# On the First Page of the Renaudineau-Shaw Spectral Sequence

Jules Chenal

► To cite this version:

Jules Chenal. On the First Page of the Renaudineau-Shaw Spectral Sequence. 2024. hal-04520683

HAL Id: hal-04520683

https://hal.science/hal-04520683

Preprint submitted on 25 Mar 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
ON THE FIRST PAGE OF THE RENAUDINEAU-SHAW
SPECTRAL SEQUENCE

Jules Chenal*

Laboratoire Paul Painlevé, UMR8524

Université de Lille

March 2024

**Abstract**

A T-hypersurface is a combinatorial hypersurface of the real locus of a projective toric variety $Y$. It is constructed from a primitive triangulation $K$ of a moment polytope $P$ of $Y$ and a 0-cochain $\varepsilon$ on $K$ with coefficients in the field with two elements $\mathbb{F}_2$, called a *sign distribution*. O. Viro showed that when $K$ is convex the T-hypersurface is ambiantly isotopic to a real algebraic hypersurface of $Y$. A. Renaudineau and K. Shaw gave upper bounds on the Betti numbers of T-hypersurfaces in terms of the Hodge numbers of a generic section of the ample line bundle $L$ associated with the moment polytope. In particular, the number of connected components of a T-hypersurface cannot exceed the geometric genus of a generic section of $L$ plus one. In this article we investigate whether this upper bound is attainable. We are able to characterise the couples $(K; \varepsilon)$ leading to T-hypersurfaces realising the Renaudineau-Shaw upper bound on the number of connected components. This theorem generalises B. Haas' theorem for T-curves. In contrast with this results we find that the upper bound is not always attainable on every primitive triangulations. For some of those on which it is not attainable we provide a sharper upper bound. Finally we use our characterisation to show that there always exist a triangulation and a sign distribution on the standard simplex that reach the Renaudineau-Shaw upper bound. We also study the growth of the expected number of connected components of a T-hypersurface as we dilate the moment polytope by $d$ (i.e. we tensorise the line bundle $d$-times with itself) and show that it is always of the order of $d^n$ where $n$ is the dimension of $P$.

**Acknowledgements**

This research was funded, in whole or in part, by l'Agence Nationale de la Recherche (ANR), project ANR-22-CE40-0014. For the purpose of open access, the author has applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

<footnote>*jules.chenal@univ-lille.fr</footnote>
# Introduction

This article is about the first page of the Renaudineau-Shaw spectral sequence of $T$-hypersurfaces. In particular we compute explicitly its boundary operators. These computations lead us to a generalisation of B. Haas' theorem<fnref n="1" />. This result is a characterisation of the triangulations and sign distributions that produce $T$-hypersurfaces having a maximal number of connected components with respect to the Renaudineau-Shaw inequality<fnref n="2" />. Recall that for all $T$-hypersurfaces $\mathbb{R}X_\epsilon$, A. Renaudineau and K. Shaw showed the following upper bound:

$$ \dim H_0(\mathbb{R}X_\epsilon; \mathbb{F}_2) \le 1 + h^{n-1,0}(X), \quad (1) $$

where $h^{n-1,0}(X)$ is the number of interior lattice points of $P$ the moment polytope of the ambient toric variety. In the following we will say that $\mathbb{R}X_\epsilon$ has a maximal number of connected components when the upper bound (1) is achieved. In order to state our result we define the notion of $\rho$-uniformity for primitive triangulations. It is a technical property involving the relative parities of the coordinates of the vertices of two maximal simplices of the triangulation sharing a face of codimension 1.

**Theorem 4.15 (Generalised Haas Theorem).** Let $P$ be a non-singular polytope of $t^*(\mathbb{R})$ endowed with a primitive triangulation $K$. For all sign distributions $\epsilon$, the $T$-hypersurface $\mathbb{R}X_\epsilon$ has a maximal number of connected components if and only if:

1. The triangulation $K$ is $\rho$-uniform;

2. The homological “inclusion” $H_1(\mathbb{R}X_\epsilon; \mathbb{F}_2) \to H_1(\mathbb{R}P; \mathbb{F}_2)$ is onto whenever $n > 2$;

3. The sign distribution $\epsilon$ satisfies:

$$ D^2\epsilon = \rho \text{ in } H_{n-1}(K; \mathcal{B}_1K; N). \quad (2) $$

4. For all simplices $\sigma^{n-2}$ of $\mathcal{B}_1K$ not contained in $\mathcal{B}_0K$, $(\mathrm{ap}(\sigma^{n-2}); \sigma^{n-2})_\epsilon$ vanishes.

A 2-dimensional triangulation is always $\rho$-uniform, hence this new hypothesis (relatively to Haas' statement) is automatically satisfied for $T$-curves. However it is not always the case in dimension bigger than 3. We can also note that since all $T$-surfaces on the same triangulation share the same Euler characteristic<fnref n="3" /> a $T$-surface is maximal as a real variety if and only if it has a maximal number of connected components with respect to the Renaudineau-Shaw inequality. In addition we define a number $\kappa(K)$ which measure the defect of $\rho$-uniformity of the triangulation $K$ and use this quantity to refine the Renaudineau-Shaw inequality on the number of connected components of $\mathbb{R}X_\epsilon$.

**Proposition 4.22.** Let $K$ be a primitive triangulation of a non-singular polytope. For all sign distributions $\epsilon \in C^0(K; \mathbb{F}_2)$ the rank of $M_{0,n-2}(\epsilon)$ is always at least equal to $\kappa(K)$. As a consequence we have the following upper bound:

$$ \dim H_0(\mathbb{R}X_\epsilon; \mathbb{F}_2) \le 1 + \dim H_{0,n-1}(X; \mathbb{F}_2) - \kappa(K). \quad (3) $$

<footnote n="1">¹c.f. [Haa97] or [RS23] Theorem 7.5 p.977.</footnote>

<footnote n="2">²c.f. [RS23] Theorem 1.4 p.947.</footnote>

<footnote n="3">³c.f. [Ber10].</footnote>
Moreover we use our computations to determine the asymptotic growth of the expected number of connected components of $\mathbb{R}X_\varepsilon$ when one chooses the sign distribution uniformly at random and considers dilatations of the moment polytope. More precisely we consider $P$ a non-singular lattice polytope and $(K_d)_{d\ge 1}$ a sequence of primitive triangulations of the dilates $dP$. Then we show that if $\varepsilon$ is chosen uniformly at random in $C^0(K_d; \mathbb{F}_2)$ we have the following property.

**Proposition 4.25.** For all sequences $(K_d)_{d\ge 1}$ of primitive triangulations of $(dP)_{d\ge 1}$ we have:

$$1 \ge \limsup_{d \to +\infty} \mathbb{E} \left[ \frac{b_0(\mathbb{R}X_\varepsilon(d))}{1 + h^{n-1,0}(X(d))} \right] \ge \liminf_{d \to +\infty} \mathbb{E} \left[ \frac{b_0(\mathbb{R}X_\varepsilon(d))}{1 + h^{n-1,0}(X(d))} \right] > 0.$$

Recall that $h^{n-1,0}(X(d)) = \text{vol}_{\mathbb{Z}}(P)d^n + o(d^n)$.

To conclude we provide examples of triangulations of the standard simplices and sign distributions for which the Renaudineau-Shaw upper bound is achieved. To do so we found inspiration in the construction of maximal T-surfaces in $\mathbb{P}^3$ by I. Itenberg, [Ite97].

**Theorem 0.1.** For all $n \ge 2$, and all $d \ge 1$, the Itenberg-Viro triangulation $IV_d^n$ is simply integrable. In particular, the Harnack distribution:

$$h^n(x_1, \dots, x_n) := \sum_{i<j} x_i x_j,$$

produces a projective T-hypersurfaces made of $h^{n-1,0}(X)$ spheres bounding disjoint balls and another connected component that intersects every toric divisors of $\mathbb{P}^n$.

# Contents

<table><tr><td>1</td><td>T-hypersurfaces and Tools to Study Them</td><td>3</td></tr><tr><td>2</td><td>Differential Geometry of Primitive Complexes</td><td>12</td></tr><tr><td>3</td><td>Computing the Renaudineau-Shaw Spectral Sequence</td><td>20</td></tr><tr><td>4</td><td>The First and Last Boundaries</td><td>28</td></tr><tr><td>5</td><td>Examples and Non-Examples</td><td>45</td></tr></table>

## 1 T-hypersurfaces and Tools to Study Them

In this first section we set some notations, remind some definitions and introduce the construction of T-hypersurfaces and their Renaudineau-Shaw spectral sequence.

**Simplicial Complexes and Cellular Homology** We recall that a finite simplicial complex $K$ is the data of a set of vertices $V$ and a set of simplices $S$ of subsets of $V$ that contains every
singleton of V and is closed under the operation of taking non-empty<fnref n="4" /> subsets. The latter condition means that if $\sigma$ belongs to $S$ and $\tau$ is a non-empty subset of $\sigma$ then it also belongs to $S$. The dimension of a simplex $\sigma$ of $K$ is by definition one less than its number of elements. We will never mention the sets $V$ or $S$ but, for all non-negative integer $p$, we will denote $\sigma^p \in K$ to indicate that $\sigma^p$ is a $p$-dimensional simplex of $K$ (also called a $p$-simplex of $K$). Also we will make no difference between a vertex $v$ and the 0-simplex $\{v\}$ it defines.

Every finite simplicial complex defines a regular CW-complex<fnref n="5" /> called is *geometric realisation*. Let $C$ be the convex hull of the vertices of $K$ in the real vector space freely spanned by said vertices. The geometric realisation of $K$ is the union of all the faces of $K$ generated by the vertices of the simplices of $K$. This is an embedding of the set of simplices of $K$ into the lattice of faces of $C$, we will not really make a difference between the abstract complex $K$ and its geometric realisation. We indicate by $\sigma^p \le \sigma^q$ that $\sigma^p$ is a face of $\sigma^q$, i.e. that the former is included in the latter when seen as a set. The cells of the geometric realisation of $K$ are thought to be open hence the simplex $\sigma^p = \{v_0; \dots; v_p\}$ correspond to:

$$ \left\{ \sum_{k=0}^{p} t_k v_k \mid \forall 0 \le k \le p, t_k > 0, \text{ and } \sum_{k=0}^{p} t_k = 1 \right\}. $$

The *star* of a simplex $\sigma^p$ is the reunion of all the simplices of which it is a face. Its *link* (denoted by $lk(\sigma^p; K)$) is the reunion of all the faces $\sigma^q$ of the members of its star whose closure $\bar{\sigma}^q$ has empty intersection with $\bar{\sigma}^p$. These concepts extends to regular CW-complexes.

<figure><img src="image_2.png" /><figcaption>Figure 1: An open star and a link.</figcaption></figure>

**Definition 1.1.** A $\Delta$-complex is a regular CW-complex in which every closed cell is isomorphic (as a CW-complex) to a simplex.

**Definition 1.2** (Barycentric and Cubical Subdivision). Let $K$ be a finite $\Delta$-complex. Its *barycentric subdivision*, sometimes denoted $Sd K$ is the simplicial complex:

- whose vertices are the simplices of $K$;

- whose simplices are the set of simplices $\{\sigma^{p_0} < \dots < \sigma^{p_k}\}$ of $K$ that can be totally ordered by adjacency.

<footnote n="4">⁴We adopt this convention as we will never have to consider the empty simplex.</footnote>

<footnote n="5">⁵A CW-complex in which the pair made of a cell and its closure is homeomorphic to an open ball in its closure.</footnote>
The geometric realisation of SdK is a regular subdivision of K. The cubical subdivision of K is a regular CW-complex subdivision of K that is coarser than its barycentric subdivision. An open cell of this subdivision is indexed by a pair of simplices [σ^p; σ^q] where σ^p is a face of σ^q. This particular cell has dimension q - p and is the reunion of the barycentric simplices {σ^p_0 < ... < σ^p_k} for which σ^p_0 is σ^p and σ^p_k is σ^q. Every closed k-dimensional cubical cell has the combinatorics of a k-dimensional cube triangulated into k! simplices of dimension k.

<figure><img src="image_3.png" /><figcaption>(a) The barycentric subdivision of a triangle.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(b) The cubical subdivision of a triangle.</figcaption></figure>

<caption>Figure 2: Subdivisions of a triangle.</caption>

**Definition 1.3** (Cellular Cosheaves and Cellular Sheaves). Let $E$ be a regular CW-complex, and denote by **Cell** $E$ the category whose objects are the cells of $E$ and whose morphisms corresponds to adjacency, i.e. there is exactly one morphism $e^p \to e^q$ if $e^p$ is a face of $e^q$ and none otherwise. A cellular cosheaf (resp. sheaf) on $E$ is a contravariant (resp. covariant) functor:

$$F : \mathrm{Cell} E \to \mathrm{Vect}_{\mathbb{F}_2}^{\mathbf{f}},$$

where $\mathrm{Vect}_{\mathbb{F}_2}^{\mathbf{f}}$ is the category of finite dimensional vector spaces over $\mathbb{F}_2$. If $e^p \le e^q$ are two cells of $E$ and $v$ is a vector of $F(e^q)$ we will denote by $v|_{e^p}^{e^q}$ its image in $F(e^p)$ by the morphism $F(e^p \to e^q)$ called the extension morphism (we use a similar notation in the case of sheaves). A morphism $f$ between two cosheaves (or two sheaves) on $E$ is a natural transformation. The kernel, image, and cokernel of $f$ are formed “cell-wise”.

When $E$ is the cubical subdivision of a $\Delta$-complex $K$ we might call a cellular cosheaf on $E$ a cubical cosheaf on $K$. Such an object is an assignment of a $\mathbb{F}_2$-vector space to every pair of adjacent simplices $\sigma^p \le \sigma^q$ that is covariant in the first coordinate and contravariant in the second.

**Definition 1.4.** Let $E$ be a regular CW-complex, $F$ be a cellular cosheaf on $E$ and $k$ be a non-negative integer. The space of cellular $k$-chains with value in $F$ is the vector space:

$$C_k(E; F) := \bigoplus_{e^k \in E} F(e^k).$$

A typical vector of this space is written $c = \sum_{e^k \in E} v_{e^k} \otimes e^k$ where $v_{e^k}$ belongs to $F(e^k)$ for all cells $e^k$. We might denote the coefficient $v_{e^k}$ by $\langle c; e^k \rangle$. When $F$ is a cubical cosheaf on the $\Delta$-complex $K$ we denote $C_k(E; F)$, where $E$ is the cubical subdivision of $K$, by $\Omega_k(K; F)$. This vector space has a canonical decomposition:

$$\Omega_k(K; F) := \bigoplus_{q-p=k} \Omega_{p,q}(K; F), \qquad (4)$$
where $\Omega_{p,q}(K; F)$ is the direct sum over all adjacent pairs of simplices $\sigma^p \le \sigma^q$ of the vector spaces $F(\sigma^p; \sigma^q)$. The vector spaces $(C_k(E; F))_{k \ge 0}$ form a chain complex for the following boundary operator $\partial$:

$$ \forall e^{k+1} \in E, \forall v \in F(e^{k+1}), \quad \partial(v \otimes e^{k+1}) := \sum_{e^k \le e^{k+1}} v|_{e^k}^{e^{k+1}} \otimes e^k. $$

The homology of $F$ is then defined as the homology $(H_k(E; F))_{k \ge 0}$ of this complex. We note that a morphism of cosheaves induces a morphism of chain complexes.

