{j=1}x_{ij}\mu _j+y_i\nu$ and 
$\tilde {\psi }_\ast (\nu )=z\nu$, where 
$\mu _1, \dots , \mu _n$ and 
$\nu$ are elements in 
$H_2(Y)$ as in (3.2). Thus, we have 
$\tilde {\psi }^{\ast }_\mathbb {Z}(u_i)=\sum ^{n}_{j=1}x_{ji}u_j$ and 
$\tilde {\psi }^{\ast }_{\mathbb {Z}_m}(\bar {v})=\sum ^{n}_{i=1}y_i\bar {u}_i+z\bar {v}$.
Let 
$f'=\tilde {\psi }\circ f$ and let 
$C_{f'}$ be its mapping cone. Then there is a diagram of cofibration sequences
where 
$\psi$ is an induced map. Since 
$W\in GL_n(\mathbb {Z})$ and 
$z\in \mathbb {Z}^{*}_m$, the middle vertical arrow 
$\tilde \psi$ induces an isomorphism in homology. By five lemma, 
$\psi _\ast$ is an isomorphism, which implies that 
$\psi$ is a homotopy equivalence. Finally, we have 
$M(\psi )=(W,\textbf {y},z)$ by the construction.□
4. The homotopy theory of complexes in $\mathscr {C}_{n,m}$
4.1 The $\mathscr {C}_{n,1}$ case
When 
$m=1$, the mapping cone 
$C_f\in \mathscr {C}_{n,1}$ is in the form 
${\bigvee} _{i=1}^{n} S^{2}_i \cup _f e^{4}$ where 
$f\colon S^{3}\to {\bigvee} _{i=1}^{n} S_i^{2}$ is the attaching map of the 4-cell. The Hilton–Milnor theorem (see for instance [Reference Selick20, theorem 7.9.4]) implies that 
$f$ is homotopic to a wedge sum
\[ \sum^{n}_{i=1}a_i\eta_i+\sum_{1\leq j< k\leq n} a_{jk}\omega_{jk}, \]
for some integers 
$a_i$'s and 
$a_{jk}$'s. Here 
$\eta _i$'s and 
$\omega _{jk}$'s are compositions
\[ \begin{array}{l} \eta_i\colon S^{3}\overset{\eta}{\to }S^{2}_i\hookrightarrow\underset{\ell=1}{\overset{n}{{\bigvee}}}S^{2}_\ell\\ \omega_{jk}\colon S^{3}\xrightarrow{[\imath_1,\imath_2]}S^{2}_j\vee S^{2}_k\hookrightarrow \underset{l=1}{\overset{n}{{\bigvee}}} S^{2}_\ell \end{array} \]
of Hopf map 
$\eta$, Whitehead product 
$[\imath _1,\imath _2]$ and canonical inclusions of 
$S^{2}_i$ and 
$S^{2}_j\vee S^{2}_k$ into 
${{\bigvee} }_{\ell =1}^{n} S^{2}_\ell$. The lemma below shows that the coefficients 
$a_i$ and 
$a_{jk}$ are determined by 
$M_{cup}(C_f)$.
Lemma 4.1 Let 
$C_f\in \mathscr {C}_{n,1}$ be the mapping cone of 
$f\simeq \sum ^{n}_{i=1}a_i\eta _i+\sum _{1\leq j< k\leq n} a_{jk}\omega _{jk}$. If
\[ M_{cup}(C_f)= \begin{pmatrix} a'_1 & a'_{12} & \cdots & a'_{1n}\\ a'_{12} & a'_2 & \cdots & a'_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a'_{1n} & a'_{2n} & \cdots & a'_n \end{pmatrix}, \]
then 
$a_i=a'_i$ and 
$a_{jk}=a'_{jk}$ for all 
$i,j$ and 
$k$.
Proof. By lemma 3.8, we may assume 
$f=\sum ^{n}_{i=1}a_i\eta _i+\sum _{1\leq j< k\leq n} a_{jk}\omega _{jk}$. For 
$n=2$, let 
$C_1, C_2$ and 
$C_{12}$ be the mapping cones of 
$a_1\eta _1,a_2\eta _2$ and 
$a_{12}\omega _{12}$. Then their cellular cup product representations are
\begin{align*} & M_{cup}(C_1)=\left( \begin{array}{@{}c c@{}} a_1 & 0\\ 0 & 0 \end{array} \right), \quad M_{cup}(C_2)=\left( \begin{array}{@{}c c@{}} 0 & 0\\ 0 & a_2 \end{array} \right), \\ & M_{cup}(C_{12})=\left( \begin{array}{@{}c c@{}} 0 & a_{12}\\ a_{12} & 0 \end{array} \right). \end{align*}By lemma 3.4, we have
\[ M_{cup}(C_f)=\left( \begin{array}{@{}c c@{}} a_1 & a_{12}\\ a_{12} & a_2 \end{array} \right). \]So, the lemma holds.
