Buchstaber Invariant of Simple Polytopes

TL;DR

This paper introduces a new combinatorial invariant for simple polytopes derived from toric topology, analyzes its properties, and computes it for polytopes with n+3 facets, including related algebraic invariants.

Contribution

It defines the s-invariant for simple polytopes, explores its properties, and provides explicit calculations for polytopes with n+3 facets, linking combinatorics and topology.

Findings

s(P) is characterized for polytopes with n+3 facets

Derived a formula for h-polynomials of these polytopes

Computed the bigraded cohomology rings of associated moment-angle complexes

Abstract

In this paper we study a new combinatorial invariant of simple polytopes, which comes from toric topology. With each simple n-polytope P with m facets we can associate a moment-angle complex Z_P with a canonical action of the torus T^m. Then s(P) is the maximal dimension of a toric subgroup that acts freely on Z_P. The problem stated by Victor M. Buchstaber is to find a simple combinatorial description of an s-number. We describe the main properties of s(P) and study the properties of simple n-polytopes with n+3 facets. In particular, we find the value of an s-number for such polytopes, a simple formula for their h-polynomials and the bigraded cohomology rings of the corresponding moment-angle complexes

Equations

$$m_1-s(K_1)⩽m_2-s(K_2),$$

$$s(P)⩾m-\gamma +\left[\frac{\gamma }{n+1}\right]⩾\left[\frac{m}{n+1}\right].$$

$$4m⩾\{\begin{array}{ll}7(n+1),& m=0\mathrm{mod}7;\\ 7(n+1)+4,& m=1\mathrm{mod}7;\\ 7(n+1)+8,& m=2\mathrm{mod}7;\\ 7(n+1)+5,& m=3\mathrm{mod}7;\\ 7(n+1)+2,& m=4\mathrm{mod}7;\\ 7(n+1)+6,& m=5\mathrm{mod}7;\\ 7(n+1)+3,& m=6\mathrm{mod}7;\end{array}$$

$$m-s(K)⩽\dim {\omega }_1+\cdots +\dim {\omega }_l=(|{\omega }_1|-1)+\cdots +(|{\omega }_l|-1).$$

$$s(P)⩾\lceil \frac{m-n}{2}\rceil +s({\Delta }_{n-1}^{\gamma -1})$$

$$s(P)⩾\lceil \frac{m-n}{k}\rceil -(k-2)n.$$

$$\begin{gathered} \mathbb{Z}[v_1^1,v_1^2,\dots ,v_1^{k_1},v_2^1,\dots ,v_2^{k_1},\dots ,v_m^1,\dots ,v_m^{k_m}]/J, \\ J=(v_{j_1}^1{v}_{j_1}^2\dots v_{j_1}^{k_{j_1}}{v}_{j_2}^1{v}_{j_2}^2\dots v_{j_2}^{k_{j_2}}\dots v_{j_l}^1{v}_{j_l}^2\dots v_{j_l}^{k_{j_l}}:v_{j_1}\dots v_{j_l}\in I). \end{gathered}$$

$$2k-1=\sum _j{\beta }^{-1,2j}({\mathcal{Z}}_{P_{a_1,\dots ,a_{2k-1}}})=\sum _j{\beta }^{-2,2j}({\mathcal{Z}}_{P_{a_1,\dots ,a_{2k-1}}}).$$

$$s(P)+1⩽s(P\#{\Delta }^n)⩽s(P)+2.$$

$$X_i\cdot X_j=0X_i\cdot Y_j={\delta }_{i+k-1,j}Z{Y}_i\cdot Y_j=0,$$

$$X_i^2=0,X_i{X}_{i+1}=-X_{i+1}{X}_i=Y_i,X_1{X}_2{X}_3=Z.$$

$$\underset{i=1}{\overset{2k-1}{\#}}{S}^{2{\phi }_i-1}\times S^{2{\psi }_{i+k-1}-2},$$

$$\mathtt{k}[P]=\mathtt{k}[v_1,\dots ,v_m]/I,$$

$$I=(\{v_{i_1},\dots ,v_{i_k}:F_{i_1}\cap \cdots \cap F_{i_k}=\varnothing \}).$$

$$H^{\ast }({\mathcal{Z}}_P/H,\mathbb{Q})\cong {\mathrm{Tor}}_{\mathbb{Q}[t_1,\dots ,t_{m-r}]}(\mathbb{Q}[P],\mathbb{Q}),$$

