The syzygy order of big polygon spaces

TL;DR

This paper determines the precise syzygy order of the equivariant cohomology of big polygon spaces, revealing new algebraic properties related to the length vector defining these manifolds.

Contribution

It provides an exact calculation of the syzygy order for big polygon spaces' equivariant cohomology based on their length vectors, using a novel characterization of syzygies.

Findings

Exact syzygy order expressed in terms of length vector

Refined characterization of syzygies in terms of linearly independent elements in H^2(BT)

Application to understanding torsion-free and reflexive properties

Abstract

Big polygon spaces are compact orientable manifolds with a torus action whose equivariant cohomology can be torsion-free or reflexive without being free as a module over $H^*(BT)$. We determine the exact syzygy order of the equivariant cohomology of a big polygon space in terms of the length vector defining it. The proof uses a refined characterization of syzygies in terms of certain linearly independent elements in $H^2(BT)$ adapted to the isotropy groups occurring in a given $T$-space.

Equations

$$0⟶H_T^{\ast }(X)⟶H_T^{\ast }(X^T)⟶H_T^{\ast +1}(X_1,X^T)$$

$$\begin{gathered} 0⟶H_T^{\ast }(X)⟶H_T^{\ast }(X_0)⟶H_T^{\ast +1}(X_1,X_0) \\ ⟶H_T^{\ast +2}(X_2,X_1)⟶\cdots ⟶H_T^{\ast +k-1}(X_{k-1},X_{k-2}) \end{gathered}$$

$$\begin{gathered} 0⟶H_T^{\ast }(X)⟶H_T^{\ast }(X_0)⟶H_T^{\ast +1}(X_1,X_0) \\ ⟶H_T^{\ast +2}(X_2,X_1)⟶\cdots ⟶H_T^{\ast +r}(X_r,X_{r-1})⟶0 \end{gathered}$$

$$\sum _{j\in J}{\ell }_j\ne \sum _{j\notin J}{\ell }_j$$

$$\{\begin{array}{rlrl}\Vert u_j{\Vert }^2+\Vert z_j{\Vert }^2& =1& & \text{for any 1≤j≤r,}\\ {\ell }_1{u}_1+\cdots +{\ell }_r{u}_r& =0,& & \end{array}$$

$$g\cdot (u,z)=(u,g_1{z}_1,\dots ,g_r{z}_r),$$

$$\mu (\ell )=\min \{{\sigma }_{\ell }(J)\mid \text{J⊂{1,…,r} is ℓ-long and σℓ(J)>0}\}$$

$${\sigma }_{\ell }(J)=\#\{j\in J\mid \text{J∖j is ℓ-short}\},$$

$$H_{T(\text{mathbfit}a)}^{\ast }(X)\cong H_T^{\ast }(X)/\text{mathbfit}a.$$

$$0⟶C_T^{\ast }(X)\stackrel{\cdot a}{⟶}{C}_T^{\ast }(X)⟶C_{T^{\prime \prime }}^{\ast }(X)⟶0.$$

$$0⟶H_T^{\ast }(X)\stackrel{\cdot a}{⟶}{H}_T^{\ast }(X)⟶H_{T^{\prime \prime }}^{\ast }(X)⟶0,$$

$$H^2(BT)$$

$$H_T^{\ast }(X)$$

$$M^{\prime }:=H_{T^{\prime }}^{\ast }(X)\cong M/\text{mathbfit}{a}^{\prime },$$

$$M^{\prime \prime }\cong H_{T^{\prime \prime }}(X).$$

$$\ell (\sigma )=\sum _{j\in \sigma }{\ell }_j.$$

$$d\gamma =\sum _{j\in \gamma }\pm t_j^q(\gamma \setminus j)$$

$$\mathcal{C}(\Delta ;\bar{R})=\mathcal{C}(S;\bar{R})\oplus \mathcal{C}(\Delta \setminus S;\bar{R})$$

$$\mathcal{D}(S;\bar{R})=\mathcal{C}(S;\bar{R})+d\mathcal{C}(S;\bar{R})\subset \mathcal{C}(\Delta ;\bar{R}).$$

$$\mathrm{supp}c=\{\sigma \in \Delta |c_{\sigma }\ne 0\}$$

$$\ell (c)=\min \{\ell (\sigma )|\text{σ∈suppc short}\}.$$

$$f_{\ell}\colon\mathcal{C}(\Delta_{+};R)\to\mathcal{C}(\Delta;R)\!\bigm{/}\!\mathcal{C}(\Delta_{+};R)\cong\mathcal{C}(\Delta_{-};R),\qquad\gamma\mapsto d\gamma.$$

$$0\to \mathrm{coker}{f}_{\ell }\to H_T^{\ast }(X(\ell ))\to (\ker f_{\ell })[-2p]\to 0.$$

$${\sigma }_{\ell }(\gamma )=\#\{j\in \gamma \mid \text{γ∖j is ℓ-short}\}$$

$$\mu(\ell)=\min\bigl{\{}\,\sigma_{\ell}(\gamma)\mid\text{$\gamma\in\Delta_{+}$ and $\sigma_{\ell}(\gamma)>0$}\,\bigr{\}}\geq 1$$

$$M(\ell)=\mathcal{C}(\Delta;R)\!\bigm{/}\!\mathcal{D}(\Delta_{+};R).$$

$$\begin{split}N&=\mathcal{C}(\Delta;R)\!\bigm{/}\!\bigl{(}\,\mathcal{D}(\Delta_{+};R)+(t_{i_{1}},\dots,t_{i_{k-1}})\,\mathcal{C}(\Delta;R)\,\bigr{)}\\ &=\mathcal{C}(\Delta;\bar{R})\,/\,\mathcal{D}(\Delta_{+};\bar{R})\end{split}$$

$$\bar{R}=R/(t_{i_{1}},\dots,t_{i_{k-1}})=\Bbbk\bigl{[}\,t_{j}\mid j\notin I\,\bigr{]}.$$

$$d\gamma =\sum _{j\in \gamma \setminus I}\pm t_j^q(\gamma \setminus j).$$

$$(a+t_i^q{a}^{\prime })+d(b+t_i^q{b}^{\prime })=t_i^q(c+a^{\prime }+d{b}^{\prime }),$$

Full text

M. F. was supported by an NSERC Discovery Grant.

1. Introduction

Let $T\cong(S^{1})^{r}$ be a torus, and let $X$ be a $T$-manifold whose cohomology $H^{*}(X)$ (with real coefficients) is finite-dimensional. A powerful tool to compute the equivariant cohomology $H_{T}^{*}(X)$ is the Chang–Skjelbred sequence

[TABLE]

where $X^{T}\subset X$ is the fixed point set and the equivariant $1$-skeleton $X_{1}$ the union of orbits of dimension at most $1$. The first map is induced by the inclusion $X^{T}\hookrightarrow X$ and the second one is the connecting homomorphism in the long exact sequence for the pair $(X_{1},X^{T})$.