For 
$n\geq 3$, let 
$\{u_1,\ldots ,u_n,e\}$ be the cellular basis of 
$H^{*}(C_f)$. We claim that
\[ u_i\cup u_i=a_i e \quad \text{and} \quad u_j\cup u_k=a_{jk}e, \]
for each 
$1\leq i \leq n$ and 
$1\leq j< k \leq n$. The composition
\[ f_{jk}\colon S^{3}\overset{f}{\to }\underset{l=1}{\overset{n}{\bigvee}} S^{2}_l\xrightarrow{\text{pinch}}S^{2}_j\vee S^{2}_k \]
is homotopic to 
$a_{j}\eta '_1+a_{k}\eta '_2+a_{jk}\omega '_{12}$, where 
$\eta '_1\colon S^{3}\overset {\eta }{\to }S^{2}_j\hookrightarrow S^{2}_j\vee S^{2}_k$ and 
$\eta '_2\colon S^{3}\overset {\eta }{\to }S^{2}_k\hookrightarrow S^{2}_j\vee S^{2}_k$ are compositions of Hopf map 
$\eta$ and canonical inclusions and 
$\omega '_{12}\colon S^{3}\to S^{2}_j\vee S^{2}_k$ is the Whitehead product. Let 
$C_{jk}$ be the mapping cone of 
$f_{jk}$. By lemma 3.8 and the above argument, we have
\[ M_{cup}(C_{jk})=\left( \begin{array}{@{}c c@{}} a_j & a_{jk}\\ a_{jk} & a_k \end{array} \right). \]
Let 
$\{u'_j,u'_k;e'\}$ be the cellular basis of 
$H^{*}(C_{jk})$ and let 
$\alpha \colon C_f\to C_{jk}$ be the map which pinches all 2-spheres in 
$C_f$ to the basepoint except for 
$S^{2}_j$ and 
$S^{2}_k$. Then
\[ \alpha^{*}(u'_j)=u_j,~ \alpha^{*}(u'_k)=u_k \text{ and } \alpha^{*}(e')=e. \]By the naturality of cup products, we have
\begin{eqnarray*} \alpha^{*}(u'_j)\cup\alpha^{*}(u'_k)&=&\alpha^{*}(u'_j\cup u'_k)\\ u_j\cup u_k&=&\alpha^{*}(a_{jk}e')\\ &=&a_{jk}e. \end{eqnarray*}
So, 
$a'_{jk}=a_{jk}$. Similarly we can show 
$a'_i=a_i$. Hence, the lemma follows.□
Now we classify the homotopy types of CW-complexes in 
$\mathscr {C}_{n,1}$ by their integral cohomology rings in the next statement.
Proposition 4.2 Let 
$f,f'\colon S^{3}\to {\bigvee} _{i=1}^{n}S^{2}_i$ be two maps and let 
$C_f,C_{f'}\in \mathscr {C}_{n,1}$ be their mapping cones. Then 
$C_f\simeq C_{f'}$ if and only if there is a ring isomorphism 
$H^{*}(C_f)\cong H^{*}(C_{f'})$.
Proof. The ‘only if’ part is trivial. Assume that 
$H^{*}(C_f)\cong H^{*}(C_{f'})$. Then there is an invertible matrix 
$W\in \text {GL}_n(\mathbb {Z})$ such that
\[ W^{t}\cdot M_{cup}(C_f)\cdot W=\varepsilon M_{cup}(C_{f'}), \]
where 
$\varepsilon$ is either 
$1$ or 
$-1$. Suppose first 
$\varepsilon =1$. By lemma 3.9, there is a CW-complex 
$\tilde {C}\in \mathscr {C}_{n,1}$ together with a homotopy equivalence 
$\psi \colon \tilde {C}\to C_f$ such that 
$M(\psi )=W$. We claim that 
$\tilde {C} \simeq C_{f'}$. By lemma 3.7, we have
\[ M_{cup}(\tilde{C})=W^{t}\cdot M_{cup}(C_f)\cdot W=M_{cup}(C_{f'}). \]
Let 
$\tilde {f}$ be the attaching map of the 4-cell in 
$\tilde {C}$ and let
\[ M_{cup}(\tilde{C})=M_{cup}(C_{f'})=\left( \begin{array}{@{}c c c c@{}} a_1 & a_{12} & \cdots & a_{1n}\\ a_{12} & a_2 & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{1n} & a_{2n} & \cdots & a_n \end{array} \right). \]
Then lemma 4.1 implies that 
$f'$ and 
$\tilde {f}$ are homotopic to the wedge sum
\[ \sum^{n}_{i=1}a_i\eta_i+\sum_{1\leq i< j\leq n}a_{ij}\omega_{ij}, \]
which means 
$C_{f'}\simeq \tilde {C}$. Therefore, 
$C_f\simeq C_{f'}$.
Suppose 
$\varepsilon =-1$. Let 
$C_{-f'}$ be the mapping cone of 
$-f':S^{3}\overset {-1}{\to }S^{3}\overset {f'}{\to }{\bigvee} ^{n}_{i=1}S^{2}_i$. Then
\[ W^{t}\cdot M_{cup}(C_f)\cdot W=M_{cup}(C_{{-}f'}) \]
and 
$C_{-f'}\simeq C_{f'}$. The above argument implies 
$C_f\simeq C_{-f'}$, so 
$C_{f}\simeq C_{f'}$. Hence, we have established the claim.□
Corollary 4.3 Two 4-dimensional toric orbifolds without torsion in (co)homology are homotopy equivalent if and only if their integral cohomology rings are isomorphic.
As 4-dimensional quasitoric manifolds always have torsion-free (co)homology, corollary 4.3 implies that the homotopy types of 4-dimensional quasitoric manifolds are classified by their cohomology rings. As we mentioned in Introduction, the homeomorphism types of 4-dimensional toric manifolds are cohomologically rigid. One can deduce the conclusion from the topological classification of 4-dimensional smooth manifolds with 
$T^{2}$-action studied in [Reference Orlik and Raymond16] together with the cohomology formula [Reference Davis and Januszkiewicz9, theorem 4.14].
We note that the method in Proposition 4.2 applies to CW-complexes in 
$\mathscr {C}_{n,1}$ which are not necessarily manifolds. We also refer to [Reference Darby, Kuroki and Song8, § 5] for the computation of the cohomology ring of toric orbifolds considered in corollary 4.3.
