Affine Stresses: The Partition of Unity and Kalai’s Reconstruction Conjectures

Abstract

Kalai conjectured that if P is a simplicial d-polytope that has no missing faces of dimension $d-1$, then the graph of P and the space of affine 2-stresses of P determine P up to affine equivalence. We propose a higher-dimensional generalization of this conjecture: if $2\le i\le d/2$ and P is a simplicial d-polytope that has no missing faces of dimension $\ge d-i+1$, then the space of affine i-stresses of P determines the space of affine 1-stresses of P. We prove this conjecture for (1) k-stacked d-polytopes with $2\le i\le k\le d/2-1$, (2) d-polytopes that have no missing faces of dimension $\ge d-2i+2$, and (3) flag PL $(d-1)$-spheres with generic embeddings (for all $2\le i\le d/2$). We also discuss several related results and conjectures. For instance, we show that if P is a simplicial d-polytope that has no missing faces of dimension $\ge d-2i+2$, then the $(i-1)$-skeleton of P and the set of sign vectors of affine i-stresses of P determine the combinatorial type of P. Along the way, we establish the partition of unity of affine stresses: for any $1\le i\le (d-1)/2$, the space of affine i-stresses of a simplicial d-polytope as well as the space of affine i-stresses of a simplicial $(d-1)$-sphere (with a generic embedding) can be expressed as the sum of affine i-stress spaces of vertex stars. This is analogous to Adiprasito’s partition of unity of linear stresses for Cohen–Macaulay complexes.

Appendix A: Proof of Conjecture  4.5

The goal of this Appendix is to sketch the proof of Conjecture 4.5.

Theorem A.1

Let $(\Delta ,p)$ be either the boundary complex of a simplicial d-polytope with its natural embedding p, or a $\mathbb {Z}/2\mathbb {Z}$-homology $(d-1)$-sphere with a generic embedding p, and let i be a natural number such that $i\le \lfloor (d-1)/2\rfloor$. Then $\mathcal {S}^a_i(\Delta ,p)=\sum _{v\in V(\Delta )} \mathcal {S}^a_i({{\,\mathrm{\textrm{st}}\,}}(v), p)$.

The proof uses the (dual) language of the Stanley–Reisner rings. Specifically, we denote by $\mathbb {R}[\Delta ]$ the Stanley–Reisner ring of $\Delta$ and, for a face $\tau$ of $\Delta$, we denote by $\mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]$ the Stanley–Reisner ring of the star of $\tau$ in $\Delta$, considered as an $\mathbb {R}[\Delta ]$-module. As in Sect. 3.1, we let $\Theta =\Theta (p)$ be the collection of $d+1$ linear forms $(\theta _1,\ldots ,\theta _d,\theta _{d+1})$ in $\mathbb {R}[X]$ determined by p. In particular, $\theta _{d+1}=\sum _{v\in V} x_v$. Finally, for a graded $\mathbb {R}[\Delta ]$-module M and any integer j, we denote by $M_j$ the j-th graded component of M. Some computations below rely on a simple observation that $\mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]_j=0$ for all $j<0$.

The statement that $\mathcal {S}^a_i(\Delta ,p)=\sum _{v\in V(\Delta )} \mathcal {S}^a_i({{\,\mathrm{\textrm{st}}\,}}(v), p)$ is easily seen to be equivalent to the statement that the map $\big (\mathbb {R}[\Delta ]/(\Theta )\big )_i \rightarrow \sum _{v\in V(\Delta )} \big (\mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(v)]/\Theta \mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(v)]\big )_i$, induced by natural surjections $\mathbb {R}[\Delta ] \rightarrow \mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(v)]$, is injective. To establish this result we adapt the proof of Theorem 33 from [3]. The key new idea is to look at the Koszul complex $K^*(\Theta )$ w.r.t. all $d+1$ elements $(\theta _1,\ldots ,\theta _d, \theta _{d+1})$ of $\Theta (p)$ rather than just w.r.t. the first d elements. Our assumptions on $(\Delta ,p)$ along with the g-theorem (see [25, 34] for the case of polytopes and [18, Thm. 1.3] for the case of spheres) imply that $\mathbb {R}[\Delta ]$ is a Cohen–Macaulay ring of Krull dimension d, that the sequence $\theta _1,\ldots ,\theta _d$ is a regular sequence on $\mathbb {R}[\Delta ]$, and that the map $\cdot \theta _{d+1}: \big (\mathbb {R}[\Delta ]/(\theta _1,\ldots ,\theta _d)\big )_{j-1} \rightarrow \big (\mathbb {R}[\Delta ]/(\theta _1,\ldots ,\theta _d)\big )_{j}$ is injective for all $j\le \lceil d/2\rceil$. Similar statements apply to $\mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]$ for any face $\tau$ of $\Delta$. Specifically, $\theta _1,\ldots ,\theta _d$ is a regular sequence on $\mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]$ and

$$\begin{aligned} \cdot \theta _{d+1}: \big (\mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]/(\theta _1,\ldots ,\theta _d)\mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]\big )_{j-1} \rightarrow \big (\mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]/(\theta _1,\ldots ,\theta _d)\mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]\big )_{j} \end{aligned}$$

is injective for all $j\le \lceil (d-|\tau |)/2\rceil$.