If $H_{T}^{*}(X)$ is free as a module over the polynomial ring $R=H^{*}(BT)$, then the Chang–Skjelbred sequence is exact [6, Prop. 2.4], which implies that $H_{T}^{*}(X)$ can be calculated out of the equivariant $1$-skeleton. In many cases of interest, $X^{T}$ is finite and $X_{1}$ a union of $2$-spheres glued together at their poles. In such a setting, this approach is called the GKM method after work of Goresky–Kottwitz–MacPherson [12, Thm. 7.2].

It is not hard to find examples of $T$-manifolds such that (1.1) is exact without $H_{T}^{*}(X)$ being free over $R$, see below. This phenomenon was studied in detail by Allday–Franz–Puppe [1], [2], who characterized those $T$-manifolds for which the Chang–Skjelbred sequence is exact; in [9] this is generalized to non-abelian Lie groups. Allday–Franz–Puppe actually proved a more general theorem that involves higher equivariant skeletons. For $-1\leq i\leq r$, we write $X_{i}\subset X$ for the union of orbits of dimension at most $i$, so that $X_{0}=X^{T}$ and $X_{r}=X$.

Theorem 1.1** (Allday–Franz–Puppe).**

*The Chang–Skjelbred sequence (1.1) is exact if and only if $H_{T}^{*}(X)$ is a reflexive $R$-module. More generally, for any $1\leq k\leq r$, the sequence *

[TABLE]

is exact if and only if $H_{T}^{*}(X)$ is a $k$-th syzygy over $R$.

See [1, Thm. 1.1]. The additional maps in the sequence above are the connecting homomorphisms for the triples $(X_{i+1},X_{i},X_{i-1})$.

Recall that an $R$-module is reflexive if the canonical map to its double-dual is an isomorphism. Syzygies are a notion from commutative algebra that interpolates between torsion-free and free modules, see Section 2 for the precise definition. The first syzygies over $R$ are exactly the torsion-free modules, the second syzygies the reflexive ones and the $r$-th syzygies the free ones.

As a corollary (cf. the comment following [1, Thm. 5.7]) we get the result of Atiyah and Bredon [5, Thm.] that the sequence

[TABLE]

is exact if and only if $H_{T}^{*}(X)$ is free over $R$, which strengthens the Chang–Skjelbred theorem.

It is not difficult to construct $T$-manifolds such that $H_{T}^{*}(X)$ is a $k$-th syzygy for $k<r$. For example, the usual rotation action of $S^{1}$ on $S^{2}$ gives an action of $T$ on $(S^{2})^{r}$ such that $H_{T}^{*}(X)$ is free over $R$. By suitably removing two fixed points, any syzygy order less than $r$ can be realized [1, Sec. 6.1].

The situation becomes much more intriguing if one looks at compact orientable $T$-manifolds. For such an $X$, another result of Allday–Franz–Puppe says that if $H_{T}^{*}(X)$ is a syzygy of order $\geq r/2$, then it is actually free over $R$ [1, Cor. 1.4].

It already appears very difficult to construct compact orientable $T$-manifolds such that $H_{T}^{*}(X)$ is torsion-free, but not free over $R$. The first such examples were the ‘mutants of compactified representations’ given by Franz–Puppe in 2008 [11, Sec. 4]. Recently, the first author found a family of $T$-manifolds, the so-called big polygon spaces, that generalize one of the mutants to arbitrary syzygies [8]. We recall the definition.

Let $\ell\in\mathbb{R}^{r}$, called a length vector in this context. We assume that $\ell$ is generic, meaning that

[TABLE]

for any subset $J\subset\{1,\dots,r\}$. Depending on which side dominates, $J$ is called *$\ell$-long *or $\ell$-short.

Let $p$, $q\geq 1$. The big polygon space $X(\ell)=X_{p,q}(\ell)$ is the real algebraic subvariety of $\mathbb{C}^{(p+q)r}$ defined by the equations

[TABLE]

where $u_{1}$, …, $u_{r}\in\mathbb{C}^{p}$ and $z_{1}$, …, $z_{r}\in\mathbb{C}^{q}$. Since $\ell$ is generic, $X(\ell)$ is a compact orientable manifold with a smooth action of $T=(S^{1})^{r}$ given by scalar multiplication of the $z$-variables,

[TABLE]

see [8, Lemma 2.1]. The fixed point set $X(\ell)^{T}$ is the ‘space of polygons’ $E_{2p}(\ell)$ studied by Farber–Fromm [7].

It turns out that $H_{T}^{*}(X(\ell))$ is never free over $R$. In fact, $H_{T}^{*}(X(\ell))$ is not a syzygy of order

[TABLE]

where

[TABLE]

see [8, Prop. 6.3]. Our main result confirms the conjecture made in [8, Conj. 6.6].

Theorem 1.2**.**

The syzygy order of $H_{T}^{*}(X(\ell))$ over $R$ equals $\mu(\ell)-1$.

Our proof of Theorem 1.2 is purely algebraic and uses the description of $H_{T}^{*}(X(\ell))$ given in [8, Lemma 4.4]. It is inspired by the proof appearing in the second author’s Ph. D. thesis [15], which in turn is based on the quotient criterion for syzygies developed in [10] and on Morse–Bott theory for manifolds with corners.

The largest possible syzygy order for $r=2m+1$ and $r=2m+2$ is $m$. It is known that this syzygy order is realized by an essentially unique length vector which for odd $r$ corresponds to the equilateral case $\ell=(1,\dots,1)$, see [8, Cor. 6.4] and also Proposition 4.2. From Theorem 1.2 we deduce that syzygies of the next smaller order are also unique or at least almost unique.

