Toric wedge induction and toric lifting property for piecewise linear spheres with a few vertices

Let $A$ be an $n\times m$ matrix over $\mathbb{Z}$ for positive integers $n\leq m$, and $I$ an $n$-subset of $[m]$. Let $A_{I}$ denote the submatrix of $A$ formed by selecting columns indexed with $i\in I$, and $A^{I}$ the submatrix of $A$ formed by selecting rows indexed with $i\in I$. Furthermore, $\overline{A}$ represents a matrix whose columns form a basis of the kernel of $A$. Note that $\overline{A}$ depends on the choice of a basis of the kernel of $A$. We introduce one important proposition, known as the linear Gale duality:

Let $K$ be an $(n-1)$-dimensional PL sphere on $[m]$, and $H$ an $r$-dimensional subtorus of $T^{m}$ freely acting on $\mathcal{Z}_{K}$ as in (2.1). If $r=m-n$, then $H$ is completely described as follows. Let us consider a map $\lambda\colon[m]\to\mathbb{Z}^{n}$, called a characteristic map over $K$, such that $\{\lambda(i_{1}),\ldots,\lambda(i_{k})\}$ is a unimodular set for any simplex $\{i_{1},\ldots,i_{k}\}$ in $K$. For convenience, we often represent this map by an $n\times m$ matrix $\lambda=\begin{bmatrix}\lambda(1)&\cdots&\lambda(m)\end{bmatrix}$ with elements in $\mathbb{Z}$. This matrix can be interpreted as a linear map $\mathbb{Z}^{m}\to\mathbb{Z}^{n}$, and concurrently, as an element in $\operatorname{Hom}(T^{m},T^{n})$. In addition, we call $\overline{\lambda}\colon[m]\to\mathbb{Z}^{m-n}$ a dual characteristic map over $K$. Similarly, we define mod $2$ characteristic maps $\lambda^{\mathbb{R}}\colon[m]\to\mathbb{Z}_{2}^{n}$ over $K$ and mod $2$ dual characteristic maps $\overline{\lambda}^{\mathbb{R}}\colon[m]\to\mathbb{Z}_{2}^{m-n}$ over $K$. In particular, an injective mod $2$ dual characteristic map is simply called an IDCM.

For an $n\times m$ matrix $\lambda$, $\overline{\lambda}$ defines a subtorus $H$ of $T^{m}$ similar to that described in $\eqref{free_action}$. By Propositions 2.1 and  3.1, $\lambda$ is a characteristic map over $K$ if and only if the corresponding subtorus $H$ of $\overline{\lambda}$ acts on $\mathcal{Z}_{K}$ freely.

When considering the toric space $\mathcal{Z}_{K}/H$, the kernel of $\lambda$ itself is essential whereas the choice of a basis of the kernel is not important. In this context, we consider the concepts of Davis-Januszkiewicz equivalence, or simply D-J equivalence, for characteristic maps and dual characteristic maps. Two characteristic maps are said to be D-J equivalent, if one is obtained by row operations from the other. Two dual characteristic maps are said to be D-J equivalent if one is obtained by column operations from the other. This also removes the ambiguity arising from the definition of $\overline{\lambda}$.

Observe that the mod $2$ reduction of a characteristic map $\lambda$ over $K$ is a mod $2$ characteristic map over $K$. Conversely, given a mod $2$ characteristic map $\lambda^{\mathbb{R}}\colon[m]\to\mathbb{Z}_{2}^{n}$ over $K$ and a characteristic map $\lambda\colon[m]\to\mathbb{Z}^{n}$ over $K$, if $\lambda^{\mathbb{R}}$ coincides with the composition of $\lambda$ and the modulo $2$ reduction map $\mathbb{Z}^{n}\to\mathbb{Z}_{2}^{n}$, then $\lambda$ is called a lift of $\lambda^{\mathbb{R}}$:

Moreover, it is called the $\{0,1\}$-lift of $\lambda^{\mathbb{R}}$ when $\lambda$ sends $[m]$ to $\{0,1\}$-vectors. Similarly, it is called a $\{0,\pm 1\}$-lift when it sends $[m]$ to $\{0,\pm 1\}$-vectors. Note that the number of $\{0,\pm 1\}$-lifts of a given mod $2$ characteristic map is finite.