The above paragraph and standard results about Koszul complexes (such as Theorem 21 in [3]) imply that for any face $\tau$ and any integer j, the following complex of vector spaces over $\mathbb {R}$

$$\begin{aligned}{} & {} 0 \rightarrow \mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]_{j-d-1} \otimes K^0(\Theta ) {\mathop {\rightarrow }\limits ^{\partial ^0}} \mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]_{j-d} \otimes K^1(\Theta ) {\mathop {\rightarrow }\limits ^{\partial ^1}}\cdots \\{} & {} \quad \rightarrow \mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]_{j-1} \otimes K^d(\Theta ) {\mathop {\rightarrow }\limits ^{\partial ^d}} \mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]_{j} \otimes K^{d+1}(\Theta ) \\{} & {} \quad \rightarrow \big (\mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]/\Theta \mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]\big )_j \rightarrow 0 \end{aligned}$$

is almost exact, namely,

• (*) all cohomologies of this complex, except possibly for $H^d$, vanish, and

• (**) if $j\le \lceil (d-|\tau |)/2\rceil$, then $H^d$ also vanishes.

We now proceed as in the proof of [3, Thm. 33]. Let $\Delta ^{(j)}$ denote the set of j-faces of $\Delta$. (In particular, $\Delta ^{(0)}=V(\Delta )$.) Let $P^*=P^*(\Delta )$ be the partition complex

$$\begin{aligned} 0\rightarrow \mathbb {R}[\Delta ]\rightarrow \bigoplus _{v\in \Delta ^{(0)}} \mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(v)] \rightarrow \cdots \rightarrow \bigoplus _{\sigma \in \Delta ^{(d-1)}} \mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\sigma )] \rightarrow 0 \end{aligned}$$

with indexing such that $P^{-1}=\mathbb {R}[\Delta ]$ and $P^j=\bigoplus _{\tau \in \Delta ^{(j)}} \mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(\tau )]$ for $j\ge 0$. Let ${\tilde{K}}^*(\Theta )$ be the augmented Koszul complex w.r.t. $\Theta =\Theta (p)$, i.e., the complex

$$\begin{aligned} K^0(\Theta ) \rightarrow \cdots \rightarrow K^{d+1}(\Theta ) \rightarrow {\tilde{K}}^{d+2}(\Theta ):=\mathbb {R}[\Delta ]/(\Theta ) \rightarrow 0. \end{aligned}$$

Finally, let $C^{*,*}$ be the double complex $P^*\otimes {\tilde{K}}^*(\Theta )$ endowed with the grading defined in [3, Sect. 5.1.1].

We fix $i\le \lfloor (d-1)/2\rfloor =\lceil (d-2)/2\rceil$ and consider the i-th graded piece of $C^{*,*}$, $C^{*,*}_{\, i}$. The outline of the proof is as follows. Properties (*) and (**) above along with [3, Lem. 8] applied in the vertical direction of $C^{*,*}_{\, i}$ imply that $H^d\big ({{\,\mathrm{\textrm{Tot}}\,}}\big (C^{*,*}_{\, i}\big )^*\big )=H^{d+1}\big ({{\,\mathrm{\textrm{Tot}}\,}}\big (C^{*,*}_{\, i}\big )^*\big )=0$. This in turn implies that $H^d\big ({{\,\mathrm{\textrm{Tot}}\,}}\big (C^{*,*\le d+1}_{\, i}\big )^*\big )$ is isomorphic to $H^{d+1}\big (C^{*-d-2,d+2}_{\, i}\big )$. It then follows that the kernel of the map $\big (\mathbb {R}[\Delta ]/(\Theta )\big )_i \rightarrow \sum _{v\in V(\Delta )} \big (\mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(v)]/\Theta \mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(v)]\big )_i$ is isomorphic to $H^d\big ({{\,\mathrm{\textrm{Tot}}\,}}\big (C^{*,*\le d+1}_{\, i}\big )^*\big )$. Now, by [3, Prop. 26], $P^*_t$ is exact for all $t\ne 0$ and since $\Delta$ is Cohen–Macaulay, the only nontrivial cohomology that $P^*_0$ has is $H^{d-1}$. Finally, applying [3, Lem. 8] in the horizontal direction of $C^{*,*\le d+1}_{\, i}$, we conclude that $H^d\big ({{\,\mathrm{\textrm{Tot}}\,}}\big (C^{*,*\le d+1}_i\big )^*\big )=0$, and so the map $\big (\mathbb {R}[\Delta ]/(\Theta )\big )_i \rightarrow \sum _{v\in V(\Delta )} \big (\mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(v)]/\Theta \mathbb {R}[{{\,\mathrm{\textrm{st}}\,}}(v)]\big )_i$ is injective. This completes the proof. $\square$