On the Voronoi Conjecture for combinatorially Voronoi parallelohedra in dimension five

TL;DR

This paper proves that all five-dimensional Voronoi parallelohedra satisfy a key combinatorial condition, advancing the understanding of the Voronoi conjecture in five dimensions and introducing a new criterion based on the Venkov complex.

Contribution

The paper demonstrates that the combinatorial condition for a parallelohedron to be affinely Voronoi holds in five dimensions, and introduces a new sufficient condition involving the Venkov complex.

Findings

The combinatorial condition holds for all five-dimensional Voronoi parallelohedra.

The Voronoi conjecture in $\

A new sufficient condition for a parallelohedron to be affinely Voronoi is proposed.

Abstract

In a recent paper Garber, Gavrilyuk and Magazinov proposed a sufficient combinatorial condition for a parallelohedron to be affinely Voronoi. We show that this condition holds for all five-dimensional Voronoi parallelohedra. Consequently, the Voronoi conjecture in $\mathbb R^5$ holds if and only if every five-dimensional parallelohedron is combinatorially Voronoi. Here, by saying that a parallelohedron $P$ is combinatorially Voronoi, we mean that the tiling $\mathcal T(P)$ by translates of $P$ is combinatorially isomorphic to some tiling $\mathcal T(P')$, where $P'$ is a Voronoi parallelohedron, and that the isomorphism naturally induces a linear isomorphism of lattices $\Lambda(P)$ and $\Lambda(P')$. We also propose a new sufficient condition implying that a parallelohedron is affinely Voronoi. The condition is based on the new notion of the Venkov complex associated with a parallelohedron.

Equations

$$g=\frac{b}{a}\text{and}a{\mathbf{e}}_{P,P_F}+b{\mathbf{e}}_{P_G,P}+c{\mathbf{e}}_{P_F,P_G}=0,$$

$$⟨\gamma ⟩:=[F_0,F_1,\dots ,F_k],$$

$$g(\gamma ):=g(⟨\gamma ⟩)=g([F_0,F_1,\dots ,F_k]):=\prod _{i=1}^{k}g(F_i,F_{i-1}).$$

$$g({\gamma }_{\delta ,1})=g({\gamma }_{\delta ,2}).$$

$$F_1=P\cap (P+{\mathbf{x}}_2-{\mathbf{x}}_3),F_2=P\cap (P+{\mathbf{x}}_1),F_3=P\cap (P+{\mathbf{x}}_2),$$

$$\{P+{\mathbf{x}}_i:i=0,1,\dots ,4\}$$

$${\mathbf{a}}_{ij}:=\mathrm{grad}{U}_j-\mathrm{grad}{U}_i.$$

$$g(F_1,F_2)=\frac{|{\mathbf{a}}_{01}|}{|{\mathbf{a}}_{14}|},g(F_2,F_3)=\frac{|{\mathbf{a}}_{02}|}{|{\mathbf{a}}_{01}|},g(F_3,F_1)=\frac{|{\mathbf{a}}_{23}|}{|{\mathbf{a}}_{02}|}.$$

$${\mathbf{a}}_{12}+{\mathbf{a}}_{23}+{\mathbf{a}}_{34}-{\mathbf{a}}_{14}=0.$$

$$g(\gamma )=g(F_1,F_2)g(F_2,F_3)g(F_3,F_1)=\frac{|{\mathbf{a}}_{01}|}{|{\mathbf{a}}_{14}|}\cdot \frac{|{\mathbf{a}}_{02}|}{|{\mathbf{a}}_{01}|}\cdot \frac{|{\mathbf{a}}_{23}|}{|{\mathbf{a}}_{02}|}=\frac{|{\mathbf{a}}_{23}|}{|{\mathbf{a}}_{14}|}=1,$$

$$\begin{gathered} O(a,b,c,a^{\prime },b^{\prime },c^{\prime }):=\{\{a,b,c\},\{a^{\prime },b^{\prime },c^{\prime }\},\{a^{\prime },b,c\},\{a,b^{\prime },c^{\prime }\},\{a,b^{\prime },c\}, \\ \{a^{\prime },b,c^{\prime }\},\{a,b,c^{\prime }\},\{a^{\prime },b^{\prime },c\}\} \end{gathered}$$

$$X(D):=O\left(\overline{a+b},\overline{a+c},\overline{a+d},\overline{c+d},\overline{b+d},\overline{b+c}\right).$$

$$X(D):=O\left(\overline{s+a},\overline{s+b},\overline{a+d},\overline{s+c},\overline{s+d},\overline{a+b}\right).$$

$$X(D):=O\bigl{(}\overline{a+b},\overline{a+c},\overline{b+c},\overline{a+e},\overline{a+f},\overline{b+f}\bigr{)}.$$

$$X(D):=\{\{\overline{a+b},\overline{a+c},\overline{b+c}\}\}.$$

$$X:=\bigcup _{D\in {\mathcal{D}}^3(P)}X(D).$$

$$X(D)=O\left(\overline{s+a},\overline{s+b},\overline{b+c},\overline{s+c},\overline{s+d},\overline{c+d}\right).$$

$$V:=\{\{F,-F\}|\text{F is a facet of P}\}.$$

$$\phi :vert(Ven(P))\to vert(V{G}_r(P))$$

$$⟨{\gamma }_{\delta }⟩=[\pm F_1,\pm F_2,\pm F_3,\pm F_1]$$

$$⟨c,\overrightarrow{x_1{x}_2}⟩:=\ln g(F_1,F_2)\text{or}⟨c,\overrightarrow{x_1{x}_2}⟩:=\ln g(F_1,-F_2),$$

$$⟨\delta c,\{x_1,x_2,x_3\}⟩=⟨c,\partial \{x_1,x_2,x_3\}⟩=⟨c,\overrightarrow{x_1{x}_2}⟩+⟨c,\overrightarrow{x_2{x}_3}⟩+⟨c,\overrightarrow{x_3{x}_1}⟩=\ln g(\gamma )=0,$$

$$[\{F_1,-F_1\},\{F_2,-F_2\},\dots ,\{F_k,-F_k\}=\{F_1,-F_1\}]$$

$$\sum _{i=1}^{k-1}⟨c,\overrightarrow{x_i{x}_{i+1}}⟩=\sum _{i=1}^{k-1}(⟨c^{\prime },x_{i+1}⟩-⟨c^{\prime },x_i⟩)=0.$$

$$\sum _{i=1}^{k-1}⟨c,\overrightarrow{x_i{x}_{i+1}}⟩=\sum _{i=1}^{k-1}\ln g(F_i,F_{i+1})=\ln g(\gamma )\ne 0.$$

$$⟨{\gamma }_{\delta }⟩=[F_1,F_2,\dots ,F_k].$$

$$[\{F_1,-F_1\},\{F_2,-F_2\},\dots ,\{F_k,-F_k\}=\{F_1,-F_1\}]\text{and}[x_1,x_2,\dots ,x_k=x_1]$$

$$A_1(c):=$$

$$A_2(c):=$$

$$B(c):=$$

$$c^{\ast }({\gamma }_{\pi }):=⟨c,\gamma ⟩.$$

Full text

We also propose a new sufficient condition implying that a parallelohedron is affinely Voronoi. The condition is based on the new notion of the Venkov complex associated with a parallelohedron.