$$\mathbb{Q}[t_1,\dots ,t_{m-r}]\to \mathbb{Q}[v_1,\dots ,v_m],t_i\to t_{i1}{v}_1+\cdots +t_{im}{v}_m.$$

$$H^{\ast }({\mathcal{Z}}_P/H,\mathbb{Q})\cong H[\Lambda [u_1,\dots ,u_{m-r}]\otimes \mathbb{Q}[P],d],$$

$$d{u}_i=t_{i1}{v}_1+\cdots +t_{im}{v}_m,d{v}_j=0.$$

$$m-s({\Delta }_{n-1}^{m-1})⩾\lceil {\log }_2(m-n+3)\rceil +n-2;$$

$$m-s({\Delta }_2^{m-1})=\lceil {\log }_{2}m\rceil +1;$$

$$s({\Delta }_{m-3}^{m-1})=1.$$

$${\mathrm{s}}_{\mathbb{R}}(m,p)=k\text{mbox}ifandonlyif{m}_{k+1}(p-1)<m⩽m_k(p-1),$$

$$\sum _{(u,v)=0}{a}_v⩽b\text{mbox}foreachu\in ({\mathbb{Z}}_2{)}^k\setminus \{0\}$$

$$(2^k-1)Q+R+2^{k-1}-2^{k-1-l}⩽m_k(b)⩽(2^k-1)Q+2R,$$

$$P=\{\text{x}\in {\mathbb{R}}^n,A_p\text{x}+{\text{b}}_p⩾0,A_p\in {\mathbb{R}}^{m\times n},{\text{b}}_p\in {\mathbb{R}}^m\}.$$

$$\text{x}\to A_p\text{x}+{\text{b}}_p$$

$$P=\{\text{y}\in {\mathbb{R}}_{+}^m:C_p\text{y}=C_p{\text{b}}_p\}.$$

$$\begin{gathered} P_{k_1,\dots ,k_m}=\{\text{y}=(y_1^1,\dots ,y_1^{k_1},\dots ,y_m^1,\dots ,y_m^{k_m})\in {\mathbb{R}}_{+}^{k_1+\cdots +k_m}: \\ \sum _{j=1}^m{C}_{ij}\left(\sum _{l=1}^{k_j}{y}_j^l\right)=\sum _{j=1}^m{C}_{ij}{b}_j,i=1,\dots ,m-n.\} \end{gathered}$$

$$P_{k_1,\dots ,k_m}=\{\text{y}\in {\mathbb{R}}_{+}^{k_1+\cdots +k_m}:y_j^1+\cdots +y_j^{k_j}={\text{a}}_j\text{x}+b_j,j=1,\dots ,m\text{mbox}forsome\text{x}\in {\mathbb{R}}^n\},$$

$$P_{k_1,\dots ,k_m}:=\{(\text{x},Y)\in {\mathbb{R}}^{n+(k_1-1)+\cdots +(k_m-1)}:{\text{a}}_j\text{x}+b_j⩾\sum _{l=1}^{k_j-1}{y}_j^l,y_j^l⩾0\}.$$

$$\mathbb{Z}[P_{k_1,\dots ,k_m}]=\mathbb{Z}[v_1^1,\dots ,v_1^{k_1},\dots ,v_m^1,\dots ,v_m^{k_m}]/J,J=(v_{i_1}^1\dots v_{i_1}^{k_{i_1}}\dots v_{i_l}^1\dots v_{i_l}^{k_{i_l}}:v_{i_1}\dots v_{i_l}\in I).$$

Full text

1 Introduction.

Let $P^{n}=\{\textit{{x}}\in\mathbb{R}^{n}\colon A_{p}\textit{{x}}+\textit{{b}}_{p}\geqslant 0\}$ be a simple polytope and $\mathfrak{F}=\{F_{1},\dots,F_{m}\}$ the set of facets of $P^{n}$. For each facet $F_{i}\in\mathfrak{F}$ denote by $T^{F_{i}}$ the one-dimensional coordinate subgroup of $T^{\mathfrak{F}}=T^{m}$. Then assign to every face $G$ the coordinate subtorus $T^{G}=\prod_{F_{i}\supset G}T^{F_{i}}\subset T^{\mathfrak{F}}$. For each point $\textit{{q}}\in P$ we denote by $G(\textit{{q}})$ a unique face containing q in its relative interior.

For any combinatorial simple polytope $P^{n}$ we introduce the identification space: $\mathcal{Z}_{P}=(T^{\mathfrak{F}}\times\nobreak P^{n})/{\sim}$ , where $(t_{1},\textit{{p}})\sim(t_{2},\textit{{q}})$ if and only if $\textit{{p}}=\textit{{q}}$ and $t_{1}t_{2}^{-1}\in T^{G(\textit{{p}})}$.