4.2 The $\mathscr {C}_{n,m}$ case
From now on, we assume 
$m=2^{s}q$, where 
$q>1$ is odd and 
$s\geq 0$. Recall from (3.3) that 
$H^{3}(C_f)\cong \mathbb {Z}_m$ for 
$C_f\in \mathscr {C}_{n,m}$. In this subsection, we discuss the homotopy type of 
$C_f$. To be more precise, we study a necessary and sufficient condition for a wedge decomposition
\begin{equation} C_f\simeq \hat{C} \vee P^{3}(q) \end{equation}
where 
$\hat {C}$ is a complex in 
$\mathscr {C}_{n,2^{s}}$ so that 
$H^{i}(C_f) \cong H^{i}(\hat {C})$ for 
$i\neq 3$ and 
$H^{3}(\hat {C})\cong \mathbb {Z}_{2^{s}}$.
Lemma 4.4 Let q be odd and greater than 
$1$. Consider
(i) a map
$g_1\colon S^{3}\to P^{3}(q)$ and its mapping cone $C_1$,(ii) a map
$g_2\colon S^{3}\to P^{4}(q)$ and the mapping cone $C_2$ of the composition
\[ S^{3}\xrightarrow{g_2}P^{4}(q)\xrightarrow{[\kappa_1,\kappa_2]}S^{2}\vee P^{3}(q), \]
where 
$[\kappa _1,\kappa _2]$ is the Whitehead product of inclusions
\[ \kappa_1\colon S^{2}\to S^{2}\vee P^{3}(q) ~\text{and}~ \kappa_2\colon P^{3}(q)\to S^{2}\vee P^{3}(q). \]
For 
$i=1$ or 
$2$, if 
$H^{*}(C_i;\mathbb {Z}_m)$ has trivial cup products, then 
$g_i$ is null homotopic.
Proof. Let 
$q=p_1^{r_1}\ldots p_{\ell }^{r_{\ell }}$ be a primary factorization of q such that 
$p_j$'s are different odd primes and all 
$r_j$'s are at least 
$1$. By the Hurewicz theorem, 
$\pi _3(P^{4}(q))\cong \mathbb {Z}_q$. By [Reference Porter17, theorem 4] and [Reference So and Theriault21, lemma 2.1],
\[ \pi_3(P^{3}(q))\cong\bigoplus^{\ell}_{j=1}\pi_3(P^{3}(p_j^{r_j}))\cong\bigoplus^{\ell}_{j=1}\mathbb{Z}_{p_j^{r_j}}\cong\mathbb{Z}_q. \]
It suffices to prove the two cases after localization at 
$p_j$.
For the 
$i=1$ case, the lemma is a special case of [Reference So and Theriault21, Proposition 4.4]. For the 
$i=2$ case, it can be proved by the argument of [Reference So and Theriault21, Proposition 3.2] and replacing 
$P^{3}(p^{t})$ by 
$S^{2}$ and the index 
$t$ by 
$\infty$, respectively.□
Lemma 4.5 Let 
$m=2^{s}q$, where q is odd and greater than 1. Let 
$f\colon S^{3}\to {\bigvee} _{i=1}^{n} S^{2}_i\vee P^{3}(m)$ be the attaching map of the 4-cell in 
$C_f$ and let 
$M_{cup}(C_f)=(A,\textbf {b},c)$. If 
$\textbf {b}\equiv (0,\ldots ,0)\pmod {q}$ and 
$c\equiv 0\pmod {q}$, then there is a CW-complex 
$\hat {C}\in \mathscr {C}_{n,2^{s}}$ such that 
$C_f\simeq \hat {C}\vee P^{3}(q)$.
Proof. Since 
$2^{s}$ and q are coprime, we have 
$P^{3}(m)\simeq P^{3}(2^{s})\vee P^{3}(q)$. By the Hilton–Milnor theorem, 
$f$ is homotopic to a wedge sum
\[ f\simeq\sum^{n}_{i=1}a_i\eta_i+\sum_{1\leq j< k\leq n}a_{jk}\omega_{jk}+\eta'+\sum^{n}_{i=1}\omega'_{i}+\eta_q+\sum_{i=1}^{n}\omega_{iq} \]
for some integers 
$a_i$'s and 
$a_{jk}$'s. Here, 
$\eta '$, 
$\omega '_{i}$, 
$\eta _q$ and 
$\omega _{iq}$ are compositions
\[ \begin{array}{l} \eta'\colon S^{3}\xrightarrow{b'} P^{3}(2^{s})\hookrightarrow\underset{j=1}{\overset{n}{\bigvee}} S^{2}_j\vee P^{3}(2^{s})\vee P^{3}(q)\\ \omega'_{i}\colon S^{3}\xrightarrow{b'_i} P^{4}(2^{s})\xrightarrow{[\kappa'_1,\kappa'_2]} S^{2}_i\vee P^{3}(2^{s})\hookrightarrow\underset{j=1}{\overset{n}{\bigvee}} S^{2}_j\vee P^{3}(2^{s})\vee P^{3}(q)\\ \eta_q\colon S^{3}\xrightarrow{b_q} P^{3}(q)\hookrightarrow\underset{j=1}{\overset{n}{\bigvee}} S^{2}_j\vee P^{3}(2^{s})\vee P^{3}(q)\\ \omega_{iq}\colon S^{3}\xrightarrow{b_{iq}} P^{4}(q)\xrightarrow{[\kappa_1,\kappa_2]} S^{2}_i\vee P^{3}(q) \hookrightarrow\underset{j=1}{\overset{n}{\bigvee}} S^{2}_j\vee P^{3}(2^{s})\vee P^{3}(q)\\ \end{array} \]
for some maps 
$b'$, 
$b'_i$ and 
$b_q$, 
$b_{iq}$. Here
\[ [\kappa'_1,\kappa'_2]:P^{4}(2^{s})\to S^{2}_i\vee P^{3}(2^{s}) \]
is the Whitehead product of inclusions 
$\kappa '_1:S^{2}_i\to S^{2}_i\vee P^{3}(2^{s})$ and 
$\kappa '_2:P^{3}(2^{s})\to S^{2}_i\vee P^{3}(2^{s})$, and
\[ [\kappa_1,\kappa_2]:P^{4}(q)\to S^{2}_i\vee P^{3}(q) \]
is the Whitehead product of inclusions 
$\kappa _1:S^{2}_i\to S^{2}_i\vee P^{3}(q)$ and 
$\kappa _2:P^{3}(q)\to S^{2}_i\vee P^{3}(q)$. If 
$\eta _q$ and 
$\omega _{iq}$'s are null homotopic, then 
$f$ factors through a map 
$\hat {f}\colon S^{3}\to {\bigvee} _{i=1}^{n} S^{2}_i\vee P^{3}(2^{s})$. Let 
$\hat {C}$ be the mapping cone of 
$\hat {f}$. Then 
$C_f\simeq \hat {C}\vee P^{3}(q)$.