1. Introduction

A convex $d$-dimensional polytope $P$ is called a parallelohedron if it tiles $\mathbb{R}^{d}$ by translations only. Consider the family of all tilings by translates of a given parallelohedron $P$. A remarkable result by Venkov [16] and, independently, McMullen [12] asserts that this family necessarily contains a face-to-face tiling. Without loss of generality, we can impose a further requirement that the face-to-face tiling contains $P$ as one of its tiles. We denote such a tiling by $\mathcal{T}(P)$.

Minkowski [13] established that all parallelohedra are centrally symmetric. Consequently, the centers of all tiles of $\mathcal{T}(P)$ form a lattice provided the origin $\mathbf{0}$ is the center of $P$. We denote this lattice by $\Lambda(P)$. Then the set of tiles of $\mathcal{T}(P)$ is exactly $\{P+t\,|\,t\in\Lambda(P)\}$.

It is easy to check that parallelograms and centrally symmetric hexagons are parallelohedra for $d=2$, and that there are no other two-dimensional parallelohedra. The full classification of parallohedra of a given dimension $d$ exists only for $d\leq 4$. The case $d=3$ is due to Fedorov [6], and the case $d=4$ is due to Delaunay [1] with a correction by Stogrin [15].

When speaking about classification of parallelohedra, it is necessary to specify the equivalence relation used to split parallelohedra in equivalence classes. We introduce the notion of equivalence in the following definition, where $\mathcal{F}(\mathcal{T}(P))$ denotes the poset of all faces of $\mathcal{T}(P)$ ordered by inclusion.

Definition 1.1**.**

Two $d$-dimensional parallelohedra, $P$ and $P^{\prime}$, are equivalent if there is an isomorphism of face posets $f:\mathcal{F}(\mathcal{T}(P))\to\mathcal{F}(\mathcal{T}(P^{\prime}))$ inducing a linear isomorphism of lattices $f_{*}:\Lambda(P)\to\Lambda(P^{\prime})$ by restricting the action of $f$ to $d$-dimensional tiles and then passing to their centers.

Remark*.*

Equivalence (in the sense of Definition 1.1) for Voronoi parallelohedra reduces to the notion of an L-type. More precisely, two Voronoi parallelohedra are equivalent if and only if they belong to the same L-type. The concept of L-types originates in the work of Voronoi [17]. See also [3, Section 3] for a modern treatment.

Of course, if $P$ and $P^{\prime}$ are equivalent as parallelohedra, then they are combinatorially equivalent as convex polytopes. In addition, the equivalence classes also retain information about the lattice structure of the tiling, in particular, about central symmetries preserving the lattice. However, the authors are not aware of an example of two non-equivalent parallelohedra that are combinatorially equivalent as convex polytopes.

One of the most famous conjectures in the theory of parallelohedra is stated by Voronoi [17] and reads as follows.

Conjecture**.**

Every parallelohedron $P$ is affinely Voronoi.

Here we call a $d$-dimensional parallelohedron Voronoi if it is a Dirichlet-Voronoi polytope for some $d$-dimensional lattice, and affinely Voronoi if it is an affine image of a Voronoi parallelohedron.

Definition 1.2**.**

We say that a parallelohedron $P$ is combinatorially Voronoi, if there is a Voronoi parallelohedron $P^{\prime}$ such that $P$ and $P^{\prime}$ are equivalent in the sense of Definition 1.1.

Note that this equivalence is stronger than just combinatorial equivalence because we require for a linear isomorphism between the corresponding lattices be induced by combinatorial bijection.

The Voronoi conjecture has been proved for some families of parallelohedra with certain combinatorial restrictions on their properties, see [17, 18, 14, 8, 10] for details, but still remains open in general. The conjecture was also checked for all $3$- and $4$-dimensional parallelohedra, but for $d\geq 5$ remains open as well.

Recently, Garber, Gavrilyuk and Magazinov [8] proposed a sufficient condition for a parallelohedron to be affinely Voronoi. In this paper we combine this condition with the complete classification five-dimensional Voronoi parallelohedra, due to Dutour Sikirić, Garber, Schürmann, and Waldmann [3], in order to prove the following main result.

Theorem 1.3**.**

A 5-dimensional parallelohedron is affinely Voronoi if and only if it is equivalent to a Voronoi parallelohedron.

Theorem 1.3 essentially reduces the Voronoi conjecture in 5 dimensions to its weaker, combinatorial, version.

Earlier, Garber [9] verified the condition of [8] for all 4-dimensional parallelohedra.

Additionaly, we propose yet another sufficient condition, also depending only on the equiavlence class, implying that a parallelohedron is affinely Voronoi. This condition generalizes both [8] and [14].

The paper is organized as follows.

In Section 2 the key concepts and statements of the paper [8] are reproduced.

In Section 3 we introduce the Venkov complex, a 2-dimensional simplicial complex $Ven(P)$ associated with the combinatorics of $P$. The notion of the Venkov complex is then used to formulate a sufficient condition (Theorem 3.7) for a parallelohedron to be affinely Voronoi.

In Section 4 the condition of [8] is reformulated in a discrete form, namely, in terms of the 1-skeleton of $Ven(P)$ (or, equivalently, the red Venkov graph $VG_{r}(P)$).

In Section 5 we describe the algorithm to verify Theorem 1.3 and provide the details of implementation.

In Section 6 we show that the condition of Theorem 3.7 implies both the conditions of [8] and [14].

Finally, in Section 7 some open questions are proposed.

2. The $\pi$-surface of a parallelohedron

We follow the exposition of [8] in order to formulate the sufficient condition for Voronoi conjecture and refer to this paper for more details.

We fix a $d$-dimensional parallelohedron $P$. As before, let $\mathcal{T}(P)$ be the face-to-face tiling of $\mathbb{R}^{d}$ with translations of $P$. Each $(d-2)$-face of the polytope $P$ can be attributed to one of two types according to the following well-known proposition.

Proposition 2.1** (See, for instance, [16] or [12]).**

Each $(d-2)$-face $F$ of $P$ is incident either exactly to 3 or exactly to 4 copies of $P$ in the tiling $\mathcal{T}(P)$.

Remark*.*

Proposition 2.1 is equivalent to the so-called Minkowski–Venkov condition on belts of parallelohedra.

Definition 2.2**.**

If a $(d-2)$-face $F$ is incident exactly to three copies of $P$ in $\mathcal{T}(P)$, then $F$ is called primitive. Otherwise $F$ is called non-primitive.

With a tiling $\mathcal{T}(P)$ we associate the dual complex $\mathcal{D}(P)$ according to the following definition.

Definition 2.3**.**

Let $G$ be a face of $\mathcal{T}(P)$ of an arbitrary dimension $0\leq k\leq d$. Denote by $D(G)$ the dual $(d-k)$-cell of $G$, which is, by definition, the centers of all translates of $P$ in $\mathcal{T}(P)$ that contain $G$ as a face. The poset of all dual cells for the faces of $\mathcal{T}(P)$ with ordering by inclusion is called the dual complex of $\mathcal{T}(P)$ and denoted by $\mathcal{D}(P)$.