**Notations.** Let $\sigma^p$ be a simplex of a $\Delta$-complex $K$, we denote by $\chi_{\sigma^p}$ the characteristic cochain of $\sigma^p$. This is the $p$-cochain with value in $\mathbb{F}_2$ whose value is 1 on $\sigma^p$ and 0 elsewhere.

**Remark 1.5.** For a more thorough exposition of the theory of cellular sheaves and cosheaves one can look at [KS10] (under the name *constructible sheaves*), [She85], or even [Che23a] where we discuss *dihomologic cosheaves*, a generalisation of cubical cosheaves. However we can still note that:

1. Whenever $E'$ is a subcomplex of $E$ we can define relative homology with coefficients in $F$ by the usual quotient of chain complexes;

2. Definition 1.4 can be dualised for a cellular sheaf. This defines cellular cohomology with coefficients in a cellular sheaf;

3. The decomposition (4) makes $(\Omega_k(K; F); \partial)_{k \ge 0}$ into the total complex of a bicomplex. For all $p, q$, the boundary $\partial$ maps $\Omega_{p,q}$ in $\Omega_{p,q-1} \oplus \Omega_{p+1,q}$;

4. Assuming that $E'$ is a regular subdivision of the regular CW-complex $E$ there is a natural subdivision functor from the category of cosheaves on $E$ to the cosheaves on $E'$. It associates to $F$ the precomposition $F \circ S$ where $S: \text{Cell } E' \to \text{Cell } E$ is the increasing map that takes a cell of $E'$ to the unique cell of $E$ that contains it. This procedure always induces a quasi-ismorphism:

$$ \begin{array}{ccc} (C_k(E; F))_{k \ge 0} & \to & (C_k(E'; F \circ S))_{k \ge 0} \\ v \otimes e^k & \mapsto & \displaystyle\sum_{(e')^k \subset e^k} v \otimes (e')^k. \end{array} $$

**Non-Singular Polytopes, Primitive Triangulations, and Tropical Cosheaves.** In the whole article $t^*(\mathbb{Z})$ will denote a free Abelian group of rank $n \ge 1$. For every commutative ring $R$, $t^*(R)$ denotes the associated free module $t^*(\mathbb{Z}) \otimes_{\mathbb{Z}} R$. $t(\mathbb{Z})$ denotes the dual lattice of the dual vector space $t(\mathbb{R})$. We say that a sub-vector space $W \subset t^*(\mathbb{R})$ is rational if there exists a sub-module $M \subset t^*(\mathbb{Z})$ satisfying $M \otimes_{\mathbb{Z}} \mathbb{R} = W$. In this case we denote by $W(\mathbb{Z})$ the lattice $W \cap t^*(\mathbb{Z})$, and for every commutative ring $R$, by $W(R)$ the free module $W(\mathbb{Z}) \otimes_{\mathbb{Z}} R$. We say that an affine sub-space $A$ of $t^*(\mathbb{R})$ is rational if its tangent space $TA$ is.

**Definition 1.6.** A non-singular polytope of $t^*(\mathbb{R})$ is a simple$^6$ convex polytope $P$ whose vertices lie in the lattice$^7 t^*(\mathbb{Z})$ and such that for every vertex $V$ of $P$ the group $\sum_{E \ge V} TE(\mathbb{Z})$,

<footnote>$^6$A polytope $P$ is simple if every face of codimension $k$ of $P$ is the intersection of $k$ faces of codimension 1.</footnote>

<footnote>$^7$We say that $P$ is integral.</footnote>
where the E's are the edges of P containing V, is the full lattice $t^*(\mathbb{Z})$. A primitive triangulation K of P is a decomposition of P into integral simplices of minimal integral volume<fnref n="8" />, i.e. $\frac{1}{n!}$.

<figure><img src="image_5.png" /><figcaption>Figure 3: Examples of integer polytopes.</figcaption></figure>

**Remark 1.7.** With this definition a non-singular polytope P has necessarily dimension $n$. Also a polytope P is non-singular if and only if its associated projective toric variety is smooth<fnref n="9" /> and the action of the torus $\text{Hom}(t^*(\mathbb{Z}); \mathbb{C}^*)$ is effective. The toric variety of a polytope is smooth if and only if the latter is non-singular in the affine space it spans.

**Definition 1.8.** Let P be a non-singular polytope of $t^*(\mathbb{R})$ endowed with a primitive triangulation $K$. The dual hypersurface $X$ of $K$ is the sub-complex of the cubical subdivision of $K$ made of all the cubes $[\sigma^p; \sigma^q]$ for which $p$ is at least 1. See Figure 4 for instance.

<figure><img src="image_6.png" /><figcaption>Figure 4: A dual hypersurface.</figcaption></figure>

**Definition 1.9 (Contraction).** Let $V$ be a finite dimensional vector space over a field $\mathbb{F}$, and $k, l \in \mathbb{N}$ be integers. For all $\alpha \in \Lambda^l V^*$ and $v \in \Lambda^{l+k} V$ the contraction $\alpha \cdot v$ is the only element of $\Lambda^k V$ satisfying:

$$ \beta(\alpha \cdot v) = (\beta \wedge \alpha)(v), $$

for all $\beta \in \Lambda^k V^*$.

**Definition 1.10.** Let $V$ be a finite dimensional $\mathbb{F}_2$-vector space. Let $k$ be the dimension of $V$. We denote by $[V]$ the unique generator of the line $\Lambda^k V$. Whenever $V$ is a subspace of $W$ we see $[V]$ as a $k$-vector of $W$. In particular, if $K$ is a primitive subdivision of a non-singular polytope $P$ of $t^*(\mathbb{R})$, we denote, for all simplices $\sigma^k$ of $K$, by $\omega(\sigma^k)$ the element of $\Lambda^k t^*(\mathbb{F}_2)$ given by $[T\sigma^k(\mathbb{F}_2)]$.

**Definition 1.11 (Tropical Cosheaves).** Let $P$ be a non-singular polytope of $t^*(\mathbb{R})$ endowed with a primitive triangulation $K$. We give some of the cosheaves of tropical homology given by I. Itenberg, L. Katzarkov, G. Mikhalkin, and I. Zharkov<fnref n="10" />:

<footnote n="8">⁸The Lebesgue mesure normalised so that a parallelogram on a basis of the lattice $t^*(\mathbb{Z})$ has volume 1.</footnote>

<footnote n="9">⁹c.f. [Ful93], §2.1 Local Properties of Toric Varieties pp.28-31.</footnote>

<footnote n="10">¹⁰c.f. [IKMZ19].</footnote>
1. The sedentarity cosheaf $\text{Sed} : \text{for a simplex } \sigma^p \text{ of } K \text{ the group } \text{Sed}(\sigma^p) \text{ is given by:}$

$$ \text{Sed}(\sigma^p) := \{ v \in t(\mathbb{F}_2) \mid \alpha(v) = 0, \forall \alpha \in (TQ \cap t^*(\mathbb{Z})) \}, $$

where $Q$ is the smallest face of $P$ containing the relative interior<fnref n="11" /> of $\sigma^p$. If $\sigma^p \le \sigma^q$ the extension morphism $\text{Sed}(\sigma^q) \to \text{Sed}(\sigma^p)$ is the inclusion. It is defined on $K$, we denote its cubical subdivision by the same symbol, hence for all $\sigma^p \le \sigma^q$, $\text{Sed}(\sigma^p; \sigma^q)$ is by definition $\text{Sed}(\sigma^q)$;

2. The cosheaf $F_1^P$, defined as the quotient: $F_1^P := t(\mathbb{F}_2)/\text{Sed}$, where $t(\mathbb{F}_2)$ is understood as the constant cosheaf. We also denote its subdivisions by the same symbol;

3. The cosheaves $F_p^P$, for all $p \in \mathbb{N}$, are the exterior powers of $F_1^P$: $F_p^P := \bigwedge^p F_1^P$. In particular $F_0^P$ is the constant cosheaf $\mathbb{F}_2$;

4. The cosheaves<fnref n="12" /> $F_p^X$, for all $p \in \mathbb{N}$, defined as the sub-cosheaf of $F_p^P$ whose groups are, for all $\sigma^k \le \sigma^l$:

$$ F_p^X(\sigma^k; \sigma^l) := \ker \left( f : F_p^P(\sigma^k; \sigma^l) \to F_{p-k}^P(\sigma^k; \sigma^l) \right), $$

where $f$ is the contraction against $\omega(\sigma^k)$: the generator of $\bigwedge^k T\sigma^k(\mathbb{F}_2)$. We have the relation: $F_k^X(\sigma^p; \sigma^q) = \sum_{\sigma^1 \le \sigma^p} F_k^X(\sigma^1; \sigma^q) = \sum_{\sigma^1 \le \sigma^p} \bigwedge^k F_1^X(\sigma^1; \sigma^q)$. See Figure 5 for an example. The morphisms between these groups are given by quotient maps and inclusions. Note that even though $F_k^X$ is defined on the whole cubical subdivision of $K$, its support<fnref n="13" /> is contained in $X$. We denote its inclusion in $F_p^P$ by $i_p: F_p^X \subset F_p^P$.

For all integers $p, q \ge 0$ we will denote by $H_{p,q}(P; \mathbb{F}_2)$ (resp. $H_{p,q}(X; \mathbb{F}_2)$) the $q$-th homology group of $F_p^P$ (resp. $F_p^X$) and by $i_{p,q}: H_{p,q}(X; \mathbb{F}_2) \to H_{p,q}(P; \mathbb{F}_2)$ the morphism induced by the inclusion $i_p$. We will denote their dual vector spaces by lifting the indices.

<figure><img src="image_7.png" /><figcaption>Figure 5: A triangle and the groups associated by $F_1^X$ to some of its cubical cells.</figcaption></figure>

<footnote n="11">¹¹i.e. the interior of $\sigma^p$ in the affine space it spans.</footnote>

<footnote n="12">¹²This definition is different from the usual one but is equivalent in the case of primitive triangulations. Such a triangulation $K$ has $\theta(K) = 1$, c.f. [Che23a] Definition 3.5.</footnote>

<footnote n="13">¹³i.e. the union of the closed cubes carrying non-trivial groups.</footnote>
**Definition 1.14.** Let $P$ be a non-singular polytope of $t^*(\mathbb{R})$ and $K$ be a primitive triangulation of $P$. We denote by $\mathbb{R}K$ the subdivision of $\mathbb{R}P$ induced by $K$, and $|\cdot|: \mathbb{R}K \to K$ the induced cellular map. Let $\sigma^p$ be a simplex of $K$. Each lift$^{18}$ $\sigma_{\mathbb{R}}^p$ of $\sigma^p$ in $\mathbb{R}K$ is uniquely represented by an element of $t(\mathbb{F}_2)/\text{Sed}(\sigma^p) = F_1^P(\sigma)$ that we call the *argument* of $\sigma^p$, and denote by $\arg(\sigma_{\mathbb{R}}^p)$. More generally the procedure $K \mapsto \mathbb{R}K$ applies to any regular subdivision of $P$. In particular if $E$ is the cubical subdivision of $K$ then $\mathbb{R}E$ is the cubical subdivision of $\mathbb{R}K$ and $|[\sigma_{\mathbb{R}}^p; \sigma_{\mathbb{R}}^q]|$ is $[|\sigma_{\mathbb{R}}^p; |\sigma_{\mathbb{R}}^q]|$ and $\arg[\sigma_{\mathbb{R}}^p; \sigma_{\mathbb{R}}^q]$ equals $\arg(\sigma_{\mathbb{R}}^q)$.

**Definition 1.15.** Let $P$ be a non-singular polytope of $t^*(\mathbb{R})$ and $K$ be a primitive triangulation of $P$ with dual hypersurface $X$. We denote by $\omega_{\mathbb{R}X}$ the closed 1-cochain defined by the formula:

$$ \forall \sigma_{\mathbb{R}}^1 \in \mathbb{R}K, \quad \omega_{\mathbb{R}X}(\sigma_{\mathbb{R}}^1) := \omega(|\sigma_{\mathbb{R}}^1|) (\arg(\sigma_{\mathbb{R}}^1)). $$

<figure><img src="image_8.png" /><figcaption>Figure 8: The edges of $\mathbb{R}K$ on which $\omega_{\mathbb{R}X}$ is 1.</figcaption></figure>

**Definition 1.16.** Let $P$ be a non-singular polytope of $t^*(\mathbb{R})$, $K$ be a primitive triangulation of $P$ with dual hypersurface $X$, and $\varepsilon$ be a 0-cochain of $K$. We denote by $|\varepsilon|^*$ the pullback of $\varepsilon$ by the cellular map $|\cdot|$. The *T-hypersurface* defined by $\varepsilon$ is the dual complex of $d|\varepsilon|^* + \omega_{\mathbb{R}X}$ in $\mathbb{R}K$. We denote it by $\mathbb{R}X_{\varepsilon}$. We note that $|\mathbb{R}X_{\varepsilon}|$ is precisely $X$.

<figure><img src="image_9.png" /><figcaption>(a) Une distribution de signe sur une subdivision convexe du triangle de taille 5.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(b) Le patchwork combinatoire associé.</figcaption></figure>

<caption>Figure 9: Le patchwork combinatoire d'une quintique plane maximale.</caption>

<footnote>18i.e. $|\sigma_{\mathbb{R}}^p| = \sigma^p$.</footnote>
**Proposition 1.17** (E. Brugallé, L. López de Medrano, and J. Rau, Proposition 4.11 p.26, [BLdMR22]). For all non-singular polytopes $P$, all primitive triangulations $K$ of $P$ and all sign distributions $\epsilon$, the T-hypersurface $\mathbb{R}X_\epsilon$ is PL-smooth.

**The Renaudineau-Shaw Spectral Sequence.** Since $\mathbb{R}P$ and $\mathbb{R}X_\epsilon$ can be respectively folded on $P$ and $X$ by $|\cdot|$ whose fibers are discrete, we can compute the cellular homology of $\mathbb{R}P$ and $\mathbb{R}X_\epsilon$ with coefficients in $\mathbb{F}_2$ as the homology of cosheaves on $P$ and $X$ respectively. Using inclusion and subdivision we can see these cosheaves as cubical cosheaves on $K$.