Hence, it suffices to show that 
$\eta _q$ and 
$\omega _{\ell q}$ are null homotopic for any 
$\ell$ with 
$1\leq \ell \leq n$. After localization away from 2, 
$P^{3}(2^{s})$ becomes contractible and the composition
\[ f_{\ell q}\colon S^{3}\overset{f}{\to }{\bigvee}^{n}_{j=1}S^{2}_j\vee P^{3}(q)\xrightarrow{\text{pinch}}S^{2}_{\ell}\vee P^{3}(q) \]
is homotopic to the wedge sum 
$a_{\ell }\tilde {\eta }_{\ell }+\jmath \circ b_q+[\kappa _1,\kappa _2]\circ b_{\ell q}$, where 
$\tilde {\eta }$ is the composition of Hopf map 
$\eta$ and inclusion 
$S^{2}_{\ell }\to S^{2}_{\ell }\vee P^{3}(q)$, and 
$\jmath \colon P^{3}(q)\to S^{2}_{\ell }\vee P^{3}(q)$ is the inclusion.
Consider the diagram of homotopy cofibration sequences
where 
$C_{\ell q}$ is the mapping cone of 
$f_{\ell q}$ and 
$\pi$ is an induced map. Let 
$\{\bar {u}_1,\ldots ,\bar {u}_n,\bar {v};\bar {e}\}$ and 
$\{\bar {u}',\bar {v}';\bar {e}'\}$ be the mod-q cellular bases of 
$H^{*}(C_f;\mathbb {Z}_{q})$ and 
$H^{*}(C_{\ell q};\mathbb {Z}_{q})$. Then
\[ \begin{array}{c c c} \pi^{*}(\bar{u}')=\bar{u}_{\ell}, & \pi^{*}(\bar{v}')=\bar{v}, & \pi^{*}(\bar{e}')=\bar{e}. \end{array} \]
By the hypothesis, 
$\pi ^{*}(\bar {u}'\cup \bar {v}')=\bar {u}_{\ell }\cup \bar {v}=0$. Since 
$\pi ^{*}\colon H^{4}(C_{\ell q};\mathbb {Z}_{q})\to H^{4}(C_f;\mathbb {Z}_{q})$ is isomorphic, we have 
$\bar {u}'\cup \bar {v}'=0$. Similarly, we have 
$\bar {v}'\cup \bar {v}'=0$ and 
$\bar {u}'\cup \bar {u}'=a_{\ell }\bar {e}$ so that 
$M_{cup}(C_{\ell q})=(a_{\ell },0,0)$. Let 
$C_1$ and 
$C_2$ be the mapping cones of 
$[\kappa _1,\kappa _2]\circ b_{\ell q}$ and 
$\jmath \circ b_q$. By lemma 3.4,
\[ M_{cup}(C_1)=M_{cup}(C_2)=(0,0,0) \]
so 
$H^{*}(C_1;\mathbb {Z}_{q})$ and 
$H^{*}(C_2;\mathbb {Z}_{q})$ have trivial cup products. By lemma 4.4, 
$b_{\ell q}$ is null homotopic and so is 
$\omega _{\ell q}$. Also, notice that 
$C_2\simeq S^{2}_{\ell }\vee C'$ where 
$C'$ is the mapping cone of 
$b_q$. So 
$H^{*}(C';\mathbb {Z}_{q})$ has trivial cup products. By lemma 4.4, 
$b_q$ is null homotopic and so is 
$\eta _q$.□
Remark 4.6 In general, 
$\hat {C}$ cannot be further decomposed into a wedge of non-contractible spaces, for example 
$\hat {C}=\Sigma \mathbb {R}\mathbb {P}^{3}$.
Notice that 
$\hat {C}\vee P^{3}(q)$ is not contained in 
$\mathscr {C}_{n,m}$, but it is homotopic to a mapping cone in 
$\mathscr {C}_{n,m}$ as follows. Since 
$2^{s}$ and q are coprime to each other, there exist integers 
$\alpha$ and 
$\beta$ such that 
$2^{s}\alpha +q\beta =1$, where the mod-q congruence class of 
$\alpha$ and the mod-
$2^{s}$ congruence class of 
$\beta$ are unique. Identify 
$\mathbb {Z}_{2^{s}}\oplus \mathbb {Z}_q$ with 
$\mathbb {Z}_m$ via the isomorphism
\begin{equation} \rho\colon \mathbb{Z}_{2^{s}}\oplus\mathbb{Z}_q\to\mathbb{Z}_{m},\quad(x,y)\mapsto q\beta x+2^{s}\alpha y. \end{equation}
It induces a homotopy equivalence 
$\rho \colon P^{3}(2^{s})\vee P^{3}(q)\to P^{3}(m)$. Consider the diagram of homotopy cofibrations
where 
$C'$ is the mapping cone of 
$(id\vee \rho )\circ (\hat {f}\vee \ast )$ and 
$\tilde {\rho }$ is an induced homotopy equivalence. Then 
$C'\simeq \hat {C}\vee P^{3}(q)$ via 
$\tilde {\rho }$.