Remark*.*

$\mathcal{D}(P)$ is dual to the poset of faces of $\mathcal{T}(P)$, since the map $G\mapsto D(G)$ inverts the relation of inclusion.

Remark*.*

We equip each dual cell $D(G)$ with a face structure by considering its subcells, i.e., all dual cells $D(G^{\prime})\subseteq D(G)$, where $G^{\prime}\supseteq G$. Therefore we are able to talk about the combinatorics of $D(G)$.

A classification of possible combinatorial types of dual 3-cells is available due to a classical result of Delaunay.

Proposition 2.4** (See [1]).**

Let $G$ be a $(d-3)$-dimensional face of $\mathcal{T}(P)$. Then

1. (1)

The dual 3-cell $D(G)$ is combinatorially equivalent to one of the following five 3-polytopes: a tetrahedron, an octahedron (a crosspolytope), a quadrangular pyramid, a triangular prism or a cube. 2. (2)

The combinatorics of $D(G)$ coincides with the combinatorics of the polytope $\mathrm{conv}\,D(G)$. 3. (3)

All quadrangular faces of $conv\,D(G)$ are parallelograms.

We proceed by recalling the definitions of the $\delta$- and $\pi$-surfaces associated with a parallelohedron.

Definition 2.5**.**

Let $P_{\delta}$, the $\delta$-surface of $P$, be the manifold obtained from $\partial P$, the surface of $P$, by removing all closed non-primitive $(d-2)$-faces. Also let the $\pi$-surface of $P$ be defined by $P_{\pi}:=P_{\delta}/(x\sim-x)$, i.e., $P_{\pi}$ is a quotient of $P_{\delta}$ obtained by identifying opposite points.

Definition 2.6**.**

Let $F$ and $G$ be two facets of $P$ such that the intersection $F\cap G$ is a primitive $(d-2)$-face. Define the gain function $g(F,G)$ as follows. It is said that $g(F,G)=g$ if there exists a triple $(a,b,c)$ of positive numbers such that

[TABLE]

where $P_{F}$ and $P_{G}$ are the tiles of $\mathcal{T}(P)$ that are adjacent to $P$ by the facets $F$ and $G$ respectively, and $\mathbf{e}_{P_{1},P_{2}}$ denotes the unit normal to a $(d-1)$-face $P_{1}\cap P_{2}$ directed from $P_{1}$ towards $P_{2}$.

Remark*.*

There is a unique linear dependence between $\mathbf{e}_{P,P_{F}}$, $\mathbf{e}_{P_{G},P}$ and $\mathbf{e}_{P_{F},P_{G}}$ up to a common positive multiplier, moreover, the coefficients of the dependence have equal signs. Hence $g(F,G)$ is defined in a consistent and unique way.

Definition 2.7**.**

A continuous piecewise linear map $\gamma:[0,1]\to\partial P$ is called a path on $\partial P$. We call $\gamma$ a generic path if

• •

The points $\gamma(0)$ and $\gamma(1)$ are interior points of some facets of $P$.

• •

No point $\gamma(t)$, $t\in[0,1]$, belongs to any of the $(d-3)$-faces of $P$.

• •

Whenever $\gamma$ intersects some $(d-2)$-face of $P$, the intersection is transversal.

If $\gamma([0,1])\subset P_{\delta}$, we call $\gamma$ a primitive path.

Given a generic primitive path $\gamma$ on $\partial P$, let

[TABLE]

where $[F_{0},F_{1},\ldots,F_{k}]$ is the sequence of facets visited by $\gamma$. More precisely, the sequence $[F_{0},F_{1},\ldots,F_{k}]$ satisfies the following description: the path $\gamma$ starts on the facet $F_{0}$, then goes to $F_{1}$ crossing exactly one primitive face of codimension $2$, …, and, finally, ends on $F_{k}$. Define

[TABLE]

Consider an arbitrary path (continuous piecewise linear map) $\gamma_{\pi}:[0,1]\to P_{\pi}$. Since $P_{\delta}$ is a twofold cover of $P_{\pi}$, there are two ways to lift $\gamma_{\pi}$ onto $P_{\delta}$. If $\gamma_{\delta,1},\gamma_{\delta,2}:[0,1]\to P_{\delta}$ are the paths obtained by such a lift, then both $\gamma_{\delta,1}$ and $\gamma_{\delta,2}$ are primitive. If $\gamma_{\delta,1}$ or $\gamma_{\delta,2}$ is generic, then so is the other. In this case we call $\gamma_{\pi}$ generic as well. Moreover, for generic $\gamma_{\pi}$ we have

[TABLE]

Indeed, if $\langle\gamma_{\delta,1}\rangle=[F_{0},F_{1},\ldots,F_{k}]$, then $\langle\gamma_{\delta,2}\rangle=[-F_{0},-F_{1},\ldots,-F_{k}]$, and the identity (1) follows from the fact that $g(F_{i},F_{i-1})=g(-F_{i},-F_{i-1})$. Thus it is natural to put $g(\gamma_{\pi}):=g(\gamma_{\delta,1})$.

The following criterion holds.

Proposition 2.8** (See [8], Lemma 2.6 and Theorem 4.6).**

The following conditions are equivalent for a parallelohedron $P$.

1. (1)

$P$* is affinely Voronoi.* 2. (2)

For every generic path $\gamma$ on $P_{\pi}$ which is closed, i.e., $\gamma(0)=\gamma(1)$, it holds that $g(\gamma)=1$.

It will be useful to list some particular cases of closed curves $\gamma$ on $P_{\pi}$ with $g(\gamma)=1$.

Lemma 2.9**.**

Let $\gamma$ be a generic closed path on $P_{\pi}$. Assume that $\gamma$ has a lift $\gamma_{\delta}$ onto $P_{\delta}$ satisfying any of the conditions (HB), (TC) or (O) below. Then $g(\gamma)=1$.

1. (HB)

$\langle\gamma_{\delta}\rangle=[F_{1},F_{2},F_{3},-F_{1}]$, where the facets $F_{1}$, $F_{2}$ and $F_{3}$ are parallel to some primitive $(d-2)$-face $G$ of $P$. 2. (TC)

$\langle\gamma_{\delta}\rangle=[F_{1},\ldots,F_{k},F_{1}]$, where all facets $F_{1},\ldots,F_{k}$ are distinct and share a common $(d-3)$-face $G$ of $P$. 3. (O)

$\langle\gamma_{\delta}\rangle=[F_{1},F_{2},F_{3},F_{1}]$, where

[TABLE]

and $\{\mathbf{0},\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3},\mathbf{x}_{1}+\mathbf{x}_{3}-\mathbf{x}_{2}\}$ is the vertex set of a pyramidal dual 3-cell $D(G)$ with apex $\mathbf{0}$.

Before we proceed with a proof, let us give a name to each type of closed paths mentioned in Lemma 2.9.

Definition 2.10**.**

In the notation of Lemma 2.9:

1. (1)

If the condition (HB) is satisfied, then $\gamma$ is called a half-belt circuit. 2. (2)

If the condition (TC) is satisfied, then $\gamma$ is called a trivially contractible circuit. 3. (3)

If the condition (O) is satisfied, then $\gamma$ is called an Ordine circuit.

Proof of Lemma 2.9.