**Definition 1.18.** Let $P$ be a non-singular polytope of $t^*(\mathbb{R})$, $K$ be a primitive triangulation of $P$ with dual hypersurface $X$, and $\epsilon$ be a 0-cochain of $K$. Let $K^{\mathbb{R}P}$ be the cosheaf that associates to every cube of $K$ the vector space freely generated by its lifts in $\mathbb{R}K$. This cosheaf is canonically isomorphic to the cosheaf of group algebras $\mathbb{F}_2[F_1^{\mathbb{P}}]$ and hence as canonical structure of cosheaf of algebras. Now we denote by $\text{Arg}_\epsilon$ the cosheaf of sets that associates to a cube of $K$ its set of lifts in $\mathbb{R}K$ that belong to $\mathbb{R}X_\epsilon$. We denote by $K^{\mathbb{R}X_\epsilon}$ the sub-cosheaf of $K^{\mathbb{R}P}$ spanned by $\text{Arg}_\epsilon$.

**Remark 1.19.** In [RS23], A. Renaudineau and K. Shaw called $K^{\mathbb{R}X_\epsilon}$ the sign cosheaf of $\mathbb{R}X_\epsilon$ and denoted it $S_\epsilon$. They also called $\text{Arg}_\epsilon$ the real phase structure of $\mathbb{R}X_\epsilon$ and denoted it $\mathcal{E}$.

**Notations.** Let $G$ be an Abelian group, in the group algebra $\mathbb{F}_2[G]$ we denote the generator associated to the element $g$ by $x^g$.

**Remark 1.20.** We can note that if $V$ is a finite dimensional vector space over $\mathbb{F}_2$ then $\mathbb{F}_2[V]$ is a local ring with maximal ideal the kernel of the augmentation map $\mathbb{F}_2[V] \to \mathbb{F}_2$ that sends all elements $x^v$ to 1. Every element not in the kernel of the augmentation is its own inverse.

**Proposition 1.21** (A. Renaudineau and K. Shaw, Proposition 3.17 of p.964, [RS23]). *The chain complexes $(\Omega_k(\mathbb{R}K; \mathbb{F}_2))_{k \ge 0}$ and $(\Omega_k(K; K^{\mathbb{R}P}))_{k \ge 0}$ are canonically isomorphic. Moreover this isomorphism sends the sub-complex $(C_k(\mathbb{R}X_\epsilon; \mathbb{F}_2))_{k \ge 0}$ to $(\Omega_k(K; K^{\mathbb{R}X_\epsilon}))_{k \ge 0}$.*

This isomorphism sends $1 \otimes [\sigma_{\mathbb{R}}^p; \sigma_{\mathbb{R}}^q]$ to $[\sigma_{\mathbb{R}}^p; \sigma_{\mathbb{R}}^q] \otimes [|\sigma_{\mathbb{R}}^p|; |\sigma_{\mathbb{R}}^q|]$ or equivalently to $x^{\text{arg}(\sigma_{\mathbb{R}}^q)} \otimes [|\sigma_{\mathbb{R}}^p|; |\sigma_{\mathbb{R}}^q|]$.

**Definition 1.22.** Let $P$ be a non-singular polytope of $t^*(\mathbb{R})$, $K$ be a primitive triangulation of $P$ with dual hypersurface $X$, and $\epsilon$ be a 0-cochain of $K$. The decreasing filtration $(K_{(p)}^{\mathbb{R}P})_{0 \le p \le n}$ is defined by the iterated powers of its maximal ideal. We denote by $(K_{(p)}^{\mathbb{R}X_\epsilon})_{0 \le p \le n}$ the induced filtration of $K^{\mathbb{R}X_\epsilon}$. We denote by $(E_{p,q}^r(\mathbb{R}P))_{p,q,r \ge 0}$ and $(E_{p,q}^r(\mathbb{R}X_\epsilon))_{p,q,r \ge 0}$ the respective spectral sequences. The inclusion $i: K^{\mathbb{R}X_\epsilon} \to K^{\mathbb{R}P}$ induces a morphism of spectral sequences ($i_{p,q}^r: E_{p,q}^r(\mathbb{R}X_\epsilon) \to E_{p,q}^r(\mathbb{R}P))_{p,q,r \ge 0}$.

**Proposition 1.23** (Lemma 4.8 and Proposition 4.10, [RS23] and Proposition 4.22, [Che23b]). Let $p \ge 0$ be an integer. There is a unique surjective morphism of cosheaves $\text{bv}_p: K_{(p)}^{\mathbb{R}P} \to F_p^P$ that sends an element $\sum_{w \in W} x^w$ to $[W]$, for every $p$-dimensional subspace $W$ of $F_1^P$. Its kernel is $K_{(p+1)}^{\mathbb{R}P}$ and it sends $K_{(p)}^{\mathbb{R}X_\epsilon}$ onto $F_p^X$. We call them the Borel-Viro morphisms. For all $p, q \ge 0$
they induce the following commutative square:

$$ 
\begin{aligned}
& E_{p,q}^{1}(\mathbb{R}P) \xrightarrow{\text{bv}_{p,q}} H_{p,q}(P; \mathbb{F}_2) \\
& \xleftarrow{i_{p,q}^{1}} \xrightarrow{\uparrow} \xleftarrow{i_{p,q}} \\
& E_{p,q}^{1}(\mathbb{R}X_{\varepsilon}) \xrightarrow{\cong} H_{p,q}(X; \mathbb{F}_2)
\end{aligned}
\qquad (5) 
$$

Hence in what follows we shall always identify $(E_{p,q}^1(\mathbb{R}P))_{p,q \ge 0}$ and $(E_{p,q}^1(\mathbb{R}X_{\varepsilon}))_{p,q \ge 0}$ with $(H_{p,q}(P; \mathbb{F}_2))_{p,q \ge 0}$ and $(H_{p,q}(X; \mathbb{F}_2))_{p,q \ge 0}$ respectively, using the Borel-Viro isomorphisms.

**Remark 1.24.** Let $\sigma^a \le \sigma^b$ be two simplices of $K$, and $v_0, \dots, v_p \in F_1^P(\sigma^a; \sigma^b)$, we have:

$$ \text{bv}_p \left( x^{v_0} \prod_{i=1}^{p} (1 + x^{v_i}) \right) = \bigwedge_{i=1}^{p} v_i. $$

## 2 Differential Geometry of Primitive Complexes

**Shellings.** In this paragraph we provide the definitions and basic properties of shellings of simplicial complexes.

**Definition 2.1.** Let $K$ be a finite simplicial complex of dimension $n \in \mathbb{N}$. The polynomial $f_K$ is given by the formula:

$$ f_K(X) = \sum_{i=0}^{n+1} f_{i-1}(K) X^{n+1-i}, $$

where $f_i(K)$ is the number of $i$-simplices of $K$.

**Definition 2.2.** A *shelling* of a finite simplicial complex $K$ of pure dimension $n \in \mathbb{N}$, is a numbering of its maximal simplices $\sigma_0^n, \dots, \sigma_k^n$ such that, for all integers $0 \le i \le k-1$, the intersection of $\sigma_{i+1}^n$ with the previous simplices $\cup_{j \le i} \sigma_j^n$ is a non-empty union of faces of codimension 1 of $\sigma_{i+1}^n$. Let $0 \le i \le k$, we call the *order* of $\sigma_i^n$, the number of its faces of codimension 1 that belong to the previous simplices. For all $0 \le j \le n+1$, we denote by $h_j(K)$ the number of simplices of order $j$ of the shelling. Moreover, we define the polynomial $h_K$ by the formula:

$$ h_K(X) = \sum_{j=0}^{n+1} h_j(K) X^{n+1-j}. $$

A simplicial complex that admits a shelling is called *shellable*.

The polynomial $h_K$ of a shellable simplicial complex $K$ does not depend on the particular shelling used to compute it. This property is a corollary of the following formula.

**Proposition 2.3** ([Zie95], p.249). Let $K$ be a shellable simplicial complex. We have the following identity:

$$ h_K(X) = f_K(X-1). \qquad (6) $$

The relation (6) implies that $h_0(K)$ equals $f_{-1}(K)$, i.e. 1, and also that $h_{n+1}(K)$ equals $f_K(-1)$, i.e. $(-1)^{n+1} + (-1)^n \chi(|K|)$ (the Euler characteristic of $|K|$).
**Proposition 2.4** ([Bjö84], Proposition 4.3 p.12). Let $K$ be a shellable simplicial complex of dimension $n$ in which every $(n-1)$-simplex is contained in at most two $n$-simplices. Then:

1. If every $(n-1)$-simplex is contained in exactly two $n$-simplices, the geometric realisation of $K$ is homeomorphic to a sphere;

2. If there is a $(n-1)$-simplex contained in exactly one $n$-simplex, the geometric realisation of $K$ is homeomorphic to a ball.

In particular, in the first case $h_{n+1}(K)$ equals 1 and in the second 0.

**Lemma 2.5.** Let $K$ be an $n$-dimensional shellable simplicial complex in which every $(n-1)$-simplex is contained in at most two $n$-simplices. Let $\sigma_0^n, \dots, \sigma_k^n$ be a shelling of $K$. For all $0 \le i \le k$, we denote by $\tau_i$ the intersection of all the codimension 1 faces of $\sigma_i^n$ along which it intersects the previous simplices. For all $0 \le i \le k$, the star of $\tau_i$ in $K$ is contained in $\cup_{j \le i} \sigma_j^n$. In other words, when we add the simplex $\sigma_i^n$ we complete the star of $\tau_i$.

*Proof.* Let $0 \le i \le k$. The link of $\tau_i$ is a pure $(n-1-\dim(\tau_i))$-dimensional simplicial complex in which every simplex of codimension 1 is contained in at most two maximal simplices. The shelling of $K$ induces a shelling of the link $\dim(\tau_i; K)$. The proof of Lemma 8.7 p.237 of [Zie95] is directly adaptable from the case of a vertex to the case of a simplex of higher dimension. By definition of $\tau_i$, this shelling has a simplex of order $n - \dim(\tau_i)$, i.e. $\dim \dim(\tau_i; K) + 1$. This is $\dim(\tau_i; \sigma_i^n)$. Hence, by Proposition 2.4, this link must be a sphere. Thus, every simplex of codimension one of this link belongs to exactly two maximal simplices. The shelling of the link can be reversed to find another shelling of the link. This would be Lemma 8.10 p.240 of [Zie95] if we were considering boundary of simplicial polytopes. However, its proof only uses the fact that every simplex of codimension one belongs to two maximal simplices, hence it is directly transposable to our case. Reversing the order changes the order of every maximal simplex by the involution $x \mapsto \dim \dim(\tau_i; K) + 1 - x$. Therefore, the simplex of order $\dim \dim(\tau_i; K) + 1$ has to be the last for the simplex of order 0 is always the first. Therefore, when we add $\sigma_i^n$ we complete the link of $\tau_i$, hence its star. $\square$

**Differential Geometry.** Some concepts of differential geometry are adaptable to the discrete context of simplicial complexes primitively embedded in affine spaces. We give the definitions of this framework.

**Definition 2.6.** We call a *primitive complex* in $t^*(\mathbb{R})$, a purely $n$-dimensional piecewise-affinely embedded simplicial complex $K \subset t^*(\mathbb{R})$ whose simplices are primitive, i.e. of minimal integral volume. The *boundary* of $K$ is the sub-complex of simplices contained in exactly one maximal simplex of $K$. We denote it by $\partial K$.

**Definition 2.7.** Let $K$ be primitive complex in $t^*(\mathbb{R})$. The $\mathbb{F}_2$-normal cosheaf $N$ of $K$ is defined for all simplices $\sigma^p$ by the formula:

$$N(\sigma^p) := (T\sigma^p)^\perp (\mathbb{F}_2),$$
with extension morphisms given by inclusions. Likewise, the $\mathbb{F}_2$-cotangent cosheaf $T^*$ of $K$ is defined for all simplices $\sigma^p$ by the formula:

$$T^*(\sigma^p) := T^*\sigma^p(\mathbb{F}_2),$$

with extension morphisms given by the quotient morphisms. We have the following exact sequence of cosheaves:

$$0 \to N \to [K; t(\mathbb{F}_2)] \to T^* \to 0,$$

where $[K; t(\mathbb{F}_2)]$ denotes the constant cosheaf on $K$ with value $t(\mathbb{F}_2)$.

**Definition 2.8.** Let $K$ be primitive complex in $t^*(\mathbb{R})$ and $\sigma^p$ be a simplex of $K$. We denote by $\omega(\sigma^p) \in \Lambda^p t^*(\mathbb{F}_2)$ the generator of the line $\Lambda^p T\sigma^p(\mathbb{F}_2)$. If $\sigma^1$ is an edge of $K$ with vertices $\alpha$ and $\beta$ then $\omega(\sigma^1)$ is the reduction modulo 2 of $\beta - \alpha$.

**Proposition 2.9.** The cochain $[\sigma^p \mapsto \omega(\sigma^p)] \in C^p(K; \Lambda^p t^*(\mathbb{F}_2))$ is closed.

*Proof.* If $p$ equals 0 then by definition $[\sigma^0 \mapsto \omega(\sigma^0)]$ is constant to 1. It is closed. Now let $p \ge 1$ be an integer, $\sigma^p$ be a simplex of $K$, and $\beta$ be a vertex of $\sigma^p$. Since $K$ is primitive, $\omega(\sigma^p)$ is given by the following formula:

$$\omega(\sigma^p) = \bigwedge_{\alpha \in \text{lk}(\beta; \sigma^p)^{(0)}} (\alpha - \beta) = \bigwedge_{\alpha \in \text{lk}(\beta; \sigma^p)^{(0)}} \alpha + \sum_{\substack{\alpha \in \text{lk}(\beta; \sigma^p)^{(0)} \\ \alpha' \neq \alpha}} \beta \wedge \bigwedge_{\substack{\alpha' \in \text{lk}(\beta; \sigma^p)^{(0)} \\ \alpha' \neq \alpha}} \alpha' \\ = \sum_{\sigma^{p-1} \le \sigma^p} \bigwedge_{\alpha \in (\sigma^{p-1})^{(0)}} \alpha.$$

If we denote by $\lambda$ the cochain $[\sigma^{p-1} \mapsto \Lambda_{\alpha \in (\sigma^{p-1})^{(0)}} \alpha]$, the cochain $\omega$ is the differential of $\lambda$. Hence, it is closed. $\square$

**Proposition 2.10.** Let $K$ be a primitive complex in $t^*(\mathbb{R})$. For all simplices $\sigma^p \in K$ and all vertices $\alpha$ of $\sigma^p$ the set $\{\beta - \alpha : \beta$ vertex of $\text{lk}(\alpha; \sigma^p)\}$ is, up to ordering, a basis of $T\sigma^p(\mathbb{Z})$. Moreover for all $q \in \mathbb{N}$, the morphism:

$$z_{\sigma^p}^q : \Lambda^q t(\mathbb{F}_2) \to Z^q(\sigma^p; \mathbb{F}_2) \\ v \mapsto [\sigma^q \mapsto (\omega(\sigma^q))(v)],$$

fits in the short exact sequence:

$$0 \to N(\sigma^p) \wedge \bigwedge^{q-1} t(\mathbb{F}_2) \to \bigwedge^q t(\mathbb{F}_2) \xrightarrow{z_{\sigma^p}^q} Z^q(\sigma^p; \mathbb{F}_2) \to 0.$$

Moreover, for all pairs $\sigma^p \le \sigma^r$ and all vectors $v \in \Lambda^q t(\mathbb{F}_2)$ the restriction of $z_{\sigma^r}^q(v)$ to $\sigma^p$ is $z_{\sigma^p}^q(v)$.