It turns out that $\mathcal{Z}_{P}$ is a smooth manifold of dimension $m+n$ with a smooth action of $T^{m}$ induced by the standard action of the torus on the first factor. Then $\mathcal{Z}_{P}/T^{m}=P$, and the stabilizer of a point $[(t,\textit{{q}})]$ is $T^{G(\textit{{q}})}$.

Definition 1** (See [BP]).**

The Buchstaber number $s(P)$ is the maximal dimension of a torus subgroup $H\cong T^{s}$ that acts freely.

The problem stated by Victor M. Buchstaber in 2002 is to find a simple combinatorial description of $s(P)$.

It is possible to define $s(K)$ for any simplicial complex in such a way that $s(P)=s(\partial P^{*})$, where $\partial P^{*}$ is a boundary complex of a dual polytope.

In this article we establish some main properties of the $s$-number, which we summarize in the following theorem (see [E]). Some definitions are given below.

Theorem**.**

The $s$-number satisfies the following properties:

$s(P)\geqslant s(Q)$* if $Q$ is obtained from $P$ by forgetting one of the inequalities $\textit{{a}}_{i}\textit{{x}}+b_{i}\geqslant 0$.* 2. 2.

$s(P)=1$* if and only if $P=\Delta^{n}$.* 3. 3.

If $n+2\leqslant m\leqslant\frac{49}{48}n+\frac{83}{48}$, then $s(C^{n}(m)^{*})=2$. In particular, for each $k\geqslant 2$ there exists a polytope with $m-n=k$ and $s(P)=2$. 4. 4.

$s(P)+s(Q)\leqslant s(P\times Q)\leqslant s(P)+s(Q)+\min\{m_{1}-n_{1}-s(P),m_{2}-n_{2}-s(Q)\}$. 5. 5.

$s(P)+s(Q)\leqslant s(P\mathop{\rm\#}\limits Q)$, where $P\mathop{\rm\#}\limits Q$ is a connected sum of polytopes along vertices. 6. 6.

If $f:K_{1}^{n-1}\to K_{2}^{n-1}$ is a non-degenerate map of simplicial complexes, then

[TABLE]

where $m_{1}$ and $m_{2}$ are the numbers of vertices of $K_{1}$ and $K_{2}$. 7. 7.

$m_{F}-s(F)\leqslant m-s(P)$, where $F$ is a facet of $P$. 8. 8.

$s(P)\geqslant m-\gamma(P)+s(\Delta^{\gamma-1}_{n-1})$, where $\gamma(P)$ is a chromatic number of $P$ and $\Delta^{\gamma-1}_{n-1}$ is a $(n-1)$-dimensional skeleton of the simplex $\Delta^{\gamma-1}$. In particular,

[TABLE] 9. 9.

• a)

$s(\Delta^{m-1}_{n-1})\geqslant 2$* if and only if $\frac{m}{n+1}\geqslant\frac{3}{2}$;*

• b)

$s(\Delta^{m-1}_{n-1})\geqslant 3$* if and only if*

[TABLE] 10. 10.

For a simplicial complex $K$ on the set of vertices $[m]=\{1,\dots,m\}$ let $\omega$ be a minimal non-simplex, if $\omega\notin K$, but any proper face $\sigma\subset\omega$ belongs to $K$. Then if $\omega_{1},\dots,\omega_{l}$ is a collection of minimal non-simplices such that $\omega_{1}\cup\dots\cup\omega_{l}=[m]$, then

[TABLE] 11. 11.

A simple polytope is called flag if any collection of facets $F_{i_{1}},\dots,F_{i_{l}}$ such that any two $F_{i_{s}},F_{i_{t}}$ of them intersect: $F_{i_{s}}\cap F_{i_{t}}\neq\varnothing$ has nonempty intersection $F_{i_{1}}\cap\dots\cap F_{i_{l}}\neq\varnothing$. Then for a flag polytope $P^{n}$ with a chromatic number $\gamma$ we have: $\gamma\leqslant[\frac{m-n}{2}]+n$, thus

[TABLE] 12. 12.

A simple polytope is called $k$-flag, if any collection of facets $F_{i_{1}},\dots,F_{i_{l}}$ such that any $k$ of them have nonempty intersection has nonempty intersection: $F_{i_{1}}\cap\dots\cap F_{i_{l}}\neq\varnothing$. Then for a $k$-flag polytope $P^{n}$:

[TABLE] 13. 13.