Consider each condition separately.

Case (HB). See [8, Lemma 4.5].

Case (TC). See [8, Lemma 3.6].

Case (O). This case follows from the existence of a local canonical scaling around a $(d-3)$-face $G$ whose dual is combinatorially equivalent to a quadrangular pyramid. We provide the proof to make the argument self-contained.

Denote $\mathbf{x}_{0}:=\mathbf{0}$ and $\mathbf{x}_{4}:=\mathbf{x}_{1}+\mathbf{x}_{3}-\mathbf{x}_{2}$. Then

[TABLE]

is the set of all parallelohedra of $\mathcal{T}(P)$ incident to $G$. By [8, Lemma 3.7], there exist affine functions $U_{i}:\mathbb{R}^{d}\to\mathbb{R}$, $i=0,1,\ldots,4$ such that if $P+\mathbf{x}_{i}$ and $P+\mathbf{x}_{j}$ share a common facet $F_{ij}$ then $U_{i}$ and $U_{j}$ coincide on the affine hull of $F_{ij}$ and nowhere else. Define

[TABLE]

Then the following identities hold.

[TABLE]

Let us prove, the first identity of (2). The $(d-2)$-face $F_{1}\cap F_{2}$ is shared by exactly three parallelohedra of $\mathcal{T}(P)$, namely, $P$, $P+\mathbf{x}_{1}$ and $P+\mathbf{x}_{2}-\mathbf{x}_{3}=P+\mathbf{x}_{1}-\mathbf{x}_{4}$. The facet $F_{2}$ is orthogonal to the vector $\mathbf{a}_{01}$. The facet $F_{1}$ is parallel to the facet $(P+\mathbf{x}_{1})\cap(P+\mathbf{x}_{4})$ and therefore is orthogonal to the vector $\mathbf{a}_{14}$. The facet $(P+\mathbf{x}_{1})\cap(P+\mathbf{x}_{1}-\mathbf{x}_{4})$ is parallel to the facet $(P+\mathbf{x}_{4})\cap P$ and is therefore orthogonal to the vector $\mathbf{a}_{04}$. But $\mathbf{a}_{01}+\mathbf{a}_{14}-\mathbf{a}_{04}=\mathbf{0}$, hence indeed $g(F_{1},F_{2})=\frac{|\mathbf{a}_{01}|}{|\mathbf{a}_{14}|}$.

The proof of the third identity of (2) is obtained from that of the first identity by interchange $\mathbf{x}_{1}$ with $\mathbf{x}_{2}$ and $\mathbf{x}_{3}$ with $\mathbf{x}_{4}$.

Concerning the second identity of (2), the $(d-2)$-face $F_{2}\cap F_{3}$ is shared by parallelohedra $P$, $P+\mathbf{x}_{1}$ and $P+\mathbf{x}_{2}$. The normals to the faces $F_{2}$, $F_{3}$ and $(P+\mathbf{x}_{1})\cap(P+\mathbf{x}_{2})$ are, respectively, $\mathbf{a}_{01}$, $\mathbf{a}_{02}$ and $\mathbf{a}_{12}$. Since $\mathbf{a}_{01}+\mathbf{a}_{02}-\mathbf{a}_{12}=\mathbf{0}$, the second identity of (2) follows.

Finally, we have the identity

[TABLE]

The vectors $\mathbf{a}_{12}$ and $\mathbf{a}_{23}$ span a 2-dimensional space, and $\mathbf{a}_{34}$ and $\mathbf{a}_{14}$ are collinear to $\mathbf{a}_{12}$ and $\mathbf{a}_{23}$ respectively. Hence $\mathbf{a}_{12}=-\mathbf{a}_{34}$ and $\mathbf{a}_{23}=-\mathbf{a}_{14}$. In particular, $|\mathbf{a}_{23}|=|\mathbf{a}_{14}|$.

Expanding the definition of $g(\gamma)$ via (2) yields

[TABLE]

finishing the proof. ∎

To conclude this section, we reproduce the main result of [8]. See Section 4 of this paper for further discussion of the approach.

Theorem 2.11** ([8], Theorem 4.6).**

If the homology group $H_{1}(P_{\pi},\mathbb{Q})$ is generated by half-belt cycles, then $P$ is affinely Voronoi.

For the sake of brevity, we will call the condition of Theorem 2.11 the GGM condition.

3. Simplicial complex approach

In this section we propose yet another sufficient condition for a parallelohedron to satisfy the Voronoi conjecture. Both GGM and Ordine conditions are, apparently, special cases of our condition.

We will introduce the notion of a Venkov complex $Ven(P)$ associated with a parallelohedron $P$. By definition, $Ven(P)$ will be a finite homogeneous 2-dimensional simplicial complex. The name is justified by the observation that the edge structure of $Ven(P)$ coincides with that of the red Venkov graph $VG_{r}(P)$. The graph $VG_{r}(P)$ may, however, have additional isolated vertices, and the number of isolated vertices is the number of 1-dimensional summands in the representation of $P$ as a direct sum of irreducible parallelohedra, i.e. those that can not be represented as direct sum of parallelohedra of smaller dimension..

Let $\mathcal{A}$ be an arbitrary set (the alphabet) of labels, $T_{m}(\mathcal{A})$ be the set of all $m$-element subsets of $\mathcal{A}$. Every finite subset $X\subseteq T_{m}(\mathcal{A})$ defines a finite homogeneous $(m-1)$-dimensional simplicial complex $\mathcal{C}(X)$. Namely, the vertices of $\mathcal{C}(X)$ are in one-to-one correspondence with the set $\bigcup\limits_{S\in X}S$ (i.e., the set of labels that are used at least once). The facets of $\mathcal{C}(X)$ are in one-to-one correspondence with elements of $X$ so that each $S\in X$ corresponds to a facet with the vertex set labeled exactly by the elements of $S$. For our purposes we set $\mathcal{A}:=\Lambda(P)/2\Lambda(P)$, i.e., the alphabet is the set of parity classes of the lattice $\Lambda(P)$. The element $x+2\Lambda(P)\in\Lambda(P)/2\Lambda(P)$, i.e., the parity class of the lattice point $x$, will be denoted by $\bar{x}$.

It will be convenient to use a shorthand notation

[TABLE]

One can see that $\mathcal{C}(O(a,b,c,a^{\prime},b^{\prime},c^{\prime}))$ is combinatorially isomorphic to the surface of an octahedron with the pairs of opposite vertices labeled as $\{a,a^{\prime}\}$, $\{b,b^{\prime}\}$ and $\{c,c^{\prime}\}$.

Definition 3.1**.**

Let $P$ be a parallelohedron of dimension $d\geq 4$. Let $\mathcal{D}^{3}(P)$ denote the set of all dual 3-cells of the tiling $\mathcal{T}(P)$. For each $D\in\mathcal{D}^{3}(P)$ define a set $X(D)\subseteq T_{3}(\mathcal{A})$, where $\mathcal{A}:=\Lambda(P)/2\Lambda(P)$, as follows:

1. (1)

If $D$ is a combinatorial tetrahedron and $V(D)=\{a,b,c,d\}$, set

[TABLE] 2. (2)

If $D$ is a combinatorial pyramid, $V(D)=\{s,a,b,c,d\}$ and $a+c=b+d$, set

[TABLE] 3. (3)

If $D$ is a combinatorial octahedron, $V(D)=\{a,b,c,d,e,f\}$ and $a+d=b+e=c+f$, set

[TABLE] 4. (4)

If $D$ is a combinatorial prism, $V(D)=\{a,b,c,a^{\prime},b^{\prime},c^{\prime}\}$ and $a-a^{\prime}=b-b^{\prime}=c-c^{\prime}$, set

[TABLE] 5. (5)

If $D$ is a combinatorial cube, set $X(D):=\varnothing$.