Lemma 4.7 Let 
$M_{cup}(C')=(A,\textbf {b},c)$. Then 
$\textbf {b}\equiv (0,\ldots ,0)$ and 
$c\equiv 0\pmod {q}.$
Proof. We prove 
$b_i\equiv 0\pmod q$. Let
•
$\bar {u}_i\in H^{2}(\hat {C};\mathbb {Z}_m)$ and $\bar {e}\in H^{4}(\hat {C};\mathbb {Z}_m)$ be the mod-m cohomology classes dual to homology classes representing $S^{2}_i$ and the 4-cell in $\hat {C}$, respectively;•
$\mu _i,\,\omega _{2^{s}},\, \omega _q\in H_2(\hat {C}\vee P^{3}(q);\mathbb {Z})$ be the homology classes representing $S_i^{2}$, the bottom cells of $P^{3}(2^{s})$ and $P^{3}(q)$;•
$\bar {w}_{2^{s}},\,\bar {w}_{q}\in H^{2}(\hat {C}\vee P^{3}(q);\mathbb {Z}_m)$ be the cohomology classes such that
\[ \begin{array}{@{}c c c@{}} \bar{w}_{2^{s}}(\omega_{2^{s}})\equiv q\beta & \bar{w}_{2^{s}}(\omega_{q})\equiv 0 & \bar{w}_{2^{s}}(\mu_i)\equiv 0\\ \bar{w}_{q}(\omega_{2^{s}})\equiv 0 & \bar{w}_{q}(\omega_{q})\equiv 2^{s}\alpha & \bar{w}_{q}(\mu_i)\equiv 0 \end{array}\pmod{m}. \]
Denote 
$\bar {v}=\bar {w}_{2^{s}}+\bar {w}_q$. Then 
$\bar {u}_1,\ldots ,\bar {u}_n$ and 
$\bar {v}$ form a basis of 
$H^{2}(\hat {C}\vee P^{3}(q);\mathbb {Z}_m)$.
The right square of (4.3) implies
\[ \tilde{\rho}^{*}(\bar{u}'_i)=\bar{u}_i,\quad\tilde{\rho}^{*}(\bar{v}')=\bar{v}=\bar{w}_{2^{s}}+\bar{w}_q. \]By the naturality of cup products, we have
\begin{eqnarray*} \tilde{\rho}^{*}(\bar{u}'_i\cup\bar{v}')&=&\tilde{\rho}^{*}(b_i\bar{e}')\\ \bar{u}_i\cup(\bar{w}_{2^{s}}+\bar{w}_q)&=&b_i\bar{e}\\ \bar{u}\cup\bar{w}_{2^{s}}&=&b_i\bar{e}. \end{eqnarray*}
Since 
$\bar {w}_{2^{s}}$ is a multiple of q and 
$\bar {e}$ is a generator, 
$b_i\equiv 0\pmod {q}$. Similarly we can show that 
$c\equiv 0\pmod {q}$.□
Proposition 4.8 Let 
$m=2^{s}q$ as before. For 
$C_f\in \mathscr {C}_{n,m}$, let 
$M_{cup}(C_f)=(A,\textbf {b},c)$ where
\[ A=\left( \begin{array}{@{}c c c@{}} a_{11} & \cdots & a_{1n}\\ \vdots & \ddots & \vdots\\ a_{n1} & \cdots & a_{nn} \end{array} \right) \text{ and } \mathbf{b}=(b_1, \dots, b_n). \]
Then 
$C_f\simeq \hat {C}\vee P^{3}(q)$ for some 
$\hat {C}\in \mathscr {C}_{n,2^{s}}$ if and only if the system of mod-q linear equations
\begin{equation} \left\{ \begin{array}{@{}c c l@{}} a_{11}y_1+\cdots+a_{1n}y_n & \equiv & -b_1\\ \vdots & & \\ a_{n1}y_1+\cdots+a_{nn}y_n & \equiv & -b_n\\ b_1y_1+\cdots+b_ny_n & \equiv & -c \end{array} \right. \quad {\pmod{q}} \end{equation}
has a solution 
$(y_1,\ldots ,y_n)\in {(\mathbb {Z}_q)^{n}}$.
Proof. Suppose 
$g\colon C'\simeq \hat {C}\vee P^{3}(q)\to C_f$ is a homotopy equivalence. Let 
$M(g)=(W,\textbf {y},z)$ and let 
$M_{cup}(C')=(A', \mathbf {b}', c')$. By lemma 3.7, we have
\begin{equation} A'=W^{t}AW, ~\textbf{b}'=\textbf{y}AW+z\textbf{b}W ~\text{ and } ~ c'=\textbf{y}A\textbf{y}^{t}+2z\textbf{y}\textbf{b}^{t}+z^{2}c. \end{equation}
Lemma 4.7 implies 
$\mathbf {b}'\equiv (0,\ldots ,0)$ and 
$c'\equiv 0$ modulo q. Since 
$W$ and 
$z$ are invertible in 
$\mathbb {Z}_q$, we can rewrite equations in (4.5) as
\[ ~z^{{-}1}\textbf{y}A\equiv{-}\textbf{b} ~\text{ and } ~ z^{{-}1}\textbf{y}\textbf{b}^{t}\equiv{-}c \pmod{q}. \]
Therefore, 
$z^{-1}\textbf {y}$ is a solution of (4.4).