Let $\mathbb{Z}[P]=\mathbb{Z}[v_{1},\dots,v_{m}]/I$, $I=(v_{j_{1}}v_{j_{2}}\dots v_{j_{l}}:F_{j_{1}}\cap\dots\cap F_{j_{l}}=\varnothing)$ be a face ring of a polytope $P$. $P_{k_{1},\dots,\,k_{m}}$ is defined as a (combinatorial) simple polytope with the face ring

[TABLE]

Then $s(P_{k_{1},\dots,\,k_{m}})=s(P)$. 14. 14.

It is known (see [Gb]) that each simple polytope $P^{n}$ with $m=n+3$ facets can be represented in terms of a regular $(2k-1)$-gon $M_{2k-1}$ and a surjective map from $\mathfrak{F}=\{F_{1},\dots,F_{n+3}\}$ to the set of vertices of $M_{2k-1}$. The facets $F_{i_{1}},\dots,F_{i_{n}}$ intersect in a vertex if and only if the triangle formed by the vertices corresponding to the rest three facets contains the center of $M_{2k-1}$.

Then for such a polytope $P_{a_{1},\dots,\,a_{2k-1}}$ we have: $s(P)=3$ if and only if $k\leqslant 4$.

Here $k$ can be expressed in terms of bigraded Betti numbers

[TABLE] 15. 15.

There are two polytopes $P$ and $Q$ with the equal $f$-vectors and chromatic numbers, but the different $s$-numbers. Nevertheless our $P$ and $Q$ have different bigraded Betti numbers. 16. 16.

If $P$ is obtained from $Q$ by one $i$-flip, $2\leqslant i\leqslant n-1$, then $|s(P)-s(Q)|\leqslant 1$.

[TABLE]

We also investigate the properties of simple polytopes with $n+3$ facets. In particular, we will prove the following fact:

Theorem**.**

For the polytope $P=P_{a_{1},\dots,\,a_{2k-1}}$ the bigraded cohomology ring $\operatorname{H}^{*,\,*}(\mathcal{Z}_{P})$ is isomorphic to the free abelian group $\mathbb{Z}\oplus\mathbb{Z}^{2k-1}\oplus\mathbb{Z}^{2k-1}\oplus\mathbb{Z}$ with the generators

$1$, $\operatorname{bideg}1=(0,0)$;

$X_{i}$, $\operatorname{bideg}X_{i}=(-1,2(a_{i}+\dots+a_{i+k-2})),i=1,\dots,2k-1$;

$Y_{j}$, $\operatorname{bideg}Y_{j}=(-2,2(a_{j}+\dots+a_{j+k-1})),j=1,\dots,2k-1$;

$Z$, $\operatorname{bideg}Z=(-3,2(n+3))$.

For $k\geqslant 3$

[TABLE]

and for $k=2$

[TABLE]

This in not surprising, since $\mathcal{Z}_{P_{a_{1},\,a_{2},\,a_{3}}}=S^{2a_{1}-1}\times S^{2a_{2}-1}\times S^{2a_{3}-1}$, and according to the results by Lopez de Medrano [LM] for $k\geqslant 3$ the manifold $\mathcal{Z}_{P_{a_{1},\dots,\,a_{2k-1}}}$ is homeomorphic to

[TABLE]

where $\varphi_{i}=a_{i}+\dots+a_{i+k-2}$, $\psi_{j}=a_{j}+\dots+a_{j+k-1}$, and indices are taken modulo $2k-1$. See also [BM]. Our result describes additionally a bigraded structure in the cohomology ring of the moment-angle manifold $\mathcal{Z}_{P_{a_{1},\dots,\,a_{2k-1}}}$.

We also describe some properties of the construction $P\to P_{k_{1},\dots,\,k_{m}}$ (see [BBCG, BBCG2, GLM]).

The author is grateful to Victor M. Buchstaber for the formulation of the problem and for his permanent attention to this work, to Taras Panov for useful advises and comments, and to Yukiko Fukukawa and Mikiya Masuda for sending their paper “Buchstaber Invariants of Skeletons of a Simplex”.

2 Some facts.

It is known that

$1\leqslant s(P)\leqslant m-n$. $S(P)=m-n$ if and only if there exists a characteristic map from the set of facets $\mathfrak{F}$ to $\mathbb{Z}^{n}$ such that for every vertex $v=F_{i_{1}}\cap F_{i_{2}}\dots\cap F_{i_{n}}$ the vectors corresponding to the facets $F_{i_{1}},\dots,F_{i_{n}}$ form a basis of $\mathbb{Z}^{n}$. In this case $\mathcal{Z}_{P}/T^{m-n}=M^{2n}$ is a quasitoric manifold. 2. 2.