For the sake of convenience, we define the dual complex $\overline{K}$ of $K$ as the simplicial complex whose facets are the cofacets of $K$. Also, we regard $\overline{\lambda}$ as a map from $[m]$ to $\mathbb{Z}^{m-n}$ such that $\overline{\lambda}(i)$ is the $i$th row of $\overline{\lambda}$, as we did for characteristic maps. Then by the linear Gale duality, $\overline{\lambda}$ is a characteristic map over $\overline{K}$.

Let $K$ be a simplicial complex on the vertex set $V$ and $\sigma$ a simplex in $K$. The link of $\sigma$ in $K$ is the simplicial complex defined by

and the deletion of $\sigma$ in $K$ is the simplicial complex defined by

For a singleton face $\{v\}$ of $K$, its link and deletion are denoted simply by $\operatorname{Lk}_{K}(v)$ and $K\setminus v$, respectively.

For another simplicial complex $L$ on a disjoint vertex set from $K$, the join $K\ast L$ of $K$ and $L$ is defined as the simplicial complex

The suspension of $K$ is given by

where $I$ is a $1$-simplex with two new vertices $v_{1}$ and $v_{2}$, and $\partial I$ is its boundary complex. In $\operatorname{\Sigma}(K)$, the pair $\{v_{1},v_{2}\}$ is referred to as a suspended pair, and each vertex in it is called a suspended vertex.

The wedge of $K$ at a vertex $v$ of $K$ is defined as

where $I$ is a $1$-simplex comprising two new vertices. It is evident that the link of a new vertex added after applying a wedge to $K$ is isomorphic to $K$. In that sense, we often use $v_{1}$ and $v_{2}$ to refer to the two copies of $v$ in $\operatorname{Wed}_{v}(K)$. Consequently, $\operatorname{Wed}_{v}(K)$ has vertex set $(V\setminus\{v\})\cup\{v_{1},v_{2}\}$. Here, two vertices $v_{1}$ and $v_{2}$ are referred to as wedged vertices of $v$, and the edge connecting them as the wedged edge of $v$. Notably, $\operatorname{\Sigma}(K)$ can be viewed as a wedge at a ghost vertex of $K$.

The wedge operation can be defined equivalently as an easy combinatorial operation on the minimal non-faces of $K$: we duplicate the vertex $v$ in each minimal non-face of $K$ it appears in. More precisely, let $\eta\subset V$ be a subset of the vertex set of $K$.

1. (1)

   If $\eta$ contains $v$, then $\eta$ is a minimal non-face of $K$ if and only if $\eta\setminus\{v\}\cup\{v_{1},v_{2}\}$ is a minimal non-face of $\operatorname{Wed}_{v}(K)$.

2. (2)

   If $\eta$ does not contain $v$, then $\eta$ is a minimal non-face of $K$ if and only if $\eta$ is a minimal non-face of $\operatorname{Wed}_{v}(K)$.

As for suspensions, one can easily prove that the minimal non-faces of $K_{1}\ast K_{2}$ is the union of the minimal non-faces of $K_{1}$ and $K_{2}$. Then the minimal non-faces of $\partial I\ast K$ is obtained by adding $I$ in the minimal non-faces of $K$. we can add a ghost vertex to $K$ which becomes a minimal non-face of $K$. With this perspective, two consecutive wedge operations and join operations, including suspension, are associative and commutative with appropriate vertex identification.