Conversely, suppose there is a solution 
$\mathbf {y}=(y_1,\ldots ,y_n)\in (\mathbb {Z}_q)^{n}$ of (4.4). By lemma 3.9, there exist 
$C''\in \mathscr {C}_{n,m}$ and a homotopy equivalence 
$g\colon C''\to C_f$ such that 
$M(g)=(I,\textbf {y}',1)$ and
\[ M_{cup}(C'')=\Big(A,~\mathbf{y}'\bar A +\mathbf{b},~\mathbf{y}'\bar A (\mathbf{y}')^{t}+2\mathbf{b}(\mathbf{y}')^{t}+c \Big), \]
where 
$\textbf {y}'=(\rho (y_1,0),\ldots ,\rho (y_n,0))\in (\mathbb {Z}_m)^{n}$ for 
$\rho$ defined in (4.2) and 
$\bar {A}$ is the mod-m image of 
$A$. Then
\[ \mathbf{y}'\bar A +\mathbf{b}\equiv 0~\text{ and }~ \mathbf{y}'\bar A (\mathbf{y}')^{t}+2\mathbf{b}(\mathbf{y}')^{t}+c\equiv 0\pmod{q}. \]
Note that lemma 4.5 implies 
$C''\simeq \hat {C}\vee P^{3}(q)$ for some 
$\hat {C}\in \mathscr {C}_{n,2^{s}}$. Consequently, we have 
$C_f\simeq \hat {C}\vee P^{3}(q)$.□
5. Odd primary local decomposition of toric orbifolds
Let 
$X=P\times T^{2}/_\sim$ be a 4-dimensional toric orbifold associated with the combinatorial data described in § 2. Since X is simply connected and 
$H^{*}(X)$ satisfies (2.6), [Reference Hatcher12, Proposition 4H.3] implies that X is in 
$\mathscr {C}_{n,m}$ up to homotopy. Let 
$m=2^{s}q$, where q is odd and 
$s\geq 0$. In this section, we show that for any odd prime p, there is a p-local equivalence
\begin{equation} X\simeq_{(p)} \hat{X} \vee P^{3}(q) \end{equation}
for a CW-complex 
$\hat {X}$ in 
$\mathscr {C}_{n,2^{s}}$ and 
$P^{3}(q)$ denotes a point if 
$q=1$.
The q-CW complex structure of X with respect to a vertex 
$v_i$ (see remark 2.2) implies that X is homotopy equivalent to the mapping cone of a map
\[ f\colon L_i\to {\bigvee}^{n}_{j=1}S^{2} \]
where 
$L_i$ is the quotient 
$S^{3}/\mathbb {Z}_{m_{i,i+1}}$ and 
$m_{i,i+1}=|\det \begin {bmatrix}\xi ^{t}_{i},\xi ^{t}_{i+1}\end {bmatrix}|$. Recall that 
$\mathbb {Z}_{m_{i, i+1}}$ is isomorphic to a subgroup 
$\ker \rho _i$ of 
$T^{2}$, where 
$\rho _i$ is defined in (2.2). The 
$\mathbb {Z}_{m_{i, i+1}}$-action on 
$S^{3}$ is given by the inclusion 
$\ker \rho _i\hookrightarrow T^{2}$ and the standard 
$T^{2}$-action on 
$S^{3}$. If 
$m_{i, i+1}=1$, then 
$L_i\cong S^{3}$ and X is in 
$\mathscr {C}_{n,1}$. So, the equivalence (5.1) holds. If 
$m_{i,i+1}>1$, then 
$L_i$ is a lens space 
$L(m_{i,i+1};k_i)$ for some 
$k_i$ coprime to 
$m_{i,i+1}$.
In the following, the p-component 
$\nu _p(t)$ of a number 
$t$ is defined to be the p-power 
$p^{r}$ such that 
$p^{r}$ divides 
$t$ but 
$p^{r+1}$ does not.
Lemma 5.1 For p odd prime, let 
$\nu _p(m_{i,i+1})=p^{r}$ and let 
$L_i=L(m_{i,i+1};k_i)$ be a lens space. Then there is a map 
$\alpha _p\colon \Sigma L_i\to S^{4}\vee P^{3}(p^{r})$ that is a p-local equivalence.
Proof. Let 
$m_{i,i+1}=p^{r}{t}$ where p and 
$t$ are coprime. Then 
$P^{3}(m_{i,i+1})\simeq P^{3}(p^{r})\vee P^{3}(t)$. Consider the diagram of homotopy cofibration sequences
where 
$\phi$ is the attaching map of the 4-cell in 
$\Sigma L_i$, 
$\phi '$ is the composition of 
$\phi$ and the pinch map, 
$C$ is the mapping cone of 
$\phi '$ and 
$\alpha _p$ is an induced map. The right column induces an exact sequence
\[ \cdots\to \tilde{H}^{i-1}(P^{3}(t);\mathbb{Z}_{p^{r}})\to\tilde{H}^{i}(C;\mathbb{Z}_{p^{r}})\xrightarrow{\alpha_p^{*}}\tilde{H}^{i}(\Sigma L_i;\mathbb{Z}_{p^{r}})\to\tilde{H}^{i}(P^{3}(t);\mathbb{Z}_{p^{r}})\to\cdots \]
Since 
$\tilde {H}^{*}(P^{3}(t);\mathbb {Z}_{p^{r}})=0$, the map 
$\alpha _p^{*}\colon H^{*}(C;\mathbb {Z}_{p^{r}})\to H^{*}(\Sigma L_i;\mathbb {Z}_{p^{r}})$ is an isomorphism. Moreover, 
$H^{*}(C;\mathbb {Z}_{p^{r}})$ has trivial cup products because 
$\Sigma L_i$ is a suspension. Now, lemma 4.4 shows that 
$\phi '$ is null homotopic, which means 
$C\simeq S^{4}\vee P^{3}(p^{r})$. Therefore, we consider 
$\alpha _p$ as a map from 
$\Sigma L_i$ to 
$S^{4}\vee P^{3}(p^{r})$. Since 
$P^{3}(t)$ is contractible after p-localization, the right column implies that 
$\alpha _p$ is a p-local equivalence.□
Lemma 5.2 Let p be a prime and let 
$H^{3}(X)\cong \mathbb {Z}_m$. Then there exists 
$i\in \{1,\ldots ,n+2\}$ such that 
$\nu _p(m_{i,i+1})=\nu _p(m)$.