*Proof.* The first statement is a direct consequence of the primitivity of the simplices of $K$. The morphism $z_{\sigma^0}^0$ is an isomorphism, $z_{\sigma^0}^q$ is null for all $q > 0$, and $N(\sigma^p)$ equals $t(\mathbb{F}_2)$. The proposition is true for $p = 0$. Let $\sigma^p$ be a simplex of $K$ of positive dimension, and $q \ge 0$ be an integer. Proposition 2.9 ensures us that $z_{\sigma^p}^q$ is well defined with $Z^q(\sigma^p; \mathbb{F}_2)$ as target space.
Let $\beta$ be a vertex of $\sigma^p$ and let us label $\sigma_1^q, \dots, \sigma_k^q$, with $k$ being $\binom{p}{q}$, the $q$-simplices of $\sigma^p$ that contain $\beta$. The first statement implies that $\{\omega(\sigma_1^q), \dots, \omega(\sigma_k^q)\}$ is a basis of $\Lambda^q T\sigma^p(\mathbb{F}_2)$. Let $\zeta$ be a $q$-cocyle of $\sigma^p$. We define $v$ to be the unique vector of $\Lambda^q t(\mathbb{F}_2)/(\Lambda^q T\sigma^p(\mathbb{F}_2))^\perp$ satisfying:

$$ \omega(\sigma_i^q)(v) = \zeta(\sigma_i^q), $$

for all $1 \le i \le k$. Let $\sigma^q$ be a simplex of $\sigma^p$ that does not contain $\beta$. The boundary of $\beta * \sigma^q$ is made of $\sigma^q$ and $\sigma_{i_1}^q, \dots, \sigma_{i_{q+1}}^q$. Since both $\omega$ and $\zeta$ are closed we have:

$$ \omega(\sigma^q)(v) = \sum_{j=1}^{q+1} \omega(\sigma_{ij}^q)(v) = \sum_{j=1}^{q+1} \zeta(\sigma_{ij}^q) = \zeta(\sigma^q). $$

Hence, for any lift $\bar{v}$ of $v$ in $\Lambda^q t(\mathbb{F}_2)$, $z_{\sigma^p}^q(\bar{v})$ equals $\zeta$, and $z_{\sigma^p}^q$ is surjective. Moreover, the kernel of $z_{\sigma^p}^q$ is $(\Lambda^q T\sigma^p(\mathbb{F}_2))^\perp$. This vector space has codimension $\binom{p}{q}$ and contains $N(\sigma^p) \Lambda \Lambda^{q-1} t(\mathbb{F}_2)$. Using a basis of $t(\mathbb{F}_2)$ whose $(n-p)$ first vectors belong to $N(\sigma^p)$, we find that the latter space has dimension $\binom{n}{q} - \binom{p}{q}$. Therefore, they must be equal. The last statement of the proposition is a direct consequence of the definition of $z_{\sigma^p}^q$. $\square$

**Proposition 2.11.** Let $K$ be primitive complex in $t^*(\mathbb{R})$ and $\sigma^n$ be a maximal simplex of $K$. For all vertices $\alpha$ of $\sigma^n$, we denote by $e_{\sigma^n}(\alpha)$ the generator of the line $N(\mathrm{lk}(\alpha; \sigma^n))$. For all vertices $\alpha$ of $\sigma^n$ the collection $\{e_{\sigma^n}(\beta) : \beta \in \mathrm{lk}(\alpha; \sigma^p)^{(0)}\}$ is the dual basis of the reduction modulo 2 of $\{\beta - \alpha : \beta \in \mathrm{lk}(\alpha; \sigma^p)^{(0)}\}$. Moreover, we have the relation:

$$ \sum_{\alpha \in (\sigma^n)^{(0)}} e_{\sigma^n}(\alpha) = 0. $$

In addition, for all simplices $\sigma^p \le \sigma^n$, the set $\{e_{\sigma^n}(\alpha) : \alpha \in \mathrm{lk}(\sigma^p; \sigma^n)^{(0)}\}$ is a basis of the vector space $N(\sigma^p)$. Also, for all simplices $\sigma^p \le \sigma^n$, with $p \ge 1$, and all vertices $\alpha \in \sigma^n$, the interior product $\iota_{e_{\sigma^n}(\alpha)}\omega(\sigma^p)$ vanishes if $\alpha$ is not a vertex of $\sigma^p$ and equals $\omega(\mathrm{lk}(\alpha; \sigma^p))$ when $\alpha$ belongs to $\sigma^p$.

*Proof.* Let $\sigma^n$ be a simplex of $K$ and $\alpha, \beta$ be distinct vertices of $\sigma^n$. If $\beta' \in \sigma^n$ is different from $\alpha$ and $\beta$, then, by definition, $(\beta' - \alpha)(e_{\sigma^n}(\beta))$ must vanish for the edge $[\beta'; \alpha]$ belongs to $\mathrm{lk}(\beta; \sigma^n)$. The number $(\beta - \alpha)(e_{\sigma^n}(\beta))$ must be 1, for otherwise $e_{\sigma^n}(\beta)$ would vanish. Indeed, $\{\beta - \alpha : \beta \in \mathrm{lk}(\alpha; \sigma^p)^{(0)}\}$ is a basis of $t^*(\mathbb{Z})$, hence its reduction modulo 2 is a basis of $t^*(\mathbb{F}_2)$. It follows that $\{e_{\sigma^n}(\beta) : \beta \in \mathrm{lk}(\alpha; \sigma^p)^{(0)}\}$ is the dual basis of $\{\beta - \alpha (\mathrm{mod} 2) : \beta \in \mathrm{lk}(\alpha; \sigma^p)^{(0)}\}$. Moreover, it implies that for any two vertices $\beta, \beta'$ of $\sigma^n$, we have:

$$ (\beta - \beta') \left( \sum_{\alpha \in (\sigma^n)^{(0)}} e_{\sigma^n}(\alpha) \right) = (\beta - \beta')(e_{\sigma^n}(\beta) + e_{\sigma^n}(\beta')) = 1 + 1 = 0. $$

Therefore, the vector $\sum_{\alpha \in (\sigma^n)^{(0)}} e_{\sigma^n}(\alpha)$ is 0. We also deduce that, for all pairs $\sigma^p \le \sigma^n$, the set $\{e_{\sigma^n}(\alpha) : \alpha \in \mathrm{lk}(\sigma^p; \sigma^n)^{(0)}\}$ is a linearly independent family of $N(\sigma^p)$. Since its cardinality is $(n-p)$ it must be a basis. As a consequence if $\alpha \in \sigma^n$ is not a vertex of $\sigma^p$, with $p \ge 1$, then $\iota_{e_{\sigma^n}(\alpha)}\omega(\sigma^p)$ vanishes. If on the contrary $\alpha$ is a vertex of $\sigma^p$, then $\omega(\sigma^p)$
equals $\omega(\mathrm{lk}(\alpha; \sigma^p)) \wedge (\alpha - \beta)$ for any other vertex $\beta$ of $\sigma^p$. It follows that $\iota_{e_{\sigma^n}(\alpha)}\omega(\sigma^p)$ equals $(\alpha - \beta)(e_{\sigma^n}(\alpha))\omega(\mathrm{lk}(\alpha; \sigma^p))$, i.e. $\omega(\mathrm{lk}(\alpha; \sigma^p))$. $\square$

**Remark 2.12.** Using Proposition 2.10 and Proposition 2.11, we find, for all simplices $\sigma^n$ of $K$ and all vertices $\beta$ of $\sigma^n$, that the image of $e_{\sigma^n}(\beta)$ by $z_{\sigma^n}^1$ is the exact cocycle $d\chi_\beta^{19}$. Moreover, every form $\alpha \in \mathfrak{t}^*(\mathbb{F}_2)$ equals $\sum_{\beta \in \sigma^n} \alpha(e_{\sigma^n}(\beta))\beta$.

**Proposition 2.13.** Let $K$ be primitive complex in $\mathfrak{t}^*(\mathbb{R})$ and $\sigma^{n-1}$ be an interior simplex of $K$, i.e. bounding two maximal simplices $\sigma_+^n$ and $\sigma_-^n$. For all vertices $\alpha \in \sigma^{n-1}$ we denote by $\rho_{\sigma^{n-1}}(\alpha)$ the difference $e_{\sigma_+^n}(\alpha) - e_{\sigma_-^n}(\alpha)$. This vector lies on the line $N(\sigma^{n-1})$ and we denote by $|\rho_{\sigma^{n-1}}(\alpha)| \in \mathbb{F}_2$ the associated number. Moreover, if $\alpha_\pm$ denotes the vertex of $\sigma_\pm^n$ opposite to $\sigma^{n-1}$ then:

$$ \alpha_+ - \alpha_- = \sum_{\beta \in \sigma^{n-1}} |\rho_{\sigma^{n-1}}(\beta)| \pmod{2}. $$

*Proof.* Let $\alpha$ be a vertex of $\sigma^{n-1}$ and $\sigma^1$ be an edge of $\sigma^{n-1}$. Following Remark 2.12, we have:

$$ \omega(\sigma^1)(\rho_{\sigma^{n-1}}(\alpha)) = \omega(\sigma^1)(e_{\sigma_+^n}(\alpha) - e_{\sigma_-^n}(\alpha)) = 2d\chi_\alpha(\sigma^1) = 0. $$

Since $\{\omega(\sigma^1) : \sigma^1 \le \sigma^{n-1}\}$ spans $T\sigma^{n-1}(\mathbb{F}_2)$, the vector $\rho_{\sigma^{n-1}}(\alpha)$ belongs to the normal line $N(\sigma^{n-1})$. By Proposition 2.11, $e_{\sigma_+^n}(\alpha_+) = e_{\sigma_-^n}(\alpha_-)$. They are the generator $[N(\sigma^{n-1})]$ of $N(\sigma^{n-1})$. Hence, for any vertex $\beta \in \sigma^{n-1}$, we have:

$$ \begin{aligned} (\alpha_+ - \alpha_-)[N(\sigma^{n-1})] &= (\alpha_+ - \beta)[N(\sigma^{n-1})] + (\alpha_- - \beta)[N(\sigma^{n-1})] \\ &= (\alpha_+ - \beta)(e_{\sigma_+^n}(\alpha_+)) + (\alpha_- - \beta)(e_{\sigma_-^n}(\alpha_-)) \\ &= 1 + 1 = 0. \end{aligned} $$

And the form $(\alpha_+ - \alpha_-)$ is tangent to $\sigma^{n-1}$ modulo 2. Therefore, by Proposition 2.11 and Remark 2.12, we have the following formula:

$$ \alpha_+ - \alpha_- = \sum_{\beta \in \sigma^{n-1}} (\alpha_+ - \alpha_-)(e_{\sigma_+^n}(\beta))\beta. $$

For all $\beta \in \sigma^{n-1}$, we have:

$$ \begin{aligned} (\alpha_+ - \alpha_-)(e_{\sigma_+^n}(\beta)) &= (\alpha_+ - \beta)(e_{\sigma_+^n}(\beta)) + (\beta - \alpha_-)(e_{\sigma_+^n}(\beta)) \\ &= 1 + (\beta - \alpha_-)(e_{\sigma_-^n}(\beta)) + (\beta - \alpha_-)(\rho_{\sigma^{n-1}}(\beta)) \\ &= 1 + 1 + |\rho_{\sigma^{n-1}}(\beta)|(\beta - \alpha_-)[N(\sigma^{n-1})] \\ &= |\rho_{\sigma^{n-1}}(\beta)|, \end{aligned} $$

since $(\beta - \alpha_-)[N(\sigma^{n-1})]$ equals 1. Indeed, the opposite would imply that $[N(\sigma^{n-1})]$ is zero. $\square$

**Proposition 2.14.** For all interior simplices $\sigma^{n-1} \in K$ we have:

$$ \sum_{\alpha \in (\sigma^{n-1})^{(0)}} \rho_{\sigma^{n-1}}(\alpha) = 0. $$

<footnote n="19">¹⁹Recall that $\chi_\beta$ is the characteristic cochain of the vertex $\beta$. Its values are 1 on $\beta$ and 0 elsewhere.</footnote>
**Proof.** Let $\sigma_+^n$ and $\sigma_-^n$ be the two maximal simplices bounded by $\sigma^{n-1}$. By Proposition 2.11, we have:

$$ \sum_{\alpha \in (\sigma^{n-1})^{(0)}} \rho_{\sigma^{n-1}}(\alpha) = e_{\sigma_+^n}(\alpha_+) + e_{\sigma_-^n}(\alpha_-). $$

However, $e_{\sigma_+^n}(\alpha_+)$ and $e_{\sigma_-^n}(\alpha_-)$ both equal the generator of $N(\sigma^{n-1})$. Hence, the proposition follows. $\square$

**Definition 2.15.** Let $\varepsilon \in C^0(K; \mathbb{F}_2)$. The first derivative $D\varepsilon \in H_n(K; \partial K; T^*)$ is defined for all $\sigma^n \in K$ by the formula:

$$ D\varepsilon_{\sigma^n} := (z_{\sigma^n}^1)^{-1}(\mathrm{d}\varepsilon|_{\sigma^n}). $$

The following proposition, which is a direct consequence of Definition 2.15, justifies the choice of terminology.

**Proposition 2.16.** For all $\varepsilon \in C^0(K; \mathbb{F}_2)$, all $\sigma^n \in K$ and all vertices $\alpha, \beta \in \sigma^n$ we have:

$$ \varepsilon(\beta) = \varepsilon(\alpha) + D\varepsilon_{\sigma^n}(\beta - \alpha). $$

Therefore, $D\varepsilon$ vanishes if and only if $\varepsilon$ is constant.