Write, finally,

[TABLE]

Then the simplicial complex $Ven(P):=\mathcal{C}(X)$ is called the Venkov complex of $P$. The faces of $Ven(P)$ are called the Venkov triangles.

Remark*.*

The cases (2), (3), and (4) of dual 3-cells have certain linear relations between vertices. These relations force linear relations between parity classes as well.

For example in case (2), the relation $a+c=b+d$ implies that $\overline{a+d}=\overline{b+c}$ and $\overline{a+b}=\overline{c+d}$, so the set $X(D)$ can be written in an equivalent way as

[TABLE]

Similarly in case (3) , the relations $a+d=b+e=c+f$ between vertices of $D$ imply the following relations between parity classes: $\overline{a+b}=\overline{d+e}$, $\overline{a+c}=\overline{d+f}$, $\overline{b+c}=\overline{e+f}$, $\overline{a+e}=\overline{b+d}$, $\overline{a+f}=\overline{c+d}$, $\overline{b+f}=\overline{c+e}$.

Finally in case (4), the relations $a-a^{\prime}=b-b^{\prime}=c-c^{\prime}$ imply $\overline{a+b}=\overline{a^{\prime}+b^{\prime}}$, $\overline{a+c}=\overline{a^{\prime}+c^{\prime}}$, $\overline{b+c}=\overline{b^{\prime}+c^{\prime}}$.

In the sequel we may switch between equivalent parity classes without saying it explicitly.

For further simplicity, we identify the vertices of $Ven(P)$ with their labels.

Let us recall the definition of Venkov graphs.

Definition 3.2** **(The Venkov graph, see, for

instance, [14])).

Let $P$ be a parallelohedron. Set

[TABLE]

In other words, $V$ is the set of pairs of opposite facets of $P$. Let now $\{F,-F\}$ and $\{F^{\prime},-F^{\prime}\}$ be two distinct elements of $V$. We say that

• •

\bigl{\{}\{F,-F\},\{F^{\prime},-F^{\prime}\}\bigr{\}}\in E_{b} if $F\cap F^{\prime}$ is a non-primitive $(d-2)$-face of $\mathcal{T}(P)$.

• •

\bigl{\{}\{F,-F\},\{F^{\prime},-F^{\prime}\}\bigr{\}}\in E_{r} if either $F\cap F^{\prime}$ or $F\cap(-F^{\prime})$ is a primitive $(d-2)$-face of $\mathcal{T}(P)$.

Then $VG(P):=(V,E_{b}\cup E_{r})$ is called the Venkov graph of $P$ and $VG_{r}(P):=(V,E_{r})$ (respectively, $VG_{b}(P):=(V,E_{b})$) is the red (respectively, blue) Venkov graph of $P$.

The next definition establishes a correspondence between the Venkov complex and the Venkov graph of a parallelohedron.

Definition 3.3**.**

Given a parallelohedron $P$ of dimension $d\geq 4$, let a map

[TABLE]

be defined as follows. For each $x\in vert(Ven(P))$ (and thus satisfying $x\in\Lambda(P)/2\Lambda(P)$) we set $\varphi(x):=\{F,-F\}$ if $F=P\cap(P+a)$, where $\overline{a}=\xi$.

Remark*.*

Each parity class of $\Lambda(P)$ (i.e., a coset in $\Lambda(P)/2\Lambda(P)$) contains either two opposite facet vectors of $P$, or does not contain facet vectors. From the definition of $Ven(P)$ it follows immediately that each parity class $\xi\in vert(Ven(P))$ contains exactly two facet vectors, which, in turn, define the pair $(F,-F)$ uniquely.

Lemma 3.4**.**

The map $\varphi$ from Definition 3.3 has the following properties.

1. (1)

$\varphi$* is injective.* 2. (2)

$\varphi$* induces a bijection between the edge sets of $VG_{r}(P)$ and $Ven(P)$.*

Proof.

Assertion (1) holds since distinct parity classes of $\Lambda(P)$ determine distinct pairs of opposite facets.

Assertion (2) is proved by verifying the properties (a) and (b) below.

(a) If $\{x,y\}$ is an edge of $Ven(P)$ then $\{\varphi(x),\varphi(y)\}$ is an edge of $VG_{r}(P)$. This property follows immediately from the definition of $Ven(P)$.

(b) If \bigl{\{}\{F,-F\},\{F^{\prime},-F^{\prime}\}\bigr{\}} is an edge of $VG_{r}(P)$, then $\varphi^{-1}(\{F,-F\})$ and $\varphi^{-1}(\{F^{\prime},-F^{\prime}\})$ exist. Moreover, they are connected with an edge of $Ven(P)$. In order to verify this property, assume, with no loss of generality, that $F\cap F^{\prime}$ is a primitive $(d-2)$-face of $P$. Then the property (b) is immediate by considering the set of triples $X(D(G))$, where $G$ is an arbitrary $(d-3)$-subface of $F\cap F^{\prime}$, and $D(G)$ is the dual 3-cell of $G$. ∎

By the following corollary, the Venkov complex is, in a sense, a 2-dimensional extension of the red Venkov graph, which justifies our terminology.

Corollary 3.5**.**

Let $P$ be a parallelohedron of dimension $d\geq 4$. Then red Venkov graph $VG_{r}(P)$ can be obtained by adding a finite number (possibly, zero) of isolated vertices to the 1-dimensional skeleton of $Ven(P)$.

Remark*.*

The number of additional isolated vertices in $VG_{r}(P)$ equals the number of 1-dimensional summands in the representation of $P$ as a direct sum of irreducible parallelohedra, see [14].

The next lemma explains our choice of the 2-dimensional face structure for the Venkov complex.

Lemma 3.6**.**

Let a triple $\{x_{1},x_{2},x_{3}\}\in T_{3}(\Lambda(P)/2\Lambda(P))$ span a 2-dimensional face of $Ven(P)$. For $i=1,2,3$ let $\varphi^{-1}(x_{i})=\{F_{i},-F_{i}\}$. Then

1. (1)

There exists a generic closed path $\gamma$ on $P_{\pi}$, whose lift $\gamma_{\delta}$ onto $P_{\delta}$ satisfies

[TABLE]

for an appropriate choice of signs. 2. (2)

Such a path $\gamma$ is either half-belt, or trivially contractible, or Ordine.

Proof.