Corollary 1.3**.**

Let $r\geq 3$, and let $\ell\in\mathbb{R}^{r}$ be a generic length vector with weakly increasing non-negative components.

1. (1)

Assume that $r=2m+1$ is odd. Then $H_{T}^{*}(X(\ell))$ is a syzygy of order $m-1$ if and only if $X(\ell)$ is equivariantly diffeomorphic to $X(0,0,1,\dots,1)$. 2. (2)

Assume that $r=2m+2$ is even. Then $H_{T}^{*}(X(\ell))$ is a syzygy of order $m-1$ if and only if $X(\ell)$ is equivariantly diffeomorphic to $X(0,0,0,1,\dots,1)$ or to $X(1,1,1,2,\dots,2)$.

To relate our algebraic reasoning with equivariant cohomology, we develop a refined criterion for syzygies in equivariant cohomology which is of independent interest. It involves the notion of a $k$-localizing subset $S\subset H^{2}(BT)$ for a given ‘nice’ $T$-space $X$, see again Section 2 for the definitions. For a big polygon space $X(\ell)$, the set $\{t_{1},\dots,t_{r}\}$ of indeterminates of $R$ is $k$-localizing for any $k$.

Theorem 1.4**.**

Let $S\subset H^{2}(BT;\mathbb{Z})$ be $k$-localizing for $X$ for some $k\geq 1$. Then $H_{T}^{*}(X)$ is a $k$-th syzygy over $R$ if and only if any linearly independent sequence in $S$ of length at most $k$ is $H_{T}^{*}(X)$-regular.

The proof of Theorem 1.4 appears in Section 2. Theorem 1.2 is proven in Section 3 and Corollary 1.3 in Section 4. In Section 5 we state versions of our results for actions of $2$-tori $(\mathbb{Z}_{2})^{r}$ and certain ‘real’ analogues of big polygon spaces which have recently been studied by Puppe [16].

Acknowledgements**.**

We thank Dirk Schütz for sharing his Java code to enumerate all chambers of generic length vectors in a given dimension. It enabled us to verify the syzygy order conjecture in dimension $10$.

2. A refined characterization of syzygies

From now on, all cohomology is taken with coefficients in a field $\Bbbk$ of characteristic [math] unless stated otherwise. Let $T\cong(S^{1})^{r}$ be a torus; we write $R=H^{*}(BT)$.

Recall that any element $a\in H^{2}(BT;\mathbb{Z})\cong H^{1}(T;\mathbb{Z})$ can be interpreted as a character $\chi_{a}\colon T\to S^{1}$. We write $t_{i}\in H^{2}(BT;\mathbb{Z})$ for the element corresponding to the $i$-th coordinate $T\to S^{1}$, so that $R=\Bbbk[t_{1},\dots,t_{r}]$. For any linearly independent sequence $\mathbfit a=(a_{1},\dots,a_{m})$ in $H^{2}(BT;\mathbb{Z})$ we write $T(\mathbfit a)\subset T$ for the identity component of the intersection of $\ker\chi_{a_{1}}$, …, $\ker\chi_{a_{m}}$, which is of codimension $m$. Given an $R$-module $M$, we also write $M/\mathbfit a=M/(a_{1},\dots,a_{m})M$.

Let $X$ be a $T$-space. We say that $X$ is nice if it is Hausdorff, second-countable, locally compact and locally contractible, see [1, Sec. 3.1]. For instance, $X$ can be a $T$-manifold or $T$-orbifold or a complex algebraic variety with an algebraic action of $(\mathbb{C}^{\times})^{r}$. We additionally assume that $H^{*}(X)$ is finite-dimensional and that only finitely many subtori of $T$ occur as identity components of isotropy groups in $X$. In the examples just mentioned, this last condition is redundant, see [9, Thm. 7.7].111The algebraic case reduces to the manifold case: Algebraic varieties have finite Betti sum and can be decomposed into finitely many smooth varieties, stable with respect to an algebraic action. (We have implicitly used this in the introduction already.)

Let $k\geq 0$. A finitely generated $R$-module $M$ is called a $k$-th syzygy if any regular sequence in $R$ of length at most $k$ is also $M$-regular. (See [1, Sec. 2.3] for equivalent definitions of syzygies.) If $M$ is a syzygy of order $k$, but not of order $k+1$, then we say that the syzygy order of $M$ equals $k$.

In our topological context, it is enough to consider sequences of linear elements.

Lemma 2.1**.**

Let $X$ be a nice $T$-space and let $k\geq 0$. Then $H_{T}^{*}(X)$ is a $k$-th syzygy if and only if every linear independent sequence in $H^{2}(BT;\mathbb{Z})$ of length at most $k$ is $H_{T}^{*}(X)$-regular.

Proof.

This is implicit in [1, Thm. 5.7]. There it is shown that $H_{T}^{*}(X)$ is a $k$-th syzygy if and only if it is free over all subrings $H^{*}(BT^{\prime\prime})\subset R$ where $T^{\prime\prime}$ is a quotient of $T$ of rank $\leq k$ (equivalently, equal to $k$ if $k\leq r$). These are exactly the subrings of $R$ that are generated by linearly independent sequences in $H^{2}(BT;\mathbb{Z})$ of length $\leq k$.

Let $\mathbfit a$ be such a sequence and let $T^{\prime\prime}$ be the corresponding quotient of $T$. Because the graded module $M=H_{T}^{*}(X)$ is bounded below, it is free over $R^{\prime\prime}=H^{*}(BT^{\prime\prime})$ if and only if $\mathbfit a$ is $M$-regular, see [14, Lemma p. 5]. (The argument given there remains valid for non-finitely generated modules.) ∎

Lemma 2.2**.**

Let $X$ be a $T$-space. If a sequence $\mathbfit a$ in $H^{2}(BT;\mathbb{Z})$ is $H_{T}^{*}(X)$-regular, then the restriction map $H_{T}^{*}(X)\to H^{*}_{T(\mathbfit a)}(X)$ induces an isomorphism

[TABLE]

Proof.

This is again contained in the proof of [1, Thm. 5.7]. By induction, we may assume that $\mathbfit a$ consists of a single element $0\neq a\in H^{2}(BT;\mathbb{Z})$. We may also assume that it is not divisible by any integer ${}>1$.