**Definition 2.17.** Let $\varepsilon \in C^0(K; \mathbb{F}_2)$. The second derivative $D^2\varepsilon \in H_{n-1}(K; \partial K; N)$ is defined by the formula:

$$ D^2\varepsilon := \partial(D\varepsilon), $$

where $\partial$ denotes the connecting morphism $H_n(K; \partial K; T^*) \to H_{n-1}(K; \partial K; N)$ of the homological long exact sequence associated with the short exact sequence of cosheaves of Definition 2.7.

**Remark 2.18.** When the support of $K$ is a closed ball the homological long exact sequence associated with the short exact sequence of cosheaves of Definition 2.7 splits around the dimension $n$. It induces the following short exact sequence:

$$ 0 \to \underbrace{H_n(K; \partial K; t(\mathbb{F}_2))}_{\cong t(\mathbb{F}_2)} \to H_n(K; \partial K; T^*) \xrightarrow{\partial} H_{n-1}(K; \partial K; N) \to 0. $$

Because $N$ vanishes on maximal simplices. We note that $H_{n-1}(K; \partial K; N)$ is a group of cycles.

**Proposition 2.19.** Let $\sigma^{n-1}$ be an interior simplex bounding two maximal simplices $\sigma_\pm^n$, and $\alpha_\pm$ be the vertex of $\sigma_\pm^n$ opposite to $\sigma^{n-1}$. The number $D^2\varepsilon_{\sigma^{n-1}}(\alpha_- - \beta)$ does not depend on the choice of vertex $\beta \in \sigma^{n-1}$. It satisfies the following property:

$$ \varepsilon(\alpha_+) = \varepsilon(\alpha_-) + D\varepsilon_{\sigma_+^n}(\alpha_+ - \alpha_-) + D^2\varepsilon_{\sigma^{n-1}}(\alpha_- - \beta). $$

Moreover, if $|K|$ is homeomorphic to a ball, $D^2\varepsilon$ vanishes if and only if $\varepsilon$ is affine.

**Proof.** Let $\beta$ be a vertex of $\sigma^{n-1}$. We have:

$$ \begin{aligned} \varepsilon(\alpha_+) + \varepsilon(\alpha_-) &= D\varepsilon_{\sigma_+^n}(\alpha_+ + \beta) + D\varepsilon_{\sigma_-^n}(\alpha_- + \beta) \\ &= D\varepsilon_{\sigma_+^n}(\alpha_+ + \alpha_-) + D^2\varepsilon_{\sigma^{n-1}}(\alpha_- + \beta). \end{aligned} $$
Hence $\varepsilon(\alpha_+)$ equals $\varepsilon(\alpha_-) + D\varepsilon_{\sigma_+^n}(\alpha_+ - \alpha_-) + D^2\varepsilon_{\sigma^{n-1}}(\alpha_- - \beta)$ and $D^2\varepsilon_{\sigma^{n-1}}(\alpha_- - \beta)$ does not depend on $\beta$. The cochain $\varepsilon$ is represented in each $n$-simplex $\sigma^n$ of $K$ by an affine function of $t^*(\mathbb{F}_2)$ whose linear part is given by $D\varepsilon_{\sigma^n}$. Using the exact sequence of Remark 2.18, we see that if $D^2\varepsilon$ vanishes then this linear part is the same on every maximal simplices of $K$ since $K$ is connected. Hence $\varepsilon$ is globally given by an affine map. $\square$

**Lemma 2.20.** If the primitive complex $K$ is shellable then both $D$ and $D^2$ are surjective. Therefore, we have the exact sequences:

$$0 \to \mathbb{F}_2 \to C^0(K; \mathbb{F}_2) \xrightarrow{D} H_n(K; \partial K; T^*) \to 0,$$

and:

$$0 \to \mathrm{Aff}(t^*(\mathbb{F}_2)) \to C^0(K; \mathbb{F}_2) \xrightarrow{D^2} H_{n-1}(K; \partial K; N) \to 0.$$

*Proof.* Let $\varepsilon \in C^0(K; \mathbb{F}_2)$ and $\sigma^n$ be a maximal simplex of $K$. By Definition 2.15, we have the following formula:

$$D\varepsilon_{\sigma^n} = \sum_{\alpha \in \sigma^n} \varepsilon(\alpha)e_{\sigma^n}(\alpha).$$

Now, let $\sigma^{n-1}$ be an interior simplex of $K$. By Definition 2.17, we have:

$$D^2\varepsilon_{\sigma^{n-1}} = (\varepsilon(\alpha_+) + \varepsilon(\alpha_-))[N(\sigma^{n-1})] + \sum_{\alpha \in \sigma^{n-1}} \varepsilon(\alpha)\rho_{\sigma^{n-1}}(\alpha),$$

where $\alpha_\pm * \sigma^{n-1}$ are the two $n$-simplices bounded by $\sigma^{n-1}$, and $[N(\sigma^{n-1})]$ is the generator of $N(\sigma^{n-1})$. We can also write it as follows:

$$D^2\varepsilon_{\sigma^{n-1}} = (\varepsilon(\alpha_+) + \varepsilon(\alpha_-) + \sum_{\alpha \in \sigma^{n-1}} \varepsilon(\alpha)|\rho_{\sigma^{n-1}}(\alpha)|)[N(\sigma^{n-1})].$$

We will use this formula to show that when $K$ is shellable, $D^2$ is surjective. Let $(\sigma_i^n)_{0 \le i \le k}$ be a shelling of $K$, and $c \in H_{n-1}(K; \partial K; N)$. We will construct a $\varepsilon$ whose image by $D^2$ is $c$. For all $0 \le i \le k$, we denote by $K_i$ the union $\cup_{0 \le j \le i} K(\sigma_j^n)$. For all interior simplices $\sigma^{n-1}$, we denote by $|\langle c; \sigma^{n-1} \rangle| \in \mathbb{F}_2$ the unique number for which $|\langle c; \sigma^{n-1} \rangle|[N(\sigma^{n-1})]$ equals $\langle c; \sigma^{n-1} \rangle$. We note that $K_{i+1}^{(0)}$ equals $K_i^{(0)}$ if the order of $\sigma_{i+1}^n$ is bigger than 1. Otherwise, $K_{i+1}^{(0)} \setminus K_i^{(0)}$ is reduced to the vertex $lk(K_i; K_{i+1})$. It induces an order $(\alpha_j)_{1 \le j \le h_1(K)}$ on the vertices of $K$ that do not belong to $\sigma_0^n$. For all $1 \le j \le h_1(K)$, we denote by $\sigma_j^{n-1}$ the simplex for which $\sigma_j^{n-1} * \alpha_j$ is the simplex of order 1 at which $\alpha_j$ appears. We also denote by $\alpha_j^-$ the other vertex of the link of $\sigma_j^{n-1}$ in $K$. Now we inductively define $\varepsilon$ by the following rule:

$$\left\{ \begin{array}{l} \varepsilon|_{\sigma_0^n} = 0 \\ \\ \varepsilon(\alpha_j) = |\langle c; \sigma_j^{n-1} \rangle| + \varepsilon(\alpha_j^-) + \sum_{\alpha \in \sigma_j^{n-1}} \varepsilon(\alpha)|\rho_{\sigma_j^{n-1}}(\alpha)|, \quad \forall 1 \le j \le h_1(K). \end{array} \right.$$

By construction, $D^2\varepsilon_{\sigma_j^{n-1}}$ equals $\langle c; \sigma_j^{n-1} \rangle$ for all $1 \le j \le h_1(K)$. We claim this is also true on the other interior $(n-1)$-simplices of $K$. We will show by induction on $0 \le i \le k$, that $D^2\varepsilon_{\sigma^{n-1}}$ equals $\langle c; \sigma^{n-1} \rangle$ for all interior $(n-1)$-simplices of $K_i$. The assertion is true on $K_1$ for $\sigma_1^n$ has
order 1. Let us assume it true on $K_i$. The only interior $(n-1)$-simplices of $K_{i+1}$ that were not already interior in $K_i$ are those contained in $\sigma_{i+1}^n$ that contain $\tau_{i+1}$ (the intersection of the contact faces of $\sigma_{i+1}^n$ with $K_i$). If $\mathrm{ord}(\sigma_{i+1}^n)$ is 1, then by construction the property is true on $K_{i+1}$. If $\mathrm{ord}(\sigma_{i+1}^n)$ is bigger than 1, let us consider two $(n-1)$-simplices $\sigma_{\pm}^{n-1}$ that contain $\tau_i$. Since $D^2\varepsilon$ and $c$ are cycles we have:

$$D^2\varepsilon_{\sigma_+^{n-1}} + D^2\varepsilon_{\sigma_-^{n-1}} = \sum_{\substack{\sigma^{n-1} \ge \sigma_+^{n-1} \cap \sigma_-^{n-1} \\ \sigma^{n-1} \ne \sigma_{\pm}^{n-1}}} D^2\varepsilon_{\sigma^{n-1}}, \qquad (7)$$

and:

$$\langle c; \sigma_+^{n-1} \rangle + \langle c; \sigma_-^{n-1} \rangle = \sum_{\substack{\sigma^{n-1} \ge \sigma_+^{n-1} \cap \sigma_-^{n-1} \\ \sigma^{n-1} \ne \sigma_{\pm}^{n-1}}} \langle c; \sigma^{n-1} \rangle. \qquad (8)$$

If $\sigma^{n-1} \in K$ contains $\sigma_+^{n-1} \cap \sigma_-^{n-1}$ and is different from $\sigma_+^{n-1}$ and $\sigma_-^{n-1}$ it cannot belong to $\sigma_{i+1}^n$ for $\sigma_+^{n-1} \cap \sigma_-^{n-1}$ has codimension 2. Hence, by Lemma 2.5, it belongs to $K_i$. Thus, the right hand sides of (7) and (8) are equal by recursion hypothesis. Therefore, we have the following relation:

$$D^2\varepsilon_{\sigma_+^{n-1}} + D^2\varepsilon_{\sigma_-^{n-1}} = \langle c; \sigma_+^{n-1} \rangle + \langle c; \sigma_-^{n-1} \rangle. \qquad (9)$$

The vector $[N(\sigma_+^{n-1})]$ is linearly independent from $[N(\sigma_-^{n-1})]$. Thus (9) implies that $D^2\varepsilon_{\sigma_{\pm}^{n-1}}$ equals $\langle c; \sigma_{\pm}^{n-1} \rangle$ for $D^2\varepsilon_{\sigma_{\pm}^{n-1}}$ and $\langle c; \sigma_{\pm}^{n-1} \rangle$ belong to the line $N(\sigma_{\pm}^{n-1})$.

Since $K$ is embedded in a real vector space of the same dimension, its support can only be homeomorphic<fnref n="20" /> to a ball. Hence, we have the exact sequence of Remark 2.18. From Proposition 2.16, Proposition 2.19, and the surjectivity of $D^2$, we have the following commutative diagram with exact rows and columns:

<figure><img src="image_11.png" /></figure>

from which derive the exact sequences of the lemma.

<figure><img src="image_12.png" /></figure>

**Definition 2.21.** We define two enlargement of the boundary of $K$:

1. the sub-complex $B_0K$ of $K$ made of all the simplices of $K$ having all of their vertices in $\partial K$;

<footnote n="20">²⁰c.f. Proposition 2.4.</footnote>
2. the sub-complex $B_1K$ of $K$ made of the union of the closed simplices $K(\sigma^p)$ of $K$ for which all of their vertices $v \in K(\sigma^p)$ satisfy $v \in \partial K$ or $\text{lk}(v; \sigma^p) \in \partial K$. See Figure 10 for an example.

We note that $B_1K$ contains $B_0K$ and all the vertices of $K$. The simplices of $B_1K$ that do not belong to $B_0K$ are joins of an interior vertex of $K$ and a simplex of $\partial K$. For such a simplex $\sigma$ we denote by $\text{ap}(\sigma)$ its interior vertex.

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Figure 10: Examples of $B_0K$ and $B_1K$.</figcaption></figure>

**Definition 2.22.** Let $\sigma^{n-1}$ be an interior simplex of $K$. we denote by $I(\sigma^{n-1})$ the set of vertices $\alpha$ of $\sigma^{n-1}$ that do not belong to $\partial K$ and whose link $\text{lk}(\alpha; \sigma^{n-1})$ is not contained in $\partial K$ either. We say that $K$ is $\rho$-uniform if for all interior simplices $\sigma^{n-1} \in K$ the restriction of $\rho_{\sigma^{n-1}}$ to $I(\sigma^{n-1})$ is constant. In particular, because of Proposition 2.14, if $\dim K \le 2$ then $K$ is $\rho$-uniform. If $K$ is not $\rho$-uniform we say that $K$ fails to be $\rho$-uniform at $\sigma^{n-1}$ if $\rho_{\sigma^{n-1}}$ is not constant on $I(\sigma^{n-1})$. For all $\rho$-uniform complexes $K$, we denote by $\rho \in C_{n-1}(K; B_0K; N)$ the chain whose value on $\sigma^{n-1} \in K \setminus B_1K$ is $\rho_{\sigma^{n-1}}(\alpha)$ for any $\alpha \in I(\sigma^{n-1})$, and whose value on $\sigma^{n-1} \in B_1K \setminus B_0K$ is $\rho_{\sigma^{n-1}}(\text{ap}(\sigma^{n-1}))$.

**Proposition 2.23.** If $K$ is $\rho$-uniform, the chain $\rho$ is a cycle. It induces a homology class denoted by the same symbol.

*Proof.* Let $\sigma^{n-2}$ be a simplex of $K$ that does not belong to $\partial K$. If $n$ equals 2 then $\langle \partial\rho; \sigma^0 \rangle$ is $\sum_{\sigma^1 \ge \sigma^0} \rho_{\sigma^1}(\sigma^0)$ since $\rho$ is constant on $\sigma^1$. This vector is zero by definition of $\rho_{\sigma^1}(\sigma^0)$ as the difference $\sum_{\sigma^2 \ge \sigma^1} e_{\sigma^2}(\sigma^0)$. Now if $n$ is bigger than 2, then either $\sigma^{n-2}$ belongs to $B_1K$ or not but either way there is an interior vertex $\alpha$ of $\sigma^{n-2}$ for which $\langle \partial\rho; \sigma^{n-2} \rangle$ equals $\sum_{\sigma^{n-1} \ge \sigma^{n-2}} \rho_{\sigma^1}(\alpha)$. This vector vanishes for the same reason as in the case $n=2$. $\square$

# 3 Computing the Renaudineau-Shaw Spectral Sequence

The aim of this section is to give an effective computation of the first page of the Renaudineau-Shaw spectral sequence for T-hypersurfaces. This spectral sequence is derived from a filtration of the cellular chain complex of the T-hypersurface $\mathbb{R}X_\epsilon$. A. Renaudineau and K. Shaw showed that the groups of its first page are isomorphic to the tropical homology groups of the tropical
counterpart $X$ of $\mathbb{R}X_\epsilon$. We will compute here the boundary operators of this first page. To do so, we will follow a usual path: in a first time we will lift a tropical homology class of $X$ as a relative cycle of $\mathbb{R}X_\epsilon$, then we will compute its boundary and reproject it as a tropical homology class of $X$. To simplify this task we will reduce ourself to subspaces of tropical homology groups made of classes having a suitable representing cycle. Following the computation of the boundary of these lifts, we will finally display the obtained information in matrices and use their ranks to express the dimensions of the groups of the second page of the sequence.