Assertion (1). There exists $\tilde{F_{2}}\in\{F_{2},-F_{2}\}$ which is adjacent to $F_{1}$ by a primitive $(d-2)$-face. Similarly, choose $\tilde{F_{3}}\in\{F_{3},-F_{3}\}$ adjacent to $\tilde{F_{2}}$ and $\tilde{F_{1}}\in\{F_{1},-F_{1}\}$ adjacent to $\tilde{F_{3}}$. Then it is possible to construct $\gamma_{\delta}$ so that $\langle\gamma_{\delta}\rangle=[F_{1},\tilde{F_{2}},\tilde{F_{3}},\tilde{F_{1}}]$ and the image of $\gamma_{\delta}$ under the natural projection $P_{\delta}\to P_{\pi}$ is a closed path $\gamma$.

Assertion (2). By definition of $Ven(P)$, we have $\{x_{1},x_{2},x_{3}\}\in X(D)$, where $D$ is a dual 3-cell for $\mathcal{T}(P)$. In particular $X(D)\neq\varnothing$, hence $D$ cannot be a combinatorial cube. The rest follows from the definition of $Ven(P)$.

Indeed, if $D$ is a combinatorial tetrahedron or a combinatorial octahedron, then $\gamma$ is either a half-belt cycle, or a trivially contractible cycle. If $D$ is a combinatorial quadrangular pyramid, then $\gamma$ is either a half-belt cycle, or an Ordine cycle. Finally, if $D$ is a combinatorial prism, then $\gamma$ is a half-belt cycle. ∎

We proceed with the main result of this section — a sufficient condition for a parallelohedron to be affinely Voronoi in terms of its Venkov complex. From now on we use the notation $C_{k}(K,R)$ (respectively, $C^{k}(K,R)$) for the spaces of chains (respectively, cochains) of a simplicial complex $K$ with coefficients in a commutative ring $R$.

Theorem 3.7**.**

Let $P$ be a parallelohedron of dimension $d\geq 4$. If the first cohomology group $H^{1}(Ven(P),\mathbb{R})$ is trivial, then $P$ is affinely Voronoi.

Proof.

Assume that $P$ is not affinely Voronoi. It suffices to construct a non-trivial cohomology class in $H^{1}(Ven(P),\mathbb{R})$. Equivalently, we will construct a cochain $c\in C^{1}(Ven(P),\mathbb{R})$ such that the coboundary operator $\delta$ vanishes on $c$, but at the same time $c$ is not a coboundary itself (i.e., $c\neq\delta c^{\prime}$ for any $c^{\prime}\in C^{0}(Ven(P),\mathbb{R})$).

Let $x_{1},x_{2}\in vert(Ven(P))$ be such that $\{x_{1},x_{2}\}$ is an edge of $Ven(P)$. For $i=1,2$ let $\{F_{i},-F_{i}\}=\varphi(x_{i})$. Then $F_{1}$ is adjacent either to $F_{2}$, or to $-F_{2}$ by a primitive $(d-2)$-face of $P$. We then set

[TABLE]

respectively.

Let us prove that $c$ is a cocycle. Consider an arbitrary 2-dimensional face $\{x_{1},x_{2},x_{3}\}$ of $Ven(P)$. Applying Lemma 3.6 to $\{x_{1},x_{2},x_{3}\}$, yields a generic closed path $\gamma$ on $P_{\pi}$. We have

[TABLE]

where the last identity is a consequence of Lemma 2.9. Since $\delta c$ vanishes on every 2-face of $Ven(P)$, $c$ is indeed a cocycle.

In turn, $c$ is not a coboundary. In order to prove that we assume the converse, i.e., $c=\delta c^{\prime}$, where $c^{\prime}$ is a [math]-cochain.

By Proposition 2.8, there exists a closed path $\gamma$ on $P_{\pi}$ such that $g(\gamma)\neq 1$. Let $\gamma_{\delta}$ be a lift of $\gamma$ onto $P_{\delta}$. Suppose that $\langle\gamma_{\delta}\rangle=[F_{1},F_{2},\ldots,F_{k}]$. Then

[TABLE]

is a cycle in $VG_{r}(P)$. Therefore $x_{i}:=\varphi^{-1}(\{F_{i},-F_{i}\})$ exists for each $i=1,2,\ldots,k$, and $x_{1}=x_{k}$. Consequently,

[TABLE]

On the other hand,

[TABLE]

A contradiction shows that indeed $c$ is not a coboundary. This concludes the proof. ∎

4. Graph approach

In this section we present a method to verify the GGM condition. This approach, using the group of cycles of the red Venkov graph $VG_{r}(P)$, was proposed in [9].

The two definitions below enable us to use the relation between the topology of $P_{\pi}$ on one side, and the topology of $VG_{r}(P)$ and $Ven(P)$ on the other side.

Definition 4.1**.**

Let $\gamma_{\pi}$ be a generic closed path on $P_{\pi}$. Let $\gamma_{\pi}$ be lifted onto $P_{\delta}$ as $\gamma_{\delta}$, and let

[TABLE]

Consider the two cases

1. (1)

Let $\gamma_{\delta}$ pass through at least two different facets of $P$. In this case $x_{i}:=\varphi^{-1}(\{F_{i},-F_{i}\})$ are well-defined for all $i=1,2,\ldots,k$. Then we say that $\gamma_{\pi}$ is identified with the cycles

[TABLE]

of the red Venkov graph $VG_{r}(P)$ and the Venkov complex $Ven(P)$ respectively. 2. (2)

If $\gamma_{\delta}$ is contained in a single facet of $P$, we say that $\gamma_{\pi}$ is identified with the empty cycle of $VG_{r}(P)$ (or $Ven(P)$).

Remark*.*

Conversely, consider an arbitrary cycle $[x_{1},x_{2},\ldots,x_{k-1},x_{k}=x_{1}]$, where $k>2$, $x_{i}\in vert(Ven(P))$ and $\overrightarrow{x_{i}x_{i+1}}$ is an oriented edge of $Ven(P)$. Then there exists a closed generic path on $P_{\pi}$ identified with this cycle.

Definition 4.2**.**

A cycle $c\in C_{1}(Ven(P))$ is called a combinatorial half-belt cycle (respectively, combinatorial trivially contractible cycle  or combinatorial Ordine cycle) if it is identified with a half-belt (respectively, trivially contractible or Ordine) cycle $\gamma$ on $P_{\pi}$.

Now we are ready to reformulate the GGM condition in terms of the Venkov complex.

Lemma 4.3**.**

The following assertions are equivalent.

1. (i)

The group $H_{1}(P_{\pi},\mathbb{Q})$ is generated by half-belt cycles. 2. (ii)

The implication \bigl{(}A_{1}(c)\land A_{2}(c)\bigr{)}\Rightarrow B(c) holds for all cochains $c\in C^{1}(Ven(P),\mathbb{Q})$, where

[TABLE]

Remark*.*

The condition $B(c)$ is equivalent to the statement that $\langle c,\gamma\rangle=0$ for every 1-cycle $\gamma$, i.e., for every $\gamma\in C_{1}(Ven(P),\mathbb{Q})$ satisfying $\partial\gamma=0$.

Proof of Lemma 4.3.

(i) $\Rightarrow$ (ii). Let $P$ be a parallelohedron $P$ satisfying (i). Consider an arbitraty cochain $c\in C^{1}(Ven(P),\mathbb{Q})$ for which both $A_{1}(c)$ and $A_{2}(c)$ are true. We claim that $B(c)$ is true as well.