Proof. By [Reference Kuwata, Masuda and Zeng14, corollary 5.1] and [Reference Fischli10, lemma 3.1], 
$m=\text {gcd}\{m_{i,j}|1\leq i< j\leq n+2\}$. If 
$n=1$, the lemma is trivial. So, we prove the lemma for 
$n\geq 2$.
Without loss of generality, suppose 
$\nu _p(m_{1,j})=\nu _p(m)=p^{r}$ for some 
$j\in \{3,\ldots ,n+1\}$. Let 
$\xi _1=(a, b)$. Since 
$a$ and 
$b$ are coprime, there exist 
$u$ and 
$v$ such that
\[ \left( \begin{array}{@{}c c@{}} u & -b\\ v & a \end{array} \right) \in SL_2(\mathbb{Z}). \]
Changing the basis of 
$\mathbb {Z}^{2}$ if necessary, we may assume 
$\xi _1=(1,0)$.
Let 
$\xi _2=(x, y)$ and 
$\xi _{j}=(z,w)$. Then
\[ m_{1,2}=\left|\det\begin{pmatrix} 1 & x\\ 0 & y \end{pmatrix}\right| =|y|, \quad m_{1,j}=\left|\det\begin{pmatrix} 1 & z\\ 0 & w \end{pmatrix}\right| =|w|. \]
Write 
$w=cp^{r}$ and 
$y=c'p^{s}$, where 
$c$ and 
$c'$ are integers coprime to p and 
$s\geq r$. If 
$s=r$, then 
$v_p(m_{1,2})=\nu _p(m)$ and consequently the lemma holds. If 
$s>r$, then
\begin{align*} m_{2,j} =|xw-yz|=|cxp^{r}-c'yp^{s}| =p^{r}|cx-c'yp^{s-r}|. \end{align*}
Since 
$x$ is coprime to 
$y$, 
$x$ is coprime to p. So, 
$cx-c'yp^{s-r}$ is coprime to p and 
$\nu _p(m_{2,j})=p^{r}$. If 
$j=3$, then we are done. If not, iterate this argument to 
$m_{2,j}$, 
$\xi _3$ and 
$\xi _j$. Then we can conclude that 
$\nu _p(m_{j-1,j})=p^{r}$.□
For any odd prime p, let 
$\nu _p(m)=p^{r}$. By lemma 5.2, there is an 
$i\in \{1,\ldots ,n+2\}$ such that 
$\nu _p(m_{i,i+1})=\nu _p(m)=p^{r}$. Pick the vertex 
$v_i$ and construct the q-CW-complex structure with respect to 
$v_i$. Then there is a homotopy cofibration sequence
\[ L_i\overset{f}{\to }{\bigvee}^{n}_{j=1}S^{2}\to X \]
with a coaction 
$c:X\to X\vee \Sigma L_i$. Furthermore, the 3-skeleton of X is 
${\bigvee} ^{n}_{j=1}S^{2}\vee P^{3}(m)$ for 
$m=2^{s}q$. Let 
$\hat {X}$ be the quotient 
$X/P^{3}(q)$ and let 
$\phi _p$ be the composition
\begin{equation} \phi_p\colon X\overset{c}{\to }X\vee\Sigma L\xrightarrow{\jmath\vee\alpha} \hat{X}\vee P^{3}(p^{r})\vee S^{4}\xrightarrow{\text{pinch}} \hat{X}\vee P^{3}(p^{r}) \end{equation}
where 
$\alpha$ is the map in lemma 5.1 and 
$\jmath \colon X\to \hat {X}$ is the quotient map.
Proposition 5.3 (p-local version of main theorem)
Let p be an odd prime. If 
$\nu _p(m)=p^{r}$ for some 
$r\geq 1$, then 
$\phi _p\colon X\to \hat {X}\vee P^{3}(p^{r})$ is a p-local equivalence.□
Proof. We claim that the map 
$\phi _p$ in (5.2) induces an isomorphism on 
$\mathbb {Z}_{(p)}$-cohomology
\begin{equation} \phi^{*}_p\colon H^{*}(\hat{X}\vee P^{3}(p^{r});\mathbb{Z}_{(p)})\to H^{*}(X;\mathbb{Z}_{(p)}) \end{equation}
where 
$\mathbb {Z}_{(p)}$ is the ring of p-local integers.
The cofibration sequence 
${P^{3}(q)}\hookrightarrow X\overset {\jmath }{\to }\hat {X}$ induces an exact sequence
\[ \cdots\to\tilde{H}^{i-1}({P^{3}(q)})\to \tilde{H}^{i}(\hat{X})\overset{\jmath^{*}}{\to}\tilde{H}^{i}(X)\to\tilde{H}^{i}({P^{3}(q)})\to\tilde{H}^{i+1}(\hat{X})\to\cdots \]
For 
$i\neq 3$, since 
$\tilde {H}^{i}(P^{3}(p^{r}))=0$ and 
$\jmath ^{\ast }\colon H^{i}(\hat {X})\to H^{i}(X)$ is an isomorphism, the map (5.3) is an isomorphism.