**Lifting Tropical Cycles.** Let $P$ be a smooth $n$-dimensional polytope of $t^*(\mathbb{R})$ endowed with a primitive triangulation $K$. For all $\sigma^p \in K$, we choose $w(\sigma^p) = w_1(\sigma^p) \wedge \dots \wedge w_p(\sigma^p)$ a totally decomposed $p$-vector of $t(\mathbb{F}_2)$ satisfying $\omega(\sigma^p)(w(\sigma^p)) = 1$.

**Definition 3.1.** Let $0 \le p \le n-1$ be an integer. We define $g^p: C^p(K; \partial K; \mathbb{F}_2) \to H_{p,n-1-p}(X; \mathbb{F}_2)$ by the following formula. For all $p$-cochain $\gamma$ we set:

$$g^p(\gamma) = \left[ \sum_{\sigma^{p+1} \le \sigma^n} d(\gamma w)(\sigma^{p+1}) \otimes [\sigma^{p+1}; \sigma^n] \right]. \qquad (10)$$

We can note that $g^0$ is known to be an isomorphism. Indeed, since $F_0^X$ is the constant cosheaf $\mathbb{F}_2$, the group $H_{0,n-1}(X; \mathbb{F}_2)$ is the (usual) homology group of $X$ with coefficients in $\mathbb{F}_2$. The space $X$ is homotopic to a wedge of $(n-1)$-spheres, one for each interior vertex of $K$. Let $p$ be an interior vertex of $K$, the class $g^0(\chi_p)$ is the fundamental class of the corresponding sphere, which is the link of $p$ in the barycentric subdivision of $K$.

**Lemma 3.2.** For all integers $0 \le p \le n-1$, the morphism $g^p$ does not depend on the choice of $w$. Moreover its image is the kernel of morphism $i_{p,n-1-p}: H_{p,n-1-p}(X; \mathbb{F}_2) \to H_{p,n-1-p}(P; \mathbb{F}_2)$.

*Proof.* In a first time we can notice the cycle :

$$z = \sum_{\sigma^{p+1} \le \sigma^n} d(\gamma w)(\sigma^{p+1}) \otimes [\sigma^{p+1}; \sigma^n] \in \Omega_{n-1-p}(K; F_p^X),$$

is the boundary of the chain :

$$b = \sum_{\sigma^p \le \sigma^n} \gamma(\sigma^p) w(\sigma^p) \otimes [\sigma^p; \sigma^n] \in \Omega_{n-p}(K; F_p^P).$$

Hence it shows that the image of $g^p(\gamma)$ in the tropical homology of the toric variety $P$ vanishes. Also, should we choose another $w$, say $w'$, the chain :

$$b - b' = \sum_{\sigma^p \le \sigma^n} \gamma(\sigma^p)(w - w')(\sigma^p) \otimes [\sigma^p; \sigma^n],$$

belongs to $\Omega_{n-p}(K; F_p^X)$ since $\omega(\sigma^p)(w(\sigma^p) - w'(\sigma^p))$ vanishes by hypotheses on $w$ and $w'$. Now the boundary of $b - b'$ is :

$$\partial(b - b') = z - \sum_{\sigma^{p+1} \le \sigma^n} d(\gamma w')(\sigma^{p+1}) \otimes [\sigma^{p+1}; \sigma^n].$$
And the class $g^p(\gamma)$ does not depend on $w$. The fact that the image of $g^p$ is the kernel of $i_{p,n-1-p}$ comes from Theorem 1 p.20 and Theorem 2 p.32 of [Che23a]. They assert that both the tropical homology of $P$ and $X$ can be computed as the cohomology of cellular sheaves on $K$. Let $Q$ denote either $P$ or $X$ and define, as in the proof of aforementioned Theorem 2, the cellular sheaves of local tropical homology:

$$G_p^Q : \sigma^q \mapsto H_n(K; K - \sigma^q; F_p^Q).$$

Then the inclusion $i_p : F_p^X \to F_p^P$ induces an inclusion $j_p : G_p^X \to G_p^P$ and we have a commutative diagram:

$$\begin{tikzcd}
H_{p,n-1-p}(X; \mathbb{F}_2) \arrow[r, "i_{p,n-1-p}"] \arrow[d, "\cong"] & H_{p,n-1-p}(P; \mathbb{F}_2) \arrow[d, "\cong"] \\
H^{p+1}(K; G_p^X) \arrow[r, "j_p^{p+1}"] & H^{p+1}(K; G_p^P)
\end{tikzcd}$$

The cokernel sheaf of $j_p$ is supported$^{21}$ on the $p$-th skeleton of $K$ and for all simplex $\sigma^p$, coker($j_p$)($\sigma^p$) is either $\mathbb{F}_2$ or 0, the latter occurring if and only if $\sigma^p$ lies on the boundary of $P$. Therefore the group of $p$-cocycles with coefficients in coker($j_p$) is precisely $C^p(K; \partial K; \mathbb{F}_2)$ and the exactness of the sequence:

$$C^p(K; \partial K; \mathbb{F}_2) \xrightarrow{g^p} H_{p,n-1-p}(X; \mathbb{F}_2) \to H_{p,n-1-p}(P; \mathbb{F}_2) \to 0,$$

comes from the exactness of the sequence:

$$H^p(K; \mathrm{coker}(j_p)) \xrightarrow{d} H^{p+1}(K; L_p^X) \to H^{p+1}(K; L_p^P) \to 0.$$

□

**Remark 3.3.** The morphism $i_{p,n-1-p} : H_{p,n-1-p}(X; \mathbb{F}_2) \to H_{p,n-1-p}(P; \mathbb{F}_2)$ is surjective. This is a consequence of the Tropical Lefschetz Theorem$^{22}$.

**Lemma 3.4 (Origin).** Let $\varepsilon \in C^0(K; \mathbb{F}_2)$ be a sign distribution, $\sigma^p \le \sigma^{p+1} \le \sigma^n$ be three simplices of $K$, and $W$ be a supplementary sub-space of $N(e^p)$ in $\mathfrak{t}(\mathbb{F}_2)$. The point:

$$\mathbf{O}_{\sigma^n}(e^p) := \mathrm{D}_{\varepsilon_{\sigma^n}} + \sum_{\sigma^0 \le \sigma^p} e_{\sigma^n}(\sigma^0) \in \mathfrak{t}(\mathbb{F}_2),$$

satisfies $\mathbf{O}_{\sigma^n}(e^p) + W \subset \mathrm{Arg}_{\varepsilon}(\sigma^{p+1}; \sigma^n)$.

*Proof.* Following Lemma 3.7 p.18 in [Che23b], the set $\mathrm{Arg}_{\varepsilon}(\sigma^{p+1}; \sigma^n)$ is described as the following set $\{v \in \mathfrak{t}(\mathbb{F}_2) | z_{\sigma^{p+1}}^1(v) \neq d\varepsilon|_{\sigma^{p+1}}\}$. By Definition 2.15, the image of the vector $\mathrm{D}_{\varepsilon_{\sigma^n}}$ by $z_{\sigma^n}^1$ is $d\varepsilon|_{\sigma^n}$. Hence, by Proposition 2.10, $z_{\sigma^{p+1}}^1(\mathrm{D}_{\varepsilon_{\sigma^n}})$ equals $d\varepsilon|_{\sigma^{p+1}}$. Let us denote by $v$ the vector:

$$v := \sum_{\beta \text{ vertex of } \sigma^p} e_{\sigma^n}(\beta).$$

<footnote>$^{21}$c.f. [Che23a] Theorem 2 p.32, especially the end of the proof.</footnote>

<footnote>$^{22}$c.f. [ARS21] Theorem 1.2 p.1349, [BLdMR22] Proposition 3.2 p.15, or [Che23a] Corollary 3 p.34.</footnote>
Proposition 2.11 ensures that $z_{\sigma^{p+1}}^1(v)$ equals $d\chi_\alpha$ where $\alpha$ is the vertex of $\sigma^{p+1}$ opposite to $\sigma^p$. If $\sigma^1$ is an edge of $\sigma^{p+1}$ that contains $\alpha$ then $z_{\sigma^{p+1}}^1(D\varepsilon_{\sigma^n} + v)(\sigma^1)$ equals $d\varepsilon(\sigma^1) + 1$, and $D\varepsilon_{\sigma^n} + v$ belongs to $\text{Arg}_\varepsilon(\sigma^{p+1}; \sigma^n)$. Let $w \in t(F_2)$ be a vector that does not belong to $N(e^p)$. The restriction of $z_{\sigma^{p+1}}^1(D\varepsilon_{\sigma^n} + v + w)$ to $\sigma^p$ is $d\varepsilon|_{\sigma^p} + z_{\sigma^p}^1(w)$. Since $w$ does not belong to $N(e^p)$, $z_{\sigma^p}^1(w)$ is different from zero and, consequently, $D\varepsilon_{\sigma^n} + v + w$ belongs to $\text{Arg}_\varepsilon(\sigma^{p+1}; \sigma^n)$. $\square$

**Lemma 3.5 (Lift).** Let $\varepsilon \in C^0(K; \mathbb{F}_2)$ be a sign distribution, $0 \le p \le n-1$ be an integer, and $\gamma$ be a cochain of $C^p(K; \partial K; \mathbb{F}_2)$. The chain:

$$c := \sum_{\sigma^{p+1} \le \sigma^n} d \left( \gamma x^{\mathbf{o}_{\sigma^n}} \prod_{i=1}^{p} (1 + x^{w_i}) \right) (\sigma^{p+1}) \otimes [\sigma^{p+1}; \sigma^n] \in \Omega_{n-1-p} (K; \mathcal{K}_{(p)}^{\mathbb{R}X_\varepsilon}),$$

has its boundary equal to:

$$\sum_{\sigma^{p+1} \le \sigma^{n-1}} d \left( \gamma G_{\sigma^{n-1}} \prod_{i=1}^{p} (1 + x^{w_i}) \right) (\sigma^{p+1}) \otimes [\sigma^{p+1}; \sigma^{n-1}],$$

where $G_{\sigma^{n-1}}(\sigma^p)$ is null on the $p$-simplices of the boundary and otherwise equals $\sum_{\sigma^n \ge \sigma^{n-1}} x^{\mathbf{o}_{\sigma^n}(\sigma^p)}$. This boundary belongs to $\Omega_{n-2-p}(K; \mathcal{K}_{(p+1)}^{\mathbb{R}X_\varepsilon})$ and $c$ represents the homology class $g^p(\gamma)$.

*Proof.* Let $0 \le p \le n-1$ be an integer. By definition, $(E_{p,q}^1(\mathbb{R}X_\varepsilon))_{q \ge 0}$ is the homology of the $p$-th graded piece of the filtered cosheaf $\mathcal{K}^{\mathbb{R}X_\varepsilon}$. By Proposition 4.22 p.29 in [Che23b], this cosheaf is isomorphic to $F_p^X$ through $bv_p^X$. Therefore, any homology class $[c] \in H_{p,q}(X; \mathbb{F}_2)$ can be represented by a relative cycle, i.e. a chain $c \in \Omega_q(K; \mathcal{K}_{(p)}^{\mathbb{R}X_\varepsilon})$ whose boundary belongs to $\Omega_{q-1}(K; \mathcal{K}_{(p+1)}^{\mathbb{R}X_\varepsilon})$. Let $\sigma^{p+1} \le \sigma^n$ be a pair of simplices of $K$, and $\mathbf{o} + \langle v_1, \dots, v_p \rangle$ be a $p$-affine subspace of $t(\mathbb{F}_2)$ contained in $\text{Arg}_\varepsilon(\sigma^{p+1}; \sigma^n)$. From Remark 1.24, we have:

$$\text{bv}_p^X(\sigma^{p+1}; \sigma^n) \left( x^{\mathbf{o}} \prod_{i=1}^{p} (1 + x^{v_i}) \right) = \bigwedge_{i=1}^{p} v.$$

Since $\omega(\sigma^p)(w(\sigma^p))$ equals 1, for all $\sigma^p \le \sigma^{p+1}$, the vector space $\langle w_1(\sigma^p), \dots, w_p(\sigma^p) \rangle$ is a supplementary sub-space of $N(\sigma^p)$ in $t(\mathbb{F}_2)$. Using Lemma 3.4, it follows that the chain:

$$c := \sum_{\sigma^{p+1} \le \sigma^n} d \left( \gamma x^{\mathbf{o}_{\sigma^n}} \prod_{i=1}^{p} (1 + x^{w_i}) \right) (\sigma^{p+1}) \otimes [\sigma^{p+1}; \sigma^n],$$

lifts the chain $\sum_{\sigma^{p+1} \le \sigma^n} d(\gamma w)(\sigma^{p+1}) \otimes [\sigma^{p+1}; \sigma^n]$. Let $\sigma^q \le \sigma^{q+n-p-2}$ be a pair of simplices of $K$. We have:

$$\langle \partial c; [\sigma^q; \sigma^{q+n-p-2}] \rangle = \sum_{\sigma^{q-1} \le \sigma^q} \langle c; [\sigma^{q-1}; \sigma^{q+n-p-2}] \rangle + \sum_{\sigma^{q+n-p-2}; \sigma^{q+n-p-1}} \langle c; [\sigma^q; \sigma^{q+n-p-1}] \rangle.$$