Let $\gamma_{\pi}\subset P_{\pi}$ be a closed generic curve. By Definition 4.1, $\gamma_{\pi}$ is identified with a cycle $\gamma\in C_{1}(Ven(P),\mathbb{Q})$. We then set

[TABLE]

Since $A_{2}(c)$ holds, the value of $c^{*}(\gamma_{\pi})$ depends only on the homptopy type of $\gamma_{\pi}$. Therefore $c^{*}$ acts as a map $c^{*}:\pi_{1}(P_{\pi})\to\mathbb{Q}$. By construction $c^{*}$ is a homomorphism, vanishes on the commutant of $\pi_{1}(P_{\pi})$ and its image lies in the field $\mathbb{Q}$ of characteristic zero. Consequently, the action of $c^{*}$ on the group $H_{1}(P_{\pi},\mathbb{Q})$ is also well-defined. By the property $A_{1}(c)$, all half-belt cycles lie in the kernel of $c^{*}$. Using (i), we conclude that $c^{*}$ acts on $H_{1}(P_{\pi},\mathbb{Q})$ trivially, which is only possible if $c$ is a coboundary. Hence the implication (i) $\Rightarrow$ (ii) holds.

(ii) $\Rightarrow$ (i). Assume that (i) is false for $P$. Let $G$ be the proper subgroup of $H_{1}(P_{\pi},\mathbb{Q})$ generated by half-belt cycles. Let $\ell:H_{1}(P_{\pi},\mathbb{Q})\to\mathbb{Q}$ be a linear map such that $G\subseteq Ker\,\ell\subsetneq H_{1}(P_{\pi},\mathbb{Q})$.

Denote

[TABLE]

Let $\gamma\in\bar{C}_{1}(Ven(P),\mathbb{Q})$. Then there exists $n\in\mathbb{N}$ such that $n\gamma=\sum\limits_{i=1}^{s}\gamma^{i}$, where $\gamma^{i}=[x^{i}_{1},x^{i}_{2},\ldots,x^{i}_{k_{i}}=x^{i}_{1}]$. For each $\gamma^{i}$ there exists a closed generic path $\gamma^{i}_{\pi}$ identified with it according to Definition 4.1. By setting

[TABLE]

where $h_{i}$ is the homology class of $\gamma^{i}_{\pi}$, we obtain a linear map $\ell_{*}:\bar{C}_{1}(Ven(P),\mathbb{Q})\to\mathbb{Q}$. Let $c\in C^{1}(Ven(P),\mathbb{Q})$ be an arbitrary continuation of $\ell_{*}$ onto $C_{1}(Ven(P),\mathbb{Q})$. Clearly, $c$ satisfies conditions $A_{1}(c)$ and $A_{2}(c)$, but not $B(c)$. ∎

The second condition of the previous lemma can be stated in terms of the group of cycles of the red Venkov graph $VG_{r}(P)$. Recall that, by Corollary 3.5, the edge structure of $VG_{r}(P)$ and the edge structure of the Venkov complex $Ven(P)$ are isomorphic.

Definition 4.4**.**

A cycle of $VG_{r}(P)$ identified either with a half-belt or with a trivially contractible cycle on $P_{\pi}$ is called a basic cycle. The set of all basic cycles is denoted by $\mathcal{C}(P)$.

Definition 4.5**.**

If we treat a finite (non-directed) graph $G$ as a one-dimensional simplicial complex, then the group $H_{1}(G,\mathbb{Q})$ is called the group of cycles of $G$.

Remark*.*

The group of cycles of $G$ is a free abelian group (or a linear space over $\mathbb{Q}$) of rank $e-v+k$ where $e$ is the number of edges, $v$ is the number of vertices, and $k$ is the number of connected components of $G$.

Then we can reformulate Lemma 4.3 in the following way.

Lemma 4.6**.**

The group $H_{1}(P_{\pi},\mathbb{Q})$ is generated by half-belt cycles if and only if the group of cycles of the red Venkov graph of $P$ is generated by $\mathcal{C}(P)$.

Proof.

Since the second condition of Lemma 4.3 does not use two-dimensional simplices of $Ven(P)$, we can substitute $Ven(P)$ with the red Venkov graph in it. Then the implication \bigl{(}A_{1}(c)\land A_{2}(c)\bigr{)}\Rightarrow B(c) means that the rank of the subgroup generated by $\mathcal{C}(P)$ is equal to the rank of the group of cycles of $VG_{r}(P)$, and $\mathcal{C}(P)$ generates the group of cycles. ∎

5. Computational results

This section describes the computer-assisted verification of the following two results.

Theorem 5.1**.**

Let a 5-dimensional parallelohedron $P$ be equivalent to some 5-dimensional Voronoi parallelohedron. Then the cohomology group $H^{1}(Ven(P),\mathbb{R})$ is trivial.

Theorem 5.2**.**

Let a 5-dimensional parallelohedron $P$ be equivalent to some 5-dimensional Voronoi parallelohedron. Then the GGM condition holds for $P$.

Theorem 1.3 is an immediate corollary of each of the above theorems, as explained below.

Proof of Theorem 1.3.

The “only if” part is straightforward.

The “if” part is a combination of either Theorems 5.1 and 3.7, or of Theorems 5.2 and 2.11. ∎

It is sufficient to verify the conclusions of Theorems 5.1 and 5.2 for a single representative of each equivalence class of 5-dimensional Voronoi parallelohedra.

The list of representatives is available due to the algorithm of [3]. Each representative $P_{i}$ ($1\leq i\leq 110\,244$) is presented in two equivalent ways:

1. (1)

As a cell of $\mathbf{0}$ in the Voronoi tessellation for $\mathbb{Z}^{5}$, where the metric is given by an explicit quadratic form $Q_{i}$, i.e., $\|a\|=Q(a,a)$. $Q_{i}$ is presented by its $5\times 5$ matrix with integer entries. 2. (2)

As a convex hull of a set of vertices given explicitly by listing the coordinates. All coordinates are rational numbers. Additionally, every face of $P_{i}$ is described by listing its vertices.

Our first goal is to compute the dual complex $\mathcal{D}(P_{i})$, which is, in this case, a Delaunay tessellation $\mathcal{D}(\mathbb{Z}^{5},Q_{i})$. We use two different approaches. The direct approach uses the algorithm of [4], which is available in [2]. The second approach uses the following proposition.

Proposition 5.3** (See, for instance, [14]).**

The following assertions hold.

1. (1)

Let $G$ be a face of parallelohedron $P$. Then

[TABLE] 2. (2)

$D$* is a dual $k$-cell of $\mathcal{D}(P)$ if and only if $D=D(G)+v$, where $G$ is a $(d-k)$-face of $P$ and $v\in\Lambda(P)$.*

Using Proposition 5.3, one can compute $\mathcal{D}(P_{i})$ from the vertex presentation of $P_{i}$. All computations are performed over the field of rationals, therefore we are not concerned about the issues with machine precision.

Now we proceed with Theorems 5.1 and 5.2 separately.