Conversely, suppose that there are two vertices $v_{1}$ and $v_{2}$ such that for any minimal non-face $\eta$ of $K$, $\{v_{1},v_{2}\}\subset\eta$ or $\{v_{1},v_{2}\}\cap\eta=\varnothing$. If $\sigma$ is a facet of $K$ containing neither $\{v_{1}\}$ nor $\{v_{2}\}$, then $\{v_{1}\}\cup\sigma$ is a non-face, so there is a minimal non-face $\eta$ of $K$ containing $v_{1}$. This contradicts to the assumption. Hence every facet of $K$ contains $v_{1}$ or $v_{2}$. By the following lemma, if $\{v_{1},v_{2}\}$ is not a minimal non-face of $K$, then it is a wedged edge of $K$, and otherwise, it is a suspended pair of $K$.

Consider the vertex set of $K$ to be $[m]=\{1,\ldots,m\}$. In light of the associative and commutative nature of wedge operations, we introduce the notation $K(J)$, termed a $J$-construction of $K$ in [3], for a positive integer $m$-tuple $J=(j_{1},j_{2},\ldots,j_{m})$. This represents the simplicial complex obtained by applying multiple wedge operations to $K$; for each $i\in[m]$, wedge operations are applied $j_{i}-1$ times to $K$ at $i$ or its copied vertices. We will often denote the copied vertices of $i$ by $i_{1},i_{2},\ldots,i_{j_{i}}$. For the sake of convenience, even when $j_{i}=1$, we treat $i$ as $i_{1}$, and we say $i_{k}$ is a wedged vertex of $i$ for each $k\geq 1$.

In addition, due to the commutativity and associativity of the operations involved, we have the relationship:

where $I$ is the $1$-simplex on $\{1,2\}$. This leads to two characterizations regarding the suspension and wedge operations.

We define the Picard number of $K$ as $m-n$. One can observe that the wedge operation preserves the Picard number of $K$ whereas the suspension increases the Picard number of $K$ by $1$. It is known that the link, wedge, and suspension operations are closed within the class of PL spheres, see [6] for details.

A PL sphere not isomorphic to some wedge of another PL sphere is termed a seed. It should be noted that any PL sphere $K$ of Picard number $p$ can be written as $L(J)$, where $L$ is a seed of the same Picard number $p$. In addition, one can easily see that $L$ is uniquely determined up to isomorphism, whereas $J$ can be different.

For our purpose, we are interested in PL spheres of dimension $n-1$ on $[m]$ whose real Buchstaber number coincides with their Picard number $m-n$. Such a PL sphere is said to be ($\mathbb{Z}_{2}^{n}$-)colorable. Ewald [16] observed that all colorable PL spheres are obtained by colorable seeds, and Choi and Park [10] proved that the number of colorable seeds with given Picard number is finite. Although obtaining the list of colorable seeds of given Picard number is a difficult problem in itself, the list up to Picard number $4$ has been established in [6].

Let $K$ be an $(n-1)$-dimensional PL sphere on $[m]$. There are operations on mod $2$ characteristic maps over $K$ corresponding to wedge and link operations on $K$. Let $\Lambda^{\mathbb{R}}$ be a mod $2$ characteristic map over $\operatorname{Wed}_{v}(K)$. Up to D-J equivalence and vertex relabeling, we may assume that

where the column indexes $v_{1}$ and $v_{2}$ stand for the associated wedged vertices, a and b are row vectors of size $m-n$, $I_{n-1}$ is the identity matrix of size $n-1$, and $A$ is a $\mathbb{Z}_{2}$-matrix of size $(n-1)\times(m-n)$. The projection of $\Lambda^{\mathbb{R}}$ with respect to a face $\sigma$ of $\operatorname{Wed}_{v}(K)$ is a map from the vertex set of $\operatorname{Lk}_{K}(\sigma)$ to $\mathbb{Z}_{2}^{n-|\sigma|}$ defined by

for each vertex $w$ of $\operatorname{Lk}_{K}(\sigma)$. If we fix a basis of $\mathbb{Z}_{2}^{n-|\sigma|}$, we can see that $\operatorname{Proj}_{\sigma}(\Lambda^{\mathbb{R}})$ is a mod $2$ characteristic map over $\operatorname{Lk}_{K}(\sigma)$. We call $\operatorname{Proj}_{\sigma}(\Lambda^{\mathbb{R}})$ the projection onto $\operatorname{Lk}_{K}(\sigma)$.