If $q$ is different from $p+1$ and $p+2$, $\langle \partial c; [\sigma^q \le \sigma^{q+n-p-2}] \rangle$ vanishes. For the remaining pieces we have:

$$\langle \partial c; [\sigma^{p+2}; \sigma^n] \rangle = \sum_{\sigma^{p+1} \le \sigma^{p+2}} d \left( \gamma x^{\mathbf{o}_{\sigma^n}} \prod_{i=1}^{p} (1 + x^{w_i}) \right) (\sigma^{p+1}) = 0,$$
and:

$$
\begin{align*}
\langle \partial c; [\sigma^{p+1}; \sigma^{n-1}] \rangle &= \sum_{\sigma^{n-1} \le \sigma^n} d \left( \gamma x^{\mathbf{O}_{\sigma^n}} \prod_{i=1}^p (1 + x^{w_i}) \right) (\sigma^{p+1}) \\
&= \sum_{\sigma^p \le \sigma^{p+1}} \gamma(\sigma^p) \sum_{\sigma^{n-1} \le \sigma^n} (x^{\mathbf{O}_{\sigma^n}(\sigma^p)}) \prod_{i=1}^p (1 + x^{w_i(\sigma^p)}) \\
&= \sum_{\sigma^p \le \sigma^{p+1}} \gamma(\sigma^p) G_{\sigma^{n-1}}(\sigma^p) \prod_{i=1}^p (1 + x^{w_i(\sigma^p)}) \\
&= d \left( \gamma G_{\sigma^{n-1}} \prod_{i=1}^p (1 + x^{w_i}) \right) (\sigma^{p+1}),
\end{align*}
$$

since $\gamma$ vanishes on the boundary of $K$. Whenever we have a triple $\sigma^p \le \sigma^{p+1} \le \sigma^{n-1}$ with $\sigma^p$ not contained in $\partial K$, there are two $n$-simplices $\sigma_{\pm}^n$ adjacent to $\sigma^{n-1}$. In this case $\text{Arg}_{\epsilon}(\sigma^{p+1}; \sigma^{n-1}) = \text{Arg}_{\epsilon}(\sigma^{p+1}; \sigma^n)$, and $\mathbf{O}_{\sigma_{\pm}^n} + \langle w_1(\sigma^p), \dots, w_p(\sigma^p) \rangle$ are two parallel affine sub-spaces contained in this set. Therefore $G_{\sigma^{n-1}}(\sigma^p) \prod_{i=1}^p (1+x^{w_i(\sigma^p)})$ belongs to $\mathcal{K}_{(p+1)}^{\mathbb{R}\epsilon}(\sigma^{p+1}; \sigma^{n-1})$. Thus $\partial c$ belongs to $\Omega_{n-2-p}(K; \mathcal{K}_{(p+1)}^{\mathbb{R}\epsilon})$. $\square$

**Proposition 3.6.** Let $\epsilon \in C^0(K; \mathbb{F}_2)$ be a sign distribution on $K$, $0 \le p \le n-1$ be an integer, and $\gamma$ be a cochain of $C^p(K; \partial K; \mathbb{F}_2)$. We have:

$$
\partial_{p,n-1-p}^{1} g^{p}(\gamma) = \left[ \sum_{\sigma^{p+1} \le \sigma^{n-1}} d \left( \gamma w \wedge (D^2 \varepsilon_{\sigma^{n-1}} + \rho_{\sigma^{n-1}}^{p}) \right) (\sigma^{p+1}) \otimes [\sigma^{p+1}; \sigma^{n-1}] \right],
$$

where $\rho_{\sigma^{n-1}}^{p} \in C^{p}(\sigma^{n-1}; N(\sigma^{n-1}))$ is defined, for all $\sigma^{p} \le \sigma^{n-1}$, by the formula:

$$
\rho_{\sigma^{n-1}}^{p}(\sigma^{p}) := \sum_{\alpha \text{ vertex of } \sigma^{p}} \rho_{\sigma^{n-1}}(\alpha).
$$

*Proof.* By Lemma 3.5, $\partial_{p,n-1-p}^1 g^p(\gamma)$ is given by the formula:

$$
\left[ \sum_{\sigma^{p+1} \le \sigma^{n-1}} \mathrm{bv}_{p+1}^{X}(\sigma^{p+1}; \sigma^{n-1}) \left( d \left( \gamma G_{\sigma^{n-1}} \prod_{i=1}^{p} (1 + x^{w_i}) \right) (\sigma^{p+1}) \right) \otimes [\sigma^{p+1}; \sigma^{n}] \right].
$$

Let $\sigma^p \le \sigma^{p+1} \le \sigma^{n-1}$ be a triple of simplices with $\sigma^p$ not contained in $\partial K$. From Remark 1.24, we have:

$$
\begin{align*}
\mathrm{bv}_{p+1}^{X}(\sigma^{p+1}; \sigma^{n-1}) \left( G_{\sigma^{n-1}}(\sigma^p) \prod_{i=1}^{p} (1 + x^{w_i(\sigma^p)}) \right) &= \left( \sum_{\sigma^n \ge \sigma^{n-1}} \mathbf{O}_{\sigma^n}(\sigma^p) \right) \wedge w(\sigma^p) \\
&= (D^2 \varepsilon_{\sigma^{n-1}} + \rho_{\sigma^{n-1}}^p(\sigma^p)) \wedge w(\sigma^p).
\end{align*}
$$
Therefore we find that:

$$ \partial_{p,n-1-p}^1 g^p(\gamma) = \left[ \sum_{\sigma^{p+1} \le \sigma^{n-1}} d(\gamma w \wedge (\mathrm{D}^2\varepsilon_{\sigma^{n-1}} + \rho_{\sigma^{n-1}}^p)) (\sigma^{p+1}) \otimes [\sigma^{p+1}; \sigma^{n-1}] \right]. \quad \square $$

**Intersection Matrices.** We use the tropical intersection product to represent the boundary operators of the first page as matrices and provide formulæfor the dimensions of the groups of the second page that take into account that the morphisms $g^p$ are not surjective in general.

**Definition 3.7.** Let $\varepsilon \in C^0(K; \mathbb{F}_2)$ be a sign distribution, and $\sigma^p$ and $\sigma^q$ be two interior simplices of $K$ with $p+q=n-2$. We denote by $(\sigma^p; \sigma^q)_\varepsilon$ the number:

$$ (\sigma^p; \sigma^q)_\varepsilon := g^p(\chi_{\sigma^p}) \cdot \partial_{q,p+1}^1 g^q(\chi_{\sigma^q}) \in H_{0,0}(X; \mathbb{F}_2) = \mathbb{F}_2, $$

where $\cdot - \cdot$ denotes the tropical intersection product. For all integers $p, q \ge 0$ for which $p+q=n-2$, let $M_{p,q}(\varepsilon)$ denote the matrix whose entries are given by the numbers $((\sigma^p; \sigma^q)_\varepsilon)_{\sigma^p, \sigma^q \in K \setminus \partial K}$.

**Definition 3.8.** We assume that $n = 2k+1$ is odd. We define the *radical* of $X$, denoted by $\mathrm{Rad}(X)$, to be the radical of the intersection form restricted to the kernel of the morphism $i_{k,k}: H_{k,k}(X; \mathbb{F}_2) \to H_{k,k}(P; \mathbb{F}_2)$. We denote by $\mathrm{rad}(X)$ the dimension of $\mathrm{Rad}(X)$.

**Proposition 3.9.** We assume that $n = 2k+1$ is odd. Let $\omega_X$ denote the Poincaré dual in $P$ of the fundamental class of $X$. The *radical* $\mathrm{Rad}(X)$ is generated by the classes $i^{k,k}(\gamma) \cap [X] \in H_{k,k}(X; \mathbb{F}_2)$ for all $\gamma \in H^{k,k}(P; \mathbb{F}_2)$ for which $\gamma \cup \omega_X$ vanishes.

*Proof.* The Poincaré duality in $X$ provides us with an isomorphism between the group $H_{k,k}(X; \mathbb{F}_2)$ and its dual $H^{k,k}(X; \mathbb{F}_2)$. The orthogonal of $\mathrm{ker}(i_{k,k})$ is the image under this isomorphism of $\mathrm{im}(i^{k,k})$ i.e. is spanned by the element of the form $i^{k,k}(\gamma) \cap [X]$ where $\gamma$ belongs to $H^{k,k}(P; \mathbb{F}_2)$. If we denote by $\omega_X$ the Poincaré dual in $P$ of $[X]$, we have:

$$ i_{k,k}(i^{k,k}(\gamma) \cap [X]) = (\gamma \cup \omega_X) \cap [P]. $$

Therefore:

$$ \mathrm{Rad}(X) = \langle i^{k,k}(\gamma) \cap [X] \in H_{k,k}(X; \mathbb{F}_2) : \gamma \in H^{k,k}(P; \mathbb{F}_2) \text{ s.t. } \gamma \cup \omega_X = 0 \rangle. $$

**Proposition 3.10.** Let $\varepsilon \in C^0(K; \mathbb{F}_2)$ be a sign distribution. For all $p+q=n-2$, $M_{p,q}(\varepsilon)$ is the transpose of $M_{q,p}(\varepsilon)$. Moreover:

1. If $n$ is even, the rank of $M_{p,q}(\varepsilon)$ equals the rank of $\partial_{q,n-1-q}^1$;

2. If $n = 2k+1$ is odd, it is also the case except when $(p, q)$ equals $(k-1, k)$ or $(k, k-1)$. In the first case the rank of $M_{k-1,k}(\varepsilon)$ equals the rank of the restriction of $\partial_{k,k}^1$ to the kernel of $i_{k,k}^1: E_{k,k}^1(\mathbb{R}X_\varepsilon) \to E_{k,k}^1(\mathbb{R}P)$. In the second one, $M_{k,k-1}(\varepsilon)$ equals the rank of the composition of $\partial_{k+1,k-1}^1$ with the projection of $\mathrm{ker}(i_{k,k}^1)$ to its quotient by $\mathrm{Rad}(X)$.
Proof. Let $\sigma^p, \sigma^q$ be two simplices of $K$ satisfying $p + q = n - 2$, i.e. $q = n - 1 - (p + 1)$. By Definition 3.7 we have:

$$ (\sigma^p; \sigma^q)_\varepsilon = g^p(\chi_{\sigma^p}) \cdot \partial_{n-1-(p+1),p+1}^1 g^q(\chi_{\sigma^q}). $$

Let $\gamma \in H^{n-1-p,p}(X; \mathbb{F}_2)$ denote the Poincaré dual$^{23}$ of $g^p(\chi_{\sigma^p})$. By definition of the tropical intersection product, we have:

$$ (\sigma^p; \sigma^q)_\varepsilon = \gamma(\partial_{n-1-(p+1),p+1}^1 g^q(\chi_{\sigma^q})) = (d_1^{n-1-p,p}\gamma)(g^q(\chi_{\sigma^q})), $$

since $(\partial_{a,b}^r)^*$ equals $d_r^{a+r,b-1}$ by duality of the spectral sequences$^{24}$. Following Theorem 4.27 p.32 and Proposition 1.6 p.9 of [Che23b], we find that $d_1^{n-1-p,p}\gamma$ is the Poincaré dual of $\partial_{p,n-1-p}^1 g^p(\chi_{\sigma^p})$. There is a sign in the aforementioned Proposition 1.6, however since we consider $\mathbb{F}_2$-vector spaces we can discard it. Finally we have:

$$ (\sigma^p; \sigma^q)_\varepsilon = \partial_{p,n-1-p}^1 g^p(\chi_{\sigma^p}) \cdot g^q(\chi_{\sigma^q}), $$

which is equal to $(\sigma^q; \sigma^p)_\varepsilon$ since the intersection product is symmetric over $\mathbb{F}_2$. Now we assume $n$ even in a first time. The family $\{g^q(\chi_{\sigma^q}) : \sigma^q \in K \setminus \partial K\}$ spans $E_{q,n-1-q}^1(\mathbb{R}X_\varepsilon)$ and the family $\{(g^p(\chi_{\sigma^p}) \cdot -): \sigma^p \in K \setminus \partial K\}$ spans $E_1^{q+1,n-2-q}(\mathbb{R}X_\varepsilon)$. As such the rank of $M_{p,q}(\varepsilon)$ equals the rank of $\partial_{q,n-1-q}^1$. If now we assume $n = 2k + 1$ odd and that $(p, q)$ is neither $(k - 1, k)$ nor $(k, k - 1)$, the family:

$$ \{g^q(\chi_{\sigma^q}) : \sigma^q \in K \setminus \partial K\}, $$

still spans $E_{q,n-1-q}^1(\mathbb{R}X_\varepsilon)$ and the family $\{(g^p(\chi_{\sigma^p}) \cdot -): \sigma^p \in K \setminus \partial K\}$ still spans $E_1^{q+1,n-2-q}(\mathbb{R}X_\varepsilon)$. The rank of $M_{p,q}(\varepsilon)$ is still equal to the rank of $\partial_{q,n-1-q}^1$. For the two remaining cases we consider the following commutative diagram with exact columns:

<figure><img src="image_15.png" /></figure>

The vector family $\{g^k(\chi_{\sigma^k}) : \sigma^k \in K \setminus \partial K\}$ spans the kernel $\ker(i_{k,k}^1)$ and the vector family $\{(g^{k-1}(\chi_{\sigma^{k-1}}) \cdot -): \sigma^{k-1} \in K \setminus \partial K\}$ spans $E_1^{k+1,k-1}(\mathbb{R}X_\varepsilon)$. Therefore the rank of $M_{k-1,k}(\varepsilon)$ equals the rank of the restriction of $\partial_{k,k}^1$ to the kernel $\ker(i_{k,k}^1)$. The family $\{(g^k(\chi_{\sigma^k}) \cdot -): \sigma^k \in K \setminus \partial K\}$

<footnote>$^{23}$c.f. [JRS18] Theorem 5.3 p.16.</footnote>

<footnote>$^{24}$c.f. [Che23b] Definition 4.19 p.28, for the dual spectral sequence.</footnote>
$K \setminus \partial K$} spans the cokernel of $i_1^{k,k}$ and the family $\{g^{k-1}(\chi_{\sigma^{k-1}}): \sigma^{k-1} \in K \setminus \partial K\}$ spans $E_{k-1,k+1}^1(\mathbb{R}X_\varepsilon)$. The cokernel of $i_1^{k,k}$ is the quotient of $E_1^{k,k}(\mathbb{R}X_\varepsilon)$ by the orthogonal of $\ker(i_1^{1,k})$. Using the Poincaré duality, the latter quotient is isomorphic to the quotient of $E_{k,k}^1(\mathbb{R}X_\varepsilon)$ by the orthogonal of $\ker(i_1^{1,k})$ with respect to the intersection product. Since $\partial_{k,k}^1$ takes values in $\ker(i_1^{1,k})$ the rank of $M_{k-1,k}(\varepsilon)$ equals the rank of the composition of $\partial_{k,k}^1$ with the quotient projection $\ker(i_1^{1,k}) \to \ker(i_1^{1,k}) / \mathrm{Rad}(X)$. $\square$

**Proposition 3.11.** Let $\varepsilon \in C^0(K; \mathbb{F}_2)$ be a sign distribution.

1. If $n$ is even, then for all integers $0 \le p \le n-1$, we have:

$$ \dim E_{p,n-1-p}^2(\mathbb{R}X_\varepsilon) = \dim H_{p,n-1-p}(X; \mathbb{F}_2) - \mathrm{rk} M_{n-2-p,p}(\varepsilon) - \mathrm{rk} M_{n-1-p,p-1}(\varepsilon). $$

2. If $n = 2k + 1$ is odd, then for all integers $0 \le p \le n-1$, different from $k-1$, $k$, and $k+1$, we have:

$$ \dim E_{p,n-1-p}^2(\mathbb{R}X_\varepsilon) = \dim H_{p,n-1-p}(X; \mathbb{F}_2) - \mathrm{rk} M_{n-2-p,p}(\varepsilon) - \mathrm{rk} M_{n-1-p,p-1}(\varepsilon). $$

In the other cases, we have:

$$ \dim E_{k-1,k+1}^2(\mathbb{R}X_\varepsilon) = \dim H_{k-1,k+1}(X; \mathbb{F}_2) - \mathrm{rk} M_{k+1,k-2}(\varepsilon) - \mathrm{rk} M_{k,k-1}(\varepsilon) - \delta, $$

$$ \dim E_{k+1,k-1}^2(\mathbb{R}X_\varepsilon) = \dim H_{k+1,k-1}(X; \mathbb{F}_2) - \mathrm{rk} M_{k-2,k+1}(\varepsilon) - \mathrm{rk} M_{k-1,k}(\varepsilon) - \delta, $$

and:

$$ \dim E_{k,k}^2(\mathbb{R}X_\varepsilon) = \dim H_{k,k}(X; \mathbb{F}_2) - \mathrm{rk} M_{k-1,k}(\varepsilon) - \mathrm{rk} M_{k,k-1}(\varepsilon) - 2\delta. $$

where $\delta$ is the dimension of the cokernel of $i_k: H_k(\mathbb{R}X_\varepsilon; \mathbb{F}_2) \to H_k(\mathbb{R}P; \mathbb{F}_2)$. Moreover, $\delta$ cannot exceed $\mathrm{rad}(X)$. We can also note that $H_k(\mathbb{R}X_\varepsilon; \mathbb{F}_2)$ is isomorphic to $E_{k,k}^2(\mathbb{R}X_\varepsilon)$.