For Theorem 5.1, we use $\mathcal{D}(P_{i})$ to construct the simplicial complex $Ven(P_{i})$. After that, we check the triviality of $H^{1}(Ven(P_{i}),\mathbb{R})$ by verifying the identity

[TABLE]

where $\delta_{0}$ and $\delta_{1}$ are restrictions of the coboundary operator $\delta$ to the spaces $C^{0}(Ven(P_{i}),\mathbb{R})$ and $C^{1}(Ven(P),\mathbb{R})$, respectively.

But $\dim(Im\,\delta_{0})=rank(\delta_{0})$ and $\dim(Ker\,\delta_{1})=f_{1}(Ven(P_{i}))-rank(\delta_{1})$. Therefore the condition of Theorem 3.7 is equivalent to the identity

[TABLE]

The identity (4) is verified by passing $Ven(P_{i})$ to the GAP package [7, 5], where the functions on the left-hand side of (4) are readily available. The scripts that process the vertex representation of five-dimensional parallelohedra into a GAP program are available on the web-page [11].

Similarly, for Theorem 5.2, the dual complex $\mathcal{D}(P_{i})$ is used to construct the graph $VG_{r}(P_{i})$ and to determine the half-belt and the trivially contractible cycles.

The group of cycles of $VG_{r}(P_{i})$ has rank $e-v+k$, where $e$ is the number of edges, $v$ is the number of vertices and $k$ is the number of connected components. Therefore, by Lemma 4.6, it is sufficient to verify that the $\mathbb{Q}$-rank of the set $\mathcal{C}(P_{i})$ equals $e-v+k$. This is done by a straightforward computation.

6. Relations between sufficient conditions

In this section we show that the cohomology condition generalizes both Ordine’s 3-irreducibility condition and the GGM condition.

First we deal with the Ordine condition. Recall that a parallellohedron $P$ is 3-irreducible if no 3-cell of the dual complex $\mathcal{D}(P)$ is equivalent to a prism or a cube. Ordine [14] proved that every 3-irreducible parallelohedron of dimension $\geq 5$ is affinely Voronoi.

Lemma 6.1**.**

Let $P$ be a 3-irreducible parallelohedron of dimension $d\geq 5$. Then the group $H^{1}(Ven(P),\mathbb{R})$ is trivial.

Proof.

Let $c\in C^{1}(Ven(P),\mathbb{R})$ be a cocycle. We need to prove that $c$ is a coboundary.

Let $F$ anf $F^{\prime}$ be two facets of $P$ such that $F\cap F^{\prime}$ is a primitive $(d-2)$-face. Then there is an edge between $\{F,-F\}$ and $\{F^{\prime},-F^{\prime}\}$ in $VG_{r}(P)$. Consequently, $\overrightarrow{xx^{\prime}}$, where $x:=\varphi^{-1}(\{F,-F\})$ and $x^{\prime}:=\varphi^{-1}(\{F^{\prime},-F^{\prime}\})$, is an oriented edge of $Ven(P)$.

Set

[TABLE]

One can check that plugging $g$ into the argument of [14] instead of the gain function for $P$ is sufficient to produce an analogue of a canonical scaling. More precisely, there exists a function $s$ mapping facets of $P$ to positive real numbers such that the identities

[TABLE]

are satisfied.

Letting $c^{\prime}(x):=\ln s(F)$, where $F\in\varphi(x)$, yields $c=\delta c^{\prime}$. Hence $c$ is indeed a coboundary. ∎

We proceed by considering the GGM condition.

Definition 6.2**.**

A 2-dimensional face of the Venkov complex $Ven(P)$ is called a Venkov triangle.

Lemma 6.3**.**

The following assertions hold.

1. (i)

Each combinatorial half-belt cycle is a boundary of a Venkov triangle. 2. (ii)

Each combinatorial trivially contractible cycle is an integer combination of boundaries of Venkov triangles.

Proof.

Assertion (i). Let $[x_{1},x_{2},x_{3},x_{1}]$ be a combinatorial half-belt cycle. Then one can choose two facets $F_{1}\in\varphi(x_{1})$ and $F_{2}\in\varphi(x_{2})$ so that $F_{1}\cap F_{2}$ is a primitive $(d-2)$-face of $P$. Choose an arbitrary $(d-3)$-face $G\subset F_{1}\cap F_{2}$. If $D$ is the dual 3-cell of $G$, then $\{x_{1},x_{2},x_{3}\}\in X(D)$, finishing the proof of the assertion.

Assertion (ii). Let $\gamma$ be a combinatorial trivially contractible cycle around a $(d-3)$-face $G$. Let $D$ be the dual 3-cell of $G$. There are three possibilities for the combinatorial type of $D$ — a tetraheron, a crosspolytope and a quadrangular pyramid. If $D$ is a tetrahedron or a crosspolytope, then $\gamma=[x_{1},x_{2},x_{3},x_{1}]$, where $\{x_{1},x_{2},x_{3}\}\in X(D)$. Hence $\gamma$ is itself a boundary of a Venkov triangle. If $D$ is a quadrangular pyramid, then $\gamma=[x_{1},x_{2},x_{3},x_{4},x_{1}]$ and goes along an equator of the octahedron $\mathcal{C}(X(D))$. Therefore $\gamma$ is a sum of boundaries of four Venkov triangles comprising one of the hemispheres of $\mathcal{C}(X(D))$. ∎

Lemma 6.4**.**

Let assertion (ii) of Lemma 4.3 hold. Then the group $H^{1}(Ven(P),\mathbb{R})$ is trivial.

Proof.

Note that the groups $H^{1}(Ven(P),\mathbb{R})$ and $H^{1}(Ven(P),\mathbb{Q})$ are either both trivial, or both non-trivial. Therefore for the rest of the proof we will be working over the field of rationals.

Let $c\in C^{1}(Ven(P),\mathbb{R})$ be a 1-cocycle, i.e., $\delta c\equiv 0$. Equivalently, we have

[TABLE]

for every Venkov triangle $\tau$.

By Lemma 6.3, every half-belt cycle and every trivially contractible cycle can be represented as a combination of boundaries of Venkov triangles. Hence, in the notation of Lemma 4.3, $A_{1}(c)$ and $A_{2}(c)$ hold. Therefore $B(c)$ holds as well, i.e., $c$ is a coboundary.

We thus conclude that every cocycle in $C^{1}(Ven(P),\mathbb{Q})$ is a coboundary. Hence the group $H^{1}(Ven(P),\mathbb{Q})$ is trivial, and so is the group $H^{1}(Ven(P),\mathbb{R})$. ∎

7. Concluding remarks

We conclude the paper by posing an open problem.

Problem 7.1**.**

Determine whether the following statements are true or false.

1. (1)

The GGM condition holds for all Voronoi parallelohedra. 2. (2)

For every Voronoi parallelohedron $P$ of dimension $d\geq 5$ the cohomology group $H^{1}(Ven(P),\mathbb{R})$ is trivial.

If any of the statements of Problem 7.1 holds, then a hypothetical counterexample to the Voronoi conjecture should be non-equivalent to any Voronoi parallelohedron.

In particular, for five-dimensional parallelohedra both these conditions are equivalent to the Voronoi conjecture, so if one wants to find a five-dimensional counterexample for the Voronoi conjecture, then it is enough to search for a parallelohedron that violates any of these conditions. So this poses one more open question.

Problem 7.2**.**

Is every five-dimensional parallelohedron combinatorially Voronoi?