The links $\operatorname{Lk}_{\operatorname{Wed}_{v}(K)}(v_{1})$ and $\operatorname{Lk}_{\operatorname{Wed}_{v}(K)}(v_{2})$ are isomorphic to $K$ by identifying $v_{2}$ and $v_{1}$ with $v$, respectively, and the projections of $\Lambda^{\mathbb{R}}$ with respect to $v_{1}$ and $v_{2}$ are written as

If we consider the first column of each matrix corresponds to the vertex $v$, then two matrices $\lambda^{\mathbb{R}}_{1}$ and $\lambda^{\mathbb{R}}_{2}$ are mod $2$ characteristic maps over $K$. Hence, $\Lambda^{\mathbb{R}}$ corresponds to a choice of two mod $2$ characteristic maps over $K$ whose first $n$ columns form an identity matrix such that their submatrices formed by deleting the first row and the first column are identical up to D-J equivalence.

Conversely, one may construct at most one $\Lambda^{\mathbb{R}}$ over $\operatorname{Wed}_{v}(K)$ from an ordered pair of mod $2$ characteristic maps $\lambda_{1}^{\mathbb{R}}$ and $\lambda_{2}^{\mathbb{R}}$ over $K$ as in (4.1). We denote $\Lambda^{\mathbb{R}}=\lambda_{1}^{\mathbb{R}}\wedge_{v}\lambda_{2}^{\mathbb{R}}$ if it exists. If $\lambda_{1}^{\mathbb{R}}=\lambda_{2}^{\mathbb{R}}=\lambda^{R}$, then $\lambda^{\mathbb{R}}\wedge_{v}\lambda^{\mathbb{R}}$ always exists for any $v$, and it is called the canonical extension of $\lambda^{\mathbb{R}}$ at $v$.

It should be noted that we can represent each characteristic map over $K(J)$ by a combination of characteristic maps over $K$. From this viewpoint, we shall introduce one powerful inductive tool to demonstrate some properties on real toric spaces, for example the existence of a lift of $\Lambda^{\mathbb{R}}$ as in this paper.

For a PL sphere $K$, let $\mathcal{X}$ be a collection of pairs $(L,\lambda^{\mathbb{R}})$ such that every $L$ is expressed as $K(J)$ for some $J$, and $\lambda^{\mathbb{R}}$ is a mod $2$ characteristic map over $L$. Then $\mathcal{X}$ is called a wedge-stable set based on $K$ if $(\operatorname{Wed}_{v}(L),\lambda_{1}^{\mathbb{R}}\wedge\lambda_{2}^{\mathbb{R}})\in\mathcal{X}$ whenever both $(L,\lambda_{1}^{\mathbb{R}})$ and $(L,\lambda_{2}^{\mathbb{R}})$ are in $\mathcal{X}$.

We present the concept of toric wedge induction which is a method employed to demonstrate the validity of a given property across $\mathcal{X}$.

The credit of original idea of toric wedge induction should be given to Choi and Park [8]. They used it for showing the projectivity of certain toric manifolds in  [8] or [9]. Later, the authors of this paper used it for classifying toric manifolds satisfying equality within an inequality regarding the number of minimal components in their rational curve space [6].

However, it is sometimes challenging to perform the inductive step. In that situation, we can relax it by strengthening the basis step. In this paper, we introduce a new, easier version of this method that helps with the toric lifting property we want to show.

In order to do it, we briefly review the notions and properties, following [10], where the reader may find a much more details about the relations between characteristic maps over $K(J)$ and puzzles explained below.