Next, consider the cofibration sequence 
$L_i\overset {f}{\to }{\bigvee} _{i=1}^{n} S^{2}\overset {\imath }{\to }X\overset {\delta }{\to }\Sigma L_i$, where 
$\imath$ is the inclusion and 
$\delta$ is the coboundary map. It induces an exact sequence
\[ \cdots\to\tilde{H}^{i}(\Sigma L_i;\mathbb{Z}_{(p)})\xrightarrow{\delta^{*}}\tilde{H}^{i}(X;\mathbb{Z}_{(p)})\to\tilde{H}^{i}\bigg(\underset{i=1}{\overset{n}{\bigvee}} S^{2};\mathbb{Z}_{(p)}\bigg)\to\tilde{H}^{i+1}(\Sigma L;\mathbb{Z}_{(p)})\to\cdots \]
Since 
$\imath ^{*}\colon H^{2}(X;\mathbb {Z}_{(p)})\to H^{2}({\bigvee} ^{n}_{i=1}S^{2}_i;\mathbb {Z}_{(p)})$ is an isomorphism, 
$\delta ^{*}\colon H^{3}(\Sigma L;\mathbb {Z}_{(p)})\to H^{3}(X;\mathbb {Z}_{(p)})$ is an isomorphism. Consider the following commutative diagram
where the composite in the upper row is 
$\phi _p$ in (5.2) and the unnamed arrows are pinch maps. The left square commutes due to the property of the coaction map. By lemma 5.1, the map 
$\alpha _p^{*}\colon H^{3}(P^{3}(p^{r})\vee S^{4};\mathbb {Z}_{(p)})\to H^{3}(\Sigma L;\mathbb {Z}_{(p)})$ is isomorphic, so the composite in the lower row induces an isomorphism 
$H^{3}(P^{3}(p^{r});\mathbb {Z}_{(p)})\to H^{3}(X;\mathbb {Z}_{(p)})$. Since 
$H^{3}(\hat {X};\mathbb {Z}_{(p)})=0$, the map (5.3) is an isomorphism for 
$i=3$. Therefore, 
$\phi ^{*}_p\colon H^{*}(\hat {X}\vee P^{3}(p^{r});\mathbb {Z}_{(p)})\to H^{*}(X;\mathbb {Z}_{(p)})$ is an isomorphism and 
$\phi _p$ is a p-local equivalence.
Lemma 5.4 Let X be a 4-dimensional toric orbifold with 
$H^{3}(X)\cong \mathbb {Z}_m$, and let 
$\nu _p(m)=p^{r}$ for some odd prime p and 
$r\geq 1$. If 
$M_{cup}(X)=(A,\textbf {b},c)$ where
\[ A=\left( \begin{array}{@{}c c c@{}} a_{11} & \cdots & a_{1n}\\ \vdots & \ddots & \vdots\\ a_{n1} & \ldots & a_{nn} \end{array} \right) \text{ and } \mathbf{b}=(b_1, \dots, b_n), \]
then the system of mod-
$p^{r}$ linear equations
\[ \left\{ \begin{array}{@{}c c l@{}} a_{11}y_1+\ldots+a_{1n}y_n & \equiv & -b_1\\ \vdots & & \\ a_{n1}y_1+\ldots+a_{nn}y_n & \equiv & -b_n\\ b_1y_1+\ldots+b_ny_n & \equiv & -c \end{array} \right. \quad \pmod{p^{r}} \]
has a solution in 
$(\mathbb {Z}_{p^{r}})^{n}$.
Proof. By Proposition 5.3 there is a map 
$\phi _p\colon X\to \hat {X}\vee P^{3}(p^{r})$ that becomes a homotopy equivalence after localized at p, where 
$\hat {X}\in \mathscr {C}_{n,2^{s}}$ is the quotient 
$X/P^{3}(q)$. Let
\[ M(\phi_p)=(W,\textbf{y},z)\in\text{Mat}_n(\mathbb{Z})\oplus(\mathbb{Z}_m)^{n}\oplus\mathbb{Z}_m \]
be the cellular map representation of 
$\phi _p$. After p-localization, 
$W$ is an invertible matrix and 
$z$ is a unit. The lemma follows from Proposition 4.8.
□
6. Proof of the main theorems
Proof of theorem 1.2. Let 
$q=p_1^{r_1}\ldots p_k^{r_k}$ be the primary factorization where 
$p_i$'s are different odd primes and 
$r_i\geq 1$. For each prime 
$p_i$, lemma 5.4 implies that the mod-
$p_i^{r_i}$ version of (4.4) has a solution. By Chinese Remainder theorem, they give a mod-q solution for (4.4). By Proposition 4.8, X is homotopy equivalent to 
$\hat {X}\vee P^{3}(q)$.□
Proof of theorem 1.3. The ‘only if’ part is trivial. To prove the ‘if’ part, let X and 
$X'$ be 4-dimensional toric orbifolds such that 
$H^{3}(X)\cong \mathbb {Z}_m$ and 
$H^{3}(X')\cong \mathbb {Z}_{m'}$ for m and 
$m'$ odd. The hypothesis implies that 
$H^{3}(X)\cong H^{3}(X')$, hence we have 
$m=m'$. By theorem 1.2, we have 
$X\simeq \hat {X}\vee P^{3}(m)$ and 
$X'\simeq \hat {X}'\vee P^{3}(m)$ for some 
$\hat {X},\hat {X}'\in \mathscr {C}_{n,1}$. Since 
$H^{i}(X)\cong H^{i}(\hat {X})$ and 
$H^{i}(X')\cong H^{i}(\hat {X}')$ for 
$i\neq 3$, we have 
$H^{*}(\hat {X})\cong H^{*}(\hat {X}')$. Then Proposition 4.2 implies that 
$\hat {X}\simeq \hat {X}'$, which yields 
$X\simeq X'$.
□
Acknowledgement
We started this project during our participation to the Thematic Program on Toric Topology and Polyhedral Products at Fields Institute. We gratefully acknowledge the support of Fields Institute and the organizers of the programme. Furthermore, we thank Mikiya Masuda, Taras Panov, Dong Youp Suh and Donald Stanley for discussing the topics, and thank Stephen Theriault for proofreading our draft and giving helpful comments.
Financial support
The first author was supported by Fields Institute and is supported by the National Research Foundation of Korea funded by the Korean Government (NRF-2019R1A2C2010989), the second author is supported by Pacific Institute for the Mathematical Sciences (PIMS) Postdoctoral Fellowship and the third author is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07048480) and a KIAS Individual Grant (MG076101) at Korea Institute for Advanced Study.