Any polygon has a characteristic map, since we can assign to it’s edges the vectors $\begin{pmatrix}0\\ 1\end{pmatrix}$, $\begin{pmatrix}1\\ 0\end{pmatrix}$, $\begin{pmatrix}1\\ 1\end{pmatrix}$ in such a way that any two consequent edges have different vectors. 3. 3.

Any $3$-dimensional simple polytope allows a characteristic map, since there is a right colouring in $4$ colours, according to the four colours theorem. We can assign the vectors $\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}$, $\begin{pmatrix}0\\ 1\\ 0\end{pmatrix}$, $\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}$, $\begin{pmatrix}1\\ 1\\ 1\end{pmatrix}$ to the facets coloured in the colours $1$, $2$, $3$, and $4$ respectively. 4. 4.

$s(P)\geqslant m-\gamma(P)$, where $\gamma(P)$ is a chromatic number of $P$ (I.Izmestiev, 2001, [Iz]). 5. 5.

$s(P)\leqslant m-\lceil\log_{2}(\gamma(P)+1)\rceil$ (A.Aizenberg, 2009, [A]). 6. 6.

In fact, there are two dual combinatorial interpretations of the $s$-number.

Proposition** (See [BP]).**

1. (a)

$s(P)$* is the maximal positive integer such that there exists the matrix $M$ of size $m\times s$ with integer entries satisfying the property: each row $\textit{{m}}_{i}$ of $M$ corresponds to the facet $F_{i}$ and if $F_{i_{1}},\dots,F_{i_{n}}$ intersect in a vertex, then the columns of the matrix $M\setminus\{\textit{{m}}_{i_{1}},\dots,\textit{{m}}_{i_{n}}\}$ form a part of some basis of $\mathbb{Z}^{m-n}$.* 2. (b)

$s(P)$* is the maximal positive integer such that there exists a mapping $\mathfrak{F}\to\mathbb{Z}^{m-s}$ satisfying the property: if $F_{i_{1}},\dots,F_{i_{n}}$ intersect in a vertex, then the corresponding vectors form a part of some basis of $\mathbb{Z}^{m-s}$.*

These two combinatorial interpretations allow us to define $s(K)$ for any simplicial complex $K$ – we should substitute vertices of $K$ for facets of $P$ and simplices of maximal dimension for vertices of $P$. Then $s(P)=s(\partial P^{*})$. 7. 7.

If $s(P)=m-n$, then it is known that $b_{2i}=\operatorname{rk}H_{2i}(M^{2n})=h_{i}(P)$ for any quasitoric manifold $M^{2n}$.

Similarly, we can consider the space $\mathcal{Z}_{P}/H$ for any toric subgroup $H\cong T^{r}$ that acts freely on $\mathcal{Z}_{P}$. If $T$ is the $(m-r)\times m$-matrix with integral entries that arises in the second combinatorial interpretation of $s(P)$, then the cohomology ring $\operatorname{H}^{*}(\mathcal{Z}_{P}/H)$ can be described in the following way.

Definition 2**.**

The Stanley-Reisner ring (or the face ring) of a simple polytope $P$ with $m$ facets $\{F_{1},\dots,F_{m}\}$ is defined as

[TABLE]

where $\mathtt{k}$ is a ring, each variable $v_{i}$ corresponds to the facet $F_{i}$, and

[TABLE]

Then we have:

Proposition** (See [BP]).**

1. (a)

There is an isomorphism of the algebras

[TABLE]

where a structure of $\mathbb{Q}[t_{1},\dots,t_{m-r}]$-module in the Stanley-Reisner ring $\mathbb{Q}[P]$ is given by the mapping

[TABLE] 2. (b)

There is an isomorphism of the algebras:

[TABLE]

[TABLE]

In some sense we can consider dimensions of the cohomology groups $\operatorname{H}^{i}(\mathcal{Z}_{P})$ as analogues of $h$-numbers depending on the subgroup $H$.

Let us continue the list of the properties of $P$: 8. 8.

Let us consider a simplicial complex $U_{k}$ (See [DJ]). Vertices of $U_{k}$ correspond to vectors in $\mathbb{Z}^{k}$. The set of vertices $v_{1},\dots,v_{l}$