The pre-diagram $D^{\prime}(K)$ of $K$ is an edge-colored non-simple graph such that

1. (1)

   the node set of $D^{\prime}(K)$ is the set of D-J classes of the mod $2$ characteristic maps over $K$,

2. (2)

   for a vertex $v$ of $K$ and two mod $2$ characteristic maps $\lambda^{\mathbb{R}}_{1}$ and $\lambda^{\mathbb{R}}_{2}$ over $K$, a pair $(\{\lambda^{\mathbb{R}}_{1},\lambda^{\mathbb{R}}_{2}\},v)$ is a colored edge of $D^{\prime}(K)$ if and only if there is a mod $2$ characteristic map over $\operatorname{Wed}_{v}(K)$ whose two projections onto $K$ are $\lambda^{\mathbb{R}}_{1}$ and $\lambda^{\mathbb{R}}_{2}$.

We denote $G(J)$ the $1$-skeleton of the simple polytope $P(J)\coloneqq\Delta^{j_{1}-1}\times\Delta^{j_{2}-1}\times\dots\times\Delta^{j_{m}-1}$. Each edge $\bm{\epsilon}$ of $G(J)$ is uniquely written as

where $\alpha_{i}$ is a vertex of $\Delta^{j_{i}-1}$, $1\leq i\leq m,i\not=v$, and $\epsilon_{v}$ is an edge of $\Delta^{j_{v}-1}$. Then color $\bm{\epsilon}$ by $v$.

Then, a mod $2$ characteristic map $\lambda^{\mathbb{R}}$ over $K(J)$ can be expressed by an edge-colored graph homomorphism $\phi\colon G(J)\to D^{\prime}(K)$; When $\bm{\alpha}$ is a vertex of $G(J)$, we can write

where $1\leq\alpha_{i}\leq j_{i}$ is a vertex of $\Delta^{j_{i}-1}$ for $1\leq i\leq m$. Observe that

does not contain any minimal non-face of $K(J)$, so it is a face of $K(J)$. Since each vertex $i_{j_{k}}$ ($j_{k}\not=\alpha_{i}$) in $\mathcal{F}(\bm{\alpha})$ is a wedged vertex of $i_{\alpha_{i}}$, $\operatorname{Lk}_{K(J)}(\mathcal{F}(\bm{\alpha}))$ is isomorphic to $K$, and the projection $\operatorname{Proj}_{\mathcal{F}(\bm{\alpha})}(\lambda^{R})$ is a mod $2$ characteristic map over $K$ by the natural bijection between $\{1,2,\ldots,m\}$ and $\{1_{\alpha_{1}},2_{\alpha_{2}},\ldots,m_{\alpha_{m}}\}$. Define $\phi$ by $\phi(\bm{\alpha})=\operatorname{Proj}_{\mathcal{F}(\bm{\alpha})}(\lambda^{\mathbb{R}})$ for each vertex $\bm{\alpha}$ of $G(J)$. Let $\epsilon$ be an edge of $G(J)$. Up to relabeling the vertices of $K$, $\epsilon$ consists of two vertices $\alpha$ and $\alpha^{\prime}\coloneqq\alpha^{\prime}_{1}\times\alpha_{2}\times\dots\times\alpha_{m}$. Then $\phi(\epsilon)=\{\operatorname{Proj}_{\mathcal{F}(\bm{\alpha})}(\lambda^{\mathbb{R}}),\operatorname{Proj}_{\mathcal{F}(\bm{\alpha^{\prime}})}(\lambda^{\mathbb{R}})\}$. By the following proposition, $\overline{\operatorname{Proj}_{\mathcal{F}(\bm{\alpha})}(\lambda^{\mathbb{R}})}$ and $\overline{\operatorname{Proj}_{\mathcal{F}(\bm{\alpha^{\prime}})}(\lambda^{\mathbb{R}})}$ are same except their first rows. Hence $\phi$ is an edge-colored graph homomorphism.

However, not all edge-colored graph homomorphism $\phi$ is obtained from a mod $2$ characteristic map over $K(J)$. If it is, $\phi$ is called a (realizable) puzzle over $K(J)$ (or, over $K$, if there is no confusion), and $\lambda_{\phi}^{\mathbb{R}}$ denotes the corresponding mod $2$ characteristic map over $K(J)$. In particular, a puzzle which does not contain any edge corresponding to a canonical extension is called irreducible.