*Proof.* The first point is a consequence of the Rank Formula and Proposition 3.10. We assume now that $n = 2k + 1$ is an odd integer. On the line $\{p + q = n - 1\}$ we have the short exact sequence of complexes:

$$ 0 \to (\ker(i_{p,q}^1); \partial_{p,q}^1) \to (E_{p,q}^1(\mathbb{R}X_\varepsilon); \partial_{p,q}^1) \to (E_{p,q}^1(\mathbb{R}P); \partial_{p,q}^1) \to 0. $$

The complex $(E_{p,q}^1(\mathbb{R}P); \partial_{p,q}^1)_{p+q=n-1}$ has only one non-trivial group: $E_{k,k}^1(\mathbb{R}P)$. Therefore, the associated long exact sequence splits into the following exact sequences:

$$ 0 \to H_{p,q}(\ker(i^1)) \to E_{p,q}^2(\mathbb{R}X_\varepsilon) \to 0, $$

for all $p + q = n - 1$ different from $(k, k)$ and $(k + 1, k - 1)$. And the exact sequence:

$$ 0 \to H_{k,k}(\ker(i^1)) \to E_{k,k}^2(\mathbb{R}X_\varepsilon) \xrightarrow{i_{k,k}^2} E_{k,k}^1(P) \to \dots \tag{11} $$

$$ \dots \to H_{k+1,k-1}(\ker(i^1)) \to E_{k+1,k-1}^2(\mathbb{R}X_\varepsilon) \to 0. $$
The Rank Formula and Proposition 3.10 imply that, for all $p+q = n-1$ except $(k, k)$ and $(k-1, k+1)$, the dimension of $H_{p,q}(\ker(i^1))$ is equal to:

$$ \dim E_{p,q}^{1}(\mathbb{R}X_{\varepsilon}) - \mathrm{rk} M_{q-1,p}(\varepsilon) - \mathrm{rk} M_{q,p-1}(\varepsilon). $$

Moreover, if we write $\mathrm{rk} \partial_{k-1,k+1}^1 = \mathrm{rk} M_{k,k-1}(\varepsilon) + \delta$, $\delta$ cannot exceed $\mathrm{rad}(X)$, and:

$$ \dim H_{k-1,k+1}(\ker(i^1)) = \dim E_{k-1,k+1}^{1}(\mathbb{R}X_{\varepsilon}) - \mathrm{rk} M_{k,k-1}(\varepsilon) - \delta - \mathrm{rk} M_{k+1,k-2}(\varepsilon), $$

and:

$$ \dim H_{k,k}(\ker(i^1)) = \dim E_{k,k}^{1}(\mathbb{R}X_{\varepsilon}) - \dim E_{k,k}^{1}(\mathbb{R}P) - \mathrm{rk} M_{k-1,k}(\varepsilon) - \mathrm{rk} M_{k,k-1}(\varepsilon) - \delta. $$

It follows that if $p+q = n-1$ is different from $(k+1, k-1)$, $(k, k)$, and $(k-1, k+1)$ we have:

$$ \begin{aligned} \dim E_{p,q}^{2}(\mathbb{R}X_{\varepsilon}) &= \dim E_{p,q}^{1}(\mathbb{R}X_{\varepsilon}) - \mathrm{rk} M_{q-1,p}(\varepsilon) - \mathrm{rk} M_{q,p-1}(\varepsilon) \\ &= \dim H_{p,q}(X; \mathbb{F}_{2}) - \mathrm{rk} M_{q-1,p}(\varepsilon) - \mathrm{rk} M_{q,p-1}(\varepsilon). \end{aligned} $$

Using Theorem 4.27 p.32 of [Che23b] and Proposition 3.10 we find that:

$$ \begin{aligned} \dim E_{k-1,k+1}^{2}(\mathbb{R}X_{\varepsilon}) &= \dim E_{k-1,k+1}^{1}(\mathbb{R}X_{\varepsilon}) - \mathrm{rk} M_{k,k-1}(\varepsilon) - \mathrm{rk} M_{k+1,k-2}(\varepsilon) - \delta \\ &= \dim H_{k-1,k+1}(X; \mathbb{F}_{2}) - \mathrm{rk} M_{k,k-1}(\varepsilon) - \mathrm{rk} M_{k+1,k-2}(\varepsilon) - \delta, \end{aligned} $$

and:

$$ \begin{aligned} \dim E_{k+1,k-1}^{2}(\mathbb{R}X_{\varepsilon}) &= \dim E_{k+1,k-1}^{1}(\mathbb{R}X_{\varepsilon}) - \mathrm{rk} M_{k-1,k}(\varepsilon) - \mathrm{rk} M_{k-2,k+1}(\varepsilon) - \delta \\ &= \dim H_{k+1,k-1}(X; \mathbb{F}_{2}) - \mathrm{rk} M_{k-1,k}(\varepsilon) - \mathrm{rk} M_{k-2,k+1}(\varepsilon) - \delta. \end{aligned} $$

From (11) we find the last formula:

$$ \dim E_{k,k}^{2}(\mathbb{R}X_{\varepsilon}) = \dim H_{k,k}(X; \mathbb{F}_{2}) - \mathrm{rk} M_{k-1,k}(\varepsilon) - \mathrm{rk} M_{k,k-1}(\varepsilon) - 2\delta. $$

By Theorem 4.26 p.31 of [Che23b] the group $E_{k,k}^2(\mathbb{R}X_\varepsilon)$ is isomorphic to $H_k(\mathbb{R}X_\varepsilon; \mathbb{F}_2)$. The map $i_{k,k}^2$ is conjugated to $i_k : H_k(\mathbb{R}X_\varepsilon; \mathbb{F}_2) \to H_k(\mathbb{R}P; \mathbb{F}_2)$. From (11) we find that the dimension of the cokernel of $i_{k,k}^2$ is $\delta$. $\square$

# 4 The First and Last Boundaries

**Twists of $\mathbb{R}X_\varepsilon$** In this paragraph we provide a geometric interpretation of the coefficients of $M_{0,n-1}(\varepsilon)$.

**Proposition 4.1.** Let $\sigma^n$ be a simplex of $K$. The lift $\sigma^n_{\mathbb{R}}$ of $\sigma^n$ in $\mathbb{R}K$, indexed by the vector $D\varepsilon_{\sigma^n}$ is the only one avoided by $\mathbb{R}X_\varepsilon$, i.e. the only lift for which $\sigma^n_{\mathbb{R}} \cap \mathbb{R}X_\varepsilon$ is empty. More generally, if $H$ is the hypersurface of $\sigma^n$ dual to the cocycle $\zeta \in Z^1(\sigma^n; \mathbb{F}_2)$, the only lift $\sigma^n_{\mathbb{R}}$ of $\sigma^n$ for which $\sigma^n_{\mathbb{R}} \cap \mathbb{R}X_\varepsilon$ corresponds to $H$ is the one indexed by the vector $D\varepsilon_{\sigma^n} + v$ where $z_{\sigma^n}^1(v)$ equals $\zeta$.
*Proof.* Let $\sigma^n_{\mathbb{R}}$ be the lift of $\sigma^n$ indexed by the vector $D\varepsilon_{\sigma^n} + v$. The hypersurface $\sigma^n_{\mathbb{R}} \cap \mathbb{R}X_\varepsilon$ is the dual hypersurface of the cocycle $\zeta: \sigma^1 \mapsto d\varepsilon(\sigma^1) + \omega(\sigma^1)(D\varepsilon_{\sigma^n} + v)$. By Definition 2.15, $\omega(\sigma^1)(D\varepsilon_{\sigma^n})$ equals $d\varepsilon(\sigma^1)$ for all edges $\sigma^1$ of $\sigma^n$. Therefore, $\zeta$ equals $z^1_{\sigma^n}(v)$. $\square$

**Definition 4.2 (Twists).** Let $\varepsilon$ be a sign distribution on $K$, and $\sigma^{n-1}$ be an interior simplex of $K$. For all vertices $\beta$ of $\sigma^{n-1}$, we say that $\mathbb{R}X_\varepsilon$ is twisted along $\sigma^{n-1}$ in the direction of $\beta$ if $D^2\varepsilon_{\sigma^{n-1}}$ is different from $\rho_{\sigma^{n-1}}(\beta)$. The vector $D^2\varepsilon_{\sigma^{n-1}} + \rho_{\sigma^{n-1}}(\beta)$ belongs to the line $N(\sigma^{n-1})$, we denote by $|D^2\varepsilon_{\sigma^{n-1}} + \rho_{\sigma^{n-1}}(\beta)| \in \mathbb{F}_2$ the corresponding number.

**Remark 4.3.** Since both $D^2\varepsilon_{\sigma^{n-1}}$ and $\rho_{\sigma^{n-1}}(\beta)$ belong to the line $N(\sigma^{n-1})$, knowing if $\mathbb{R}X_\varepsilon$ is twisted or not along a simplex in a given direction is a numerical question. The hypersurface $\mathbb{R}X_\varepsilon$ is twisted along $\sigma^{n-1}$ in the direction of $\beta$ if and only if $\varepsilon(\alpha_+) + \varepsilon(\alpha_-) + \sum_{\alpha \in \sigma^{n-1}} \varepsilon(\alpha)|\rho_{\sigma^{n-1}}(\alpha)| + |\rho_{\sigma^{n-1}}(\beta)| = 1$, where $\sigma^{n-1} * \alpha_\pm$ are the two $n$-simplices of $K$ adjacent to $\sigma^{n-1}$.

Let $\sigma^{n-1}$ be an interior simplex of $K$. The set of directions in which $\mathbb{R}X_\varepsilon$ can be twisted along $\sigma^{n-1}$ is strongly influenced by the geometry of $K$. The function $\rho_{\sigma^{n-1}}$ partitions the vertices into two sets. The set of those on which the function vanishes and the set of the others. For all sign distributions $\varepsilon$, the hypersurface $\mathbb{R}X_\varepsilon$ is then either twisted along $\sigma^{n-1}$ in all the directions of the former set or in all the directions of the latter. Let $S$ denote the closed star of $\sigma^{n-1}$. Since $\sigma^{n-1}$ is interior it has exactly $2^n$ different lifts in $\mathbb{R}K$ and so does $S$. A lift of $S$ is indexed by a vector $v \in t(\mathbb{F}_2)$ and we denote it by $S_v$. The intersection of $\mathbb{R}X_\varepsilon$ with $S_v$ is the dual hypersurface of $[\sigma^1 \mapsto d\varepsilon(\sigma^1) + \omega(\sigma^1)(v)]$. Using Proposition 2.10, we see that the restrictions of these cocycles to $\sigma^{n-1}$ describe the whole set of cocycle $Z^1(\sigma^{n-1}; \mathbb{F}_2)$. Therefore $\mathbb{R}X_\varepsilon$ intersects $\sigma^{n-1}$ in every possible way. By reflecting back all of the intersections $\mathbb{R}X_\varepsilon \cap S_v$ into $S$, we get a collection of $2^n$ hypersurfaces of $S$ (some could be empty). Exactly $2^n - 2$ of them intersect $\sigma^{n-1}$ and are connected. For the remaining two we have the following alternative: one is empty and the other consists of two connected components that are “parallel” to $\sigma^{n-1}$, or none of them is empty and each is a “parallel” copy of $\sigma^{n-1}$. Some examples are depicted in Examples 4.5. Either way we have a collection of $2^n$ connected hypersurfaces of $S$. These hypersurfaces are naturally paired by their intersection with $\sigma^{n-1}$. We call the pair of hypersurfaces that does not intersect $\sigma^{n-1}$ the parallel pair. Beside the parallel pair there are also $n$ distinguished pairs of hypersurfaces. For all vertices $\beta$ of $\sigma^{n-1}$, there is exactly one pair $\{H_1(\beta); H_2(\beta)\}$ whose intersection with $\sigma^{n-1}$ separates $\beta$ from the other vertices of $\sigma^{n-1}$.

**Proposition 4.4.** *The hypersurface $\mathbb{R}X_\varepsilon$ is not twisted along $\sigma^{n-1}$ in the direction of $\beta$ if and only if one of the two hypersurfaces $\{H_1(\beta); H_2(\beta)\}$ separates $\beta$ from all the other vertices of $S$.*

*Proof.* Let $\sigma^\pm_\pm$ denote the two $n$-simplices of $S$. Following Proposition 4.1, the lift of $\sigma^\pm_\pm$ in which the lift of $\beta$ is separated from all the other vertices by $\mathbb{R}X_\varepsilon$, is indexed by the vector $v_\pm = D\varepsilon_{\sigma^\pm_\pm} + e_{\sigma^\pm_\pm}(\beta)$. Indeed, $z^1_{\sigma^\pm_\pm}(e_{\sigma^\pm_\pm}(\beta))$ equals $d\chi_\beta$. Let us denote by $H_\pm$ the hypersurface of $S$ corresponding to $