Let $C_{T}^{*}(X)$ be the singular Cartan model for $X$, cf. [1, Sec. 3.2]. We may assume that $a$ is contained in the basis of $H^{2}(BT;\mathbb{Z})$ chosen in the definition of the singular Cartan model. We then have a short exact sequence

[TABLE]

Because $a$ is $H_{T}^{*}(X)$-regular, this induces the exact sequence

[TABLE]

proving the claim. ∎

For any $x\in X$, the kernel of the restriction map $p_{x}\colon H^{2}(BT)\to H^{2}(BT_{x})$ has dimension equal to the codimension of $T_{x}$ in $T$. We say that a subset $S\subset H^{2}(BT)$ is $k$-localizing for $X$ if for any $x\in X$ at least $\min(k,\operatorname{codim}T_{x})$ linearly independent elements from $S$ lie in $\ker p_{x}$. This notion behaves well with respect to subtori:

Lemma 2.3**.**

Let $X$ be a $T$-space, and let $S\subset H^{2}(BT)$ be $k$-localizing for $X$ for some $k\geq 0$. For any subtorus $T^{\prime}\subset T$ of codimension $l\leq k$, the image of $S$ in $H^{2}(BT^{\prime})$ is $(k-l)$-localizing for $X$, considered as a $T^{\prime}$-space.

Proof.

Let $x\in X$ and consider the commutative diagram of surjections

[TABLE]

If $S$ contains a basis for $\ker p_{x}$, then $\pi(S)$ contains one for $\ker p^{\prime}_{x}$. If $S$ contains $k$ linearly independent elements from $\ker p_{x}$, then $\pi(S)$ contains $k-l$ linearly independent elements from $\ker p^{\prime}_{x}$. In either case, we have found enough linearly independent elements in $\pi(S)\cap\ker p^{\prime}_{x}$, which proves the claim. ∎

Theorem 2.4**.**

Let $X$ be a nice $T$-space, and let $S\subset H^{2}(BT;\mathbb{Z})$ be $k$-localizing for $X$ for some $k\geq 0$. Then $H_{T}^{*}(X)$ is a $k$-th syzygy over $R$ if and only if any linearly independent sequence in $S$ of length at most $k$ is $H_{T}^{*}(X)$-regular.

Proof.

The ‘only if’ direction follows from the definition of syzygies given above. We prove the converse by induction on $k$. Note that we may assume $0\notin S$.

We consider first the case $k=1$. Because $S$ is $1$-localizing for $X$, we can, for any $x\notin X^{T}$, find an element in $S$ lying in the kernel of the restriction map $H^{*}(BT)\to H^{*}(BT_{x})$. By the localization theorem in equivariant cohomology [4, Thm. 3.2.6], this implies that the bottom arrow in the commutative diagram

[TABLE]

is an isomorphism, where $\hat{S}\subset R$ is the multiplicative subset generated by $S$.

By assumption, no element in $\hat{S}$ is a zero-divisor for $H_{T}^{*}(X)$, so the left localization map in the diagram is injective. It follows that the top arrow is also injective, meaning that the equivariant cohomology of $X$ embeds into that of the fixed point set. Since $H_{T}^{*}(X^{T})\cong H^{*}(X^{T})\otimes R$ is free over $R$, $H_{T}^{*}(X)$ must be torsion-free.

We now consider the case $k>1$ and assume that $M\coloneqq H_{T}^{*}(X)$ is not a $k$-th syzygy. By Lemma 2.1 this means that there is an $R$-regular sequence $\mathbfit a$ of length at most $k$ in $H^{2}(BT;\mathbb{Z})$ that is not $M$-regular. We are going to show that there is another such sequence contained in $S$. If $\mathbfit a=(a_{1},\dots,a_{m})$ is of length $m<k$ or if $m=k$ and $\mathbfit a^{\prime}=(a_{1},\dots,a_{k-1})$ is not $M$-regular, then $M$ is not a syzygy of order $k-1$, and we are done by induction.

So we can assume $m=k$ and that $\mathbfit a^{\prime}$ is $M$-regular. We write $T^{\prime}=T(\mathbfit a^{\prime})$, $R^{\prime}=H^{*}(BT^{\prime})=R/\mathbfit a^{\prime}$ as well as $\pi^{\prime}$ for the canonical projection $R\to R^{\prime}$. By Lemma 2.2 we have an isomorphism

[TABLE]

and $\pi^{\prime}(a_{k})\in R^{\prime}$ is a zero-divisor for this module. Moreover, Lemma 2.3 implies that $\pi^{\prime}(S)\subset R^{\prime}$ is $1$-localizing for the $T^{\prime}$-space $X$. By the already established base case, there is a zero-divisor $\pi^{\prime}(b)\neq 0$ in $\pi^{\prime}(S)$, hence $0\neq b\in S$ is also a zero-divisor for $M^{\prime}$. Therefore, the sequence $(\mathbfit a^{\prime},b)$ is not $M$-regular.

We may assume that $b$ is not a zero-divisor for $M$ for otherwise we would be done as $M$ would not be a first syzygy. Because $M$ is graded and bounded below and the sequence $(\mathbfit a^{\prime},b)$ is made of homogeneous elements, we can rearrange it [14, Prop., p. 1] to obtain $(b,\mathbfit a^{\prime})$, which is again $R$-regular, but not $M$-regular. Since $b$ is not a zero-divisor for $M$, this means that $\mathbfit a^{\prime}$ is not regular for $M^{\prime\prime}=M/b$.

We write $T^{\prime\prime}=T(b)$ and define $R^{\prime\prime}$ and $\pi^{\prime\prime}$ accordingly. Again by Lemma 2.3, $\pi^{\prime\prime}(S)$ is $(k-1)$-localizing for $X$, considered as a $T^{\prime\prime}$-space. Appealing once more to Lemma 2.2, we get an isomorphism

[TABLE]

Given that $\mathbfit a^{\prime}$ is not $M^{\prime\prime}$-regular, $M^{\prime\prime}$ cannot be a $(k-1)$-st syzygy over $R^{\prime\prime}$. By induction, we can therefore find a sequence $\pi^{\prime\prime}(\mathbfit c)$ of length at most $k-1$ in $\pi^{\prime\prime}(S)$ that is regular for $R^{\prime\prime}$, but not for $M^{\prime\prime}$. Thus, $(b,\mathbfit c)$ is an $R$-regular sequence in $S$ of length at most $k$ that is not $M$-regular, as desired. ∎

3. Big polygon spaces

Let $r\geq 1$. We write $[r]=\{1,\dots,r\}$ and $\Delta$ for the simplex with vertex set $[r]$, considered as a simplicial complex. We call a length vector $\ell\in\mathbb{R}^{r}$ strongly generic if $\ell(\sigma)\neq\ell(\tau)$ for any two distinct simplices $\sigma$, $\tau$ in $\Delta$, where

[TABLE]

Two generic length vectors are called equivalent if they induce the same notion of ‘long’ and ‘short’ on subsets of $[r]$. The equivalence classes of generic length vectors $\ell$ are open polyhedral cones in $\mathbb{R}^{r}$ which are the connected components of the complement of a hyperplane arrangement. Because strong genericity means that certain additional hyperplanes are avoided, any generic length vector is equivalent to a strongly generic one. Two equivalent generic length vectors give rise to equivariantly diffeomorphic big polygon spaces, hence to isomorphic equivariant cohomologies. Moreover, there is no loss of generality if one assumes $\ell$ to be positive and weakly increasing, see [8, Sec. 2]. In this case, non-equivalent generic length vectors give rise to big polygon spaces which even non-equivariantly are not diffeomorphic [8, Prop. 3.7]. For the rest of this section, $\ell\in\mathbb{R}^{r}$ denotes a strongly generic length vector with positive coordinates.

For any $R$-algebra $\bar{R}$, we write $\mathcal{C}(\Delta;\bar{R})$ for the Koszul complex with coefficients in $\bar{R}$. That is, $\mathcal{C}(\Delta;\bar{R})$ is a free $\bar{R}$-module with basis $\Delta$ and differential

[TABLE]

for $\gamma\in\Delta$, cf. [8, Sec. 5]. (Note that we sometimes omit braces, as in $\gamma\setminus j$.) We introduce a grading by giving each generator $t_{i}\in R$ the degree $2$ and each $\gamma\in\Delta$ the degree $(2p+2q-1)\cdot\#\gamma$. The differential (3.2) then has degree $1-2p$.

Let $S\subset\Delta$ be a subset. We define $S_{+}$ and $S_{-}$ to be the set of $\ell$-long and $\ell$-short simplices in $S$, respectively. We write $\mathcal{C}(S;\bar{R})$ for the $\bar{R}$-submodule of $\mathcal{C}(\Delta;\bar{R})$ with basis $S$ so that

[TABLE]

as $\bar{R}$-modules. If $S$ is a simplicial subcomplex of $\Delta$, then $\mathcal{C}(S;\bar{R})$ and $\mathcal{C}(S_{-};\bar{R})$ are subcomplexes of $\mathcal{C}(\Delta;\bar{R})$, but $\mathcal{C}(S_{+};\bar{R})$ is not in general. For any $S$ we define the subcomplex

[TABLE]

For any $c=\sum_{\sigma\in\Delta}c_{\sigma}\sigma\in\mathcal{C}(\Delta;\bar{R})$, we write

[TABLE]

for its support and, assuming $c\notin\mathcal{C}(\Delta_{+})$,

[TABLE]

Lemma 3.1**.**

Consider the differential as a map

[TABLE]

Then there is a short exact sequence of graded $R$-modules

[TABLE]

In particular, the syzygy order of $H_{T}^{*}(X(\ell))$ over $R$ equals that of $\operatorname{coker}f_{\ell}$.

Here “$[-2p]$” denotes a degree shift by $2p$ downwards. The sequence actually splits by a result of Puppe [16, Lemma 3.12].

Proof.

See [8, Sec. 4, Lemma 6.2]. Note that we have indexed the basis elements in a form more convenient for our purposes. ∎

For any $\gamma\in\Delta_{+}$ we define

[TABLE]

and

[TABLE]

as in [8, eqs. (6.6)–(6.7)].

Theorem 3.2**.**

The syzygy order of $H_{T}^{*}(X(\ell))$ over $R$ is $\mu(\ell)-1$.

In [8, Prop. 6.3] it is shown that $\mu(\ell)-1$ is an upper bound for the syzygy order, and it was conjectured that one has equality [8, Conj. 6.6].

Proof.

According to Lemma 3.1, the syzygy order of $H_{T}^{*}(X(\ell))$ equals that of

[TABLE]

By what we have just said, we only have to show that $M(\ell)$ is a syzygy of order at least $\mu(\ell)-1$.

The isotropy subgroups appearing in $X(\ell)=X_{p,q}(\ell)$ are the coordinate subtori of $T=(S^{1})^{r}$. Hence for any $k$ the set $S=\{t_{1},\dots,t_{r}\}\subset H^{2}(BT;\mathbb{Z})$ is $k$-localizing for $X$. By Theorem 2.4, it suffices to show that for any $k<\mu(\ell)$ and any pairwise distinct elements $i_{1}$, …, $i_{k}$ the sequence $(t_{i_{1}},\dots,t_{i_{k}})$ is $M(\ell)$-regular.

We proceed by induction on $k$, the case $k=0$ being void. For $k>0$, we know by induction that the sequence $t_{i_{1}}$, …, $t_{i_{k-1}}$ is $M(\ell)$-regular. It remains to show that $t_{i_{k}}$ is not a zero-divisor in $N=M(\ell)/(t_{i_{1}},\dots,t_{i_{k-1}})$ or, equivalently, that $t_{i_{k}}^{q}$ is not a zero-divisor in $N$. (Recall that $M(\ell)$ is graded and bounded below, so that $t_{i_{k}}N\neq N$.) We write $I=\{i_{1},\dots,i_{k-1}\}$ and $i=i_{k}$.

We start by observing that

[TABLE]

where

[TABLE]

In this case the differential (3.2) takes the form

[TABLE]

Assume that our claim is false. Then there is a $c\in\mathcal{C}(\Delta;\mathbb{R})$ such that $t_{i}^{q}c$ is contained in $\mathcal{D}(\Delta_{+};\bar{R})$ while $c$ itself is not. We can write $t_{i}^{q}c=a+db$ for some $a$, $b\in\mathcal{C}(\Delta_{+};\bar{R})$. Since

[TABLE]

we may assume $a$ and $b$ to be $t_{i}^{q}$-free. By this we mean that in the canonical monomial basis for $\bar{R}$, no monomial divisible by $t_{i}^{q}$ appears in the non-zero coefficients $a_{\gamma}$ and $b_{\gamma}$ of $a$ and $b$ with respect to the basis $\Delta$ of $\mathcal{C}(\Delta;\bar{R})$. It implies that $c$ is $t_{i}^{q}$-free.

Let $\Gamma\subset\Delta$ be the simplicial complex of the facet not containing $i$, so that $\Delta$ is the cone over $\Gamma$ with vertex $i$. Recall that we have a bijection between the simplices $\gamma\in\Gamma$ and those in $\Delta\setminus\Gamma$ given by $\gamma\mapsto{\hat{\gamma\mkern 2.0mu}\mkern-2.0mu}{}\coloneqq\gamma\cup i$.

Using (3.3), we can decompose $a$, $b$ and $c$ as

[TABLE]

By inspection of (3.12) we see that $db_{1}\in\mathcal{C}(\Gamma;\bar{R})$ is $t_{i}^{q}$-free and

[TABLE]

where $e_{1}$ is divisible by $t_{i}^{q}$ and $e_{2}$ is $t_{i}^{q}$-free. Hence

[TABLE]

This implies

[TABLE]

Hence $c\in\mathcal{C}(\Gamma;\bar{R})$, and we can write it in the form

[TABLE]

We additionally assume that $c$ is a counterexample maximizing $\ell(c)$. The simplex $\sigma\in\operatorname{supp}c$ realizing $\ell(c)$ is necessarily short since $c\notin\mathcal{C}(\Delta_{+};\bar{R})\subset\mathcal{D}(\Delta_{+};\bar{R})$. We finally require that among all these counterexamples we pick one with the fewest monomials appearing in $c_{\sigma}\in\bar{R}$.

Figure 1 may help the reader to visualize the simplices constructed in the following arguments.

Being short, $\sigma$ cannot be contained in $\operatorname{supp}a$. Hence $\hat{\sigma}\in\operatorname{supp}b$ and $b_{\hat{\sigma}}=\pm c_{\sigma}$ by (3.19) and (3.12). In particular, $\hat{\sigma}$ is long. Since $\sigma$ is short and $\mu(\ell)\geq k+1$, we conclude that $\hat{\sigma}$ has $k+1$ short facets. Hence there is a short facet of the form

[TABLE]

given that $\#I=k-1$. Let us write

[TABLE]

(where we have used our assumption that $\ell$ is strongly generic) and

[TABLE]

By (3.21), we may assume $j=j_{p}$ in (3.20) because replacing $j$ by $j_{p}$ can only decrease the length of $\hat{\rho}$.

Looking at (3.12), we have $\hat{\rho}\in\operatorname{supp}d\hat{\sigma}$ since $j_{p}\notin I$. Given that $\hat{\rho}$ is short and not contained in $\Gamma$, it cannot appear in $db=t_{i}^{q}c-a$. Hence there must be a (necessarily long) simplex $\hat{\tau}\neq\hat{\sigma}$ appearing in $b$ and having $\hat{\rho}$ as a facet.

We have

[TABLE]

The contribution of $d(b_{\hat{\sigma}}\hat{\sigma})=\pm d(c_{\sigma}\hat{\sigma})$ to the coefficient $(db)_{\hat{\rho}}$ of $\hat{\rho}$ in $db$ is $\pm t_{j_{p}}^{q}c_{\sigma}$, and that of $d(b_{\hat{\tau}}\hat{\tau})$ likewise is $\pm t_{j^{\prime}}^{q}b_{\hat{\tau}}$. Since all monomials appearing in $(dc_{\sigma}\hat{\sigma})_{\hat{\rho}}$ must somehow be compensated for by other simplices appearing in $b$, we may choose $\hat{\tau}$ such that $t_{j^{\prime}}^{q}$ divides a monomial appearing in $c_{\sigma}$. Because $\tau$ appears in $db$ and $\sigma$ is the shortest simplex appearing there, we additionally have $\ell(\tau)>\ell(\sigma)$ or, in other words, $\ell_{j^{\prime}}>\ell_{j_{p}}$, again by strong genericity.

Now $\hat{\tau}$ is a long facet of

[TABLE]

The other facets of $\hat{\beta}$ different from $\beta$ are obtained from $\hat{\tau}$ by substituting $j_{p}$ for some $j_{q}\in\tau$. If $q<p$, we get another long facet by (3.21). Hence $\hat{\beta}$ has at most $\#I+1=k$ short facets by (3.22), including possibly $\beta$. But $\mu(\ell)\geq k+1$, so that all facets of $\hat{\beta}$ are long.

Since we have

[TABLE]

we can write $t_{i}^{q}\,\tilde{c}=d\,\tilde{b}$ with

[TABLE]

Consider now all monomials appearing in $c_{\sigma}$ that are divisible by $t_{j^{\prime}}^{q}$ and write their sum as $t_{j^{\prime}}^{q}x$ with $x\in\bar{R}$. Then $x\neq 0$ by our choice of $\hat{\tau}$, and no monomial appearing in it is divisible by $t_{i}^{q}$ since $c$ is $t_{i}^{q}$-free. The preceding discussion implies

[TABLE]

where both $a$ and $b+x\,\tilde{b}\in\mathcal{C}(\Delta_{+}\setminus\Gamma_{+};\bar{R})$ are $t_{i}^{q}$-free and $c+x\,\tilde{c}\in\mathcal{C}(\Gamma;\bar{R})$. In particular, $c+x\,\tilde{c}$ is another counterexample of the form (3.19) to our claim that $t_{i}^{q}$ is not a zero-divisor in $M(\ell)$.

Since $\ell_{j^{\prime}}>\ell_{j_{p}}$ and $j_{q}\in I$ for $p<q\leq m$, the simplex $\sigma=\beta\setminus j^{\prime}$ is the shortest one appearing in the sum (3.26). Hence

[TABLE]

The coefficient of $\sigma$ in $c+x\,\tilde{c}$ is of the form

[TABLE]

If it vanishes, then we have a strict inequality in (3.29) since $\ell$ is strongly generic. This would contradict our choice of $c$ with maximal $\ell(c)$. If it does not vanish, then it is still obtained from $c_{\sigma}$ be removing certain monomials. As such, it contains fewer monomials than $c_{\sigma}$, again contradicting our choice of $c$.

We conclude that no counterexample exists. ∎

4. Classification of high syzygies

Using the result of the previous section, we can extend the classification of big polygon spaces with high syzygies in their equivariant cohomology. Throughout this section, $\ell\in\mathbb{R}^{r}$ denotes a generic length vector with positive and weakly increasing coefficients.

Maximal syzygies, that is, those of order $m$ for $r=2m+1$ odd or $r=2m+2$ even were determined in [8]. We are going to rephrase the proof in our setting and extend the result to syzygies of order $m-1$.

Lemma 4.1**.**

If there is a long subset $J\subset[r]$ of size $\#J=\mu(\ell)$, then

[TABLE]

Proof.

We may assume $\ell$ to be strongly generic and set $\mu(\ell)=k$. Note that all subsets $I\subset[r]$ with fewer than $k$ elements are short for otherwise we would get the contradiction

[TABLE]

for a (necessarily nonempty) long set $I$ of minimal size.

Among all long subsets $J\subset[r]$ of size $k$, we pick the one with minimal $\ell(J)$. We set $j_{\mathrm{min}}=\min(J)$ and $j_{\mathrm{max}}=\max(J)$. By what we have just said, $J\setminus j_{\mathrm{max}}$ is short.

Let $I\subset[r]$ be the set of those values $j\notin J\setminus j_{\mathrm{max}}$ such that $(J\setminus j_{\mathrm{max}})\cup j$ is long. This set contains $j_{\mathrm{max}}$ and therefore is non-empty. Hence $\#I\geq k$ and $\#(I\setminus j_{\mathrm{max}})\geq k-1$.

For any $i\in I\setminus j_{\mathrm{max}}$, the set $J_{i}=(J\setminus j_{\mathrm{max}})\cup i$ is long and of size $k$, and $J_{i}\setminus i$ is short. Hence

[TABLE]

This implies $\ell(J_{i})>\ell(J)$ for otherwise $\ell(J)$ would not be minimal. In summary, $I$ consists of $j_{\mathrm{max}}$ and $k-1$ values larger than $j_{\mathrm{max}}$.

Consider the remaining $r-\#(J\cup I)=r-2k+1$ elements of $[r]\setminus(J\cup I)$. If one of them were greater than $j_{\mathrm{min}}$, then we would have $\ell([r]\setminus J)>\ell(J)$, contradicting the assumption that $J$ is long. These elements therefore are smaller than $j_{\mathrm{min}}$, and we conclude that

[TABLE]

This implies that any subset of $[r]$ containing $k$ values $\geq j_{\mathrm{min}}=r-2k+2$ is long. These sets are exactly the long sets for the length vector

[TABLE]

We conclude that they comprise half of all subsets and therefore that $\ell$ and $\ell^{\prime}$ induce the same notion of ‘long’ and ‘short’. ∎

Proposition 4.2** ([8, Cor. 6.4]).**

Let $r\geq 1$.

1. (1)

Assume that $r=2m+1$ is odd. Then $H_{T}^{*}(X(\ell))$ is a syzygy of order $m$ if and only if $\ell\sim(1,\dots,1)$. 2. (2)

Assume that $r=2m+2$ is even. Then $H_{T}^{*}(X(\ell))$ is a syzygy of order $m$ if and only if $\ell\sim(0,1,\dots,1)$.

Proof.

By Theorem 3.2, the condition on the syzygy order translates into $\mu(\ell)=m+1$. In both cases it is immediate to check that this is satisfied by the given length vectors. It remains to show the ‘only if’ direction.

As in the previous proof, any subset with fewer than $\mu(\ell)=m+1$ elements is short. Hence there must be a long subset of size $m+1$ for otherwise more than half of all subsets would be short. The claim now follows from Lemma 4.1. ∎

Proposition 4.3**.**

Let $r\geq 3$.

1. (1)

Assume that $r=2m+1$ is odd. Then $H_{T}^{*}(X(\ell))$ is a syzygy of order $m-1$ if and only if $\ell\sim(0,0,1,\dots,1)$. 2. (2)

Assume that $r=2m+2$ is even. Then $H_{T}^{*}(X(\ell))$ is a syzygy of order $m-1$ if and only if $\ell\sim(0,0,0,1,\dots,1)$ or $\ell\sim(1,1,1,2,\dots,2)$.

We can restrict ourselves to $r\geq 3$ here since $H_{T}(X(\ell))$ is always torsion for $r\leq 2$.

Proof.

This time the condition on the syzygy order translates into $\mu(\ell)=m$. In all cases it is elementary to verify that it is satisfied by the given length vectors. It remains to show the ‘only if’ direction.

Let $J\subset[r]$ be a long subset of minimal size. Since half of all subsets are long, we have $1\leq\#J\leq m+1$ and also $m=\mu(\ell)\leq\sigma_{\ell}(J)=\#J$, hence $m\leq\#J\leq m+1$.

Assume that $r=2m+1$ is odd. If $\#J=m+1$, then all subsets of size at most $m$ are short. Since these are already half of all subsets, those having at least $m+1$ elements are long. This implies $\mu(\ell)=m+1$, contrary to our assumption. Hence $\#J=m=\mu(\ell)$, and we can appeal to Lemma 4.1.

Now let $r=2m+2$ be even. The case $\#J=m$ is dealt with as before. So we can assume that long sets have at least $m+1$ elements, and we have to show $\ell\sim\ell^{\prime}=(1,1,1,2,\dots,2)$.

For the purpose of this proof, call a subset $J\subset[r]$ distinguished if it is of size $m+1$ and contains $2$ and $3$. Assume that there is a distinguished long set $J$. We claim that in this case all distinguished sets are long.

In order to prove this, choose a $j\notin J$ such that $j>3$. (This is possible because there are $m+1\geq 2$ elements not in $J$.) Then both $J^{\prime}=J\cup j$ and $J=J^{\prime}\setminus j$ are long, hence so are $J^{\prime}\setminus 2$ and $J^{\prime}\setminus 3$. Thus,

[TABLE]

Since $\mu(\ell)=m$, this implies $\sigma_{\ell}(J^{\prime})=0$. In other words, replacing any element of $J$ by an element J\mathrel{\mathchoice{{\ooalign{\hfil\displaystyle/\mkern 1.0mu\hfil$\crcr$\displaystyle\owns}}}{{\ooalign{\hfil\textstyle/\mkern 1.0mu\hfil$\crcr$\textstyle\owns}}}{{\ooalign{\hfil\scriptstyle/\mkern 1.0mu\hfil$\crcr$\scriptstyle\owns}}}{{\ooalign{\hfil\scriptscriptstyle/\mkern 1.0mu\hfil$\crcr$\scriptscriptstyle\owns}}}}j>3 leads to another long set. Applying this procedure repeatedly, we can transform $J$ into any other distinguished set while keeping it long, which proves the claim.

Given a distinguished set, we can also replace $2$ and $3$ by larger elements without making the set short. So we see that any subset $J\subset[r]$ of size $m+1$ not containing $1$ is long, as are all subsets of larger size (because their complements, being of size at most $m$, are short). These sets are exactly the long subsets for $\ell^{\prime\prime}=(0,1,\dots,1)$ and therefore all $\ell$-long subset. But this is impossible as $\mu(\ell^{\prime\prime})=m+1\neq m$.

We conclude that any distinguished set is short. Hence so is any subset $J$ of size $m+1$ such that $\#(J\cap\{1,2,3\})\geq 2$. Together with the subsets of smaller size, these are exactly the $\ell^{\prime}$-short subsets. So they are also exactly the $\ell$-short subsets, which shows $\ell\sim\ell^{\prime}$. ∎

There seems to be no easy description of syzygies of lower order. For instance, computer experiments show that for $r=9=2\cdot 4+1$ there are, up to equivalence, $5$ length vectors $\ell$ (out of $175,428$) such that $H_{T}^{*}(X(\ell))$ has syzygy order $2=4-2$, and for $r=10=2\cdot 4+2$ there are $18$ (out of $52,980,624$). (The numbers of non-equivalent length vectors can be found in [13, Sec. 10.3.1].)

5. The real case

Let $\ell\in\mathbb{R}^{r}$ be a generic length vector, and let $p$, $q\geq 1$. The real big polygon space $Y(\ell)=Y_{p,q}(\ell)$ is defined by restricting all variables to the reals, that is, by

[TABLE]

where $u_{1}$, …, $u_{r}\in\mathbb{R}^{p}$ and $z_{1}$, …, $z_{r}\in\mathbb{R}^{q}$. It is the fixed point set of $X(\ell)$ under complex conjugation of all variables, hence again smooth. The $2$-torus $G=(\mathbb{Z}_{2})^{r}$ of rank $r$ acts on $Y(\ell)$ by reversing the signs of the $z$-variables. We assume now that $\Bbbk$ is a field of characteristic $2$. The $G$-equivariant cohomology of real big polygon spaces (and more general spaces) with coefficients in $\Bbbk$ has been studied by Puppe [16], following the work of Hausmann [13, Sec. 10.3].

The fixed point set $Y(\ell)^{G}$ is empty for $p=1$, by the very definition of a generic length vector. By the localization theorem, this implies that $H_{G}^{*}(Y(\ell))$ is a torsion module over $R=H^{*}(BG)$ in this case. For $p>1$, however, the theory parallels the one for the complex case. In particular, the equivariant cohomology $H_{G}^{*}(Y(\ell))$ is given by a formula analogous to Lemma 3.1, which allows us to proceed in the same way as before. We content ourselves with sketching the arguments.

Lemma 5.1**.**

Grade $\mathcal{C}(\Delta;R)$ by setting $|t_{i}|=1$ for each generator $t_{i}\in R$ and $|\gamma|=(p+q-1)\cdot\#\gamma$ for each $\gamma\in\Delta$. With $f_{\ell}$ defined as in Lemma 3.1, there is a short exact sequence of graded $R$-modules

[TABLE]

In particular, the syzygy order of $H_{T}^{*}(X(\ell))$ over $R$ equals that of $\operatorname{coker}f_{\ell}$.

Proof.

See [16, Sec. 3], in particular Proposition 3.11 and equations (3.19)–(3.22). Alternatively, one could adapt the proof given in [8]. ∎

Let $X$ be a $G$-space, and for $x\in X$ let $p_{x}\colon H^{1}(BG)\to H^{1}(BG_{x})$ be the restriction map. For $0\leq k\leq r$, we call a subset $S\subset H^{1}(BG)$ $k$-localizing if for any $x\in X$ at least $\min(k,\operatorname{corank}G_{x})$ linearly independent elements from $S$ lie in $\ker p_{x}$.

We need the following analogue of Lemma 2.1 from [3]. A nice $G$-space is defined in analogy with the torus case.

Lemma 5.2**.**

Let $X$ be a nice $G$-space and let $k\geq 0$. Then $H_{G}^{*}(X)$ is a $k$-th syzygy if and only if every linear independent sequence in $H^{1}(BG;\mathbb{F}_{2})$ of length at most $k$ is $H_{G}^{*}(X)$-regular.

Arguing as in Sections 2 and 3, we obtain the following.

Theorem 5.3**.**

Let $X$ be a nice $G$-space, and let $S\subset H^{1}(BG;\mathbb{F}_{2})$ be $k$-localizing for $X$ for some $k\geq 0$. Then $H_{G}^{*}(X)$ is a $k$-th syzygy over $R$ if and only if any linearly independent sequence in $S$ of length at most $k$ is $H_{G}^{*}(X)$-regular.

Theorem 5.4**.**

In the case $p>1$ the syzygy order of $H_{G}^{*}(Y(\ell))$ over $R$ is $\mu(\ell)-1$. In particular, it is the same as the syzygy order of $H_{T}(X(\ell);\mathbb{Q})$ over $H^{*}(BT;\mathbb{Q})$.

As a consequence, the characterizations of high syzygies derived in Section 4 carry over to real big polygon spaces. The analogue of Proposition 4.2 has already been established by Puppe [16, Prop. 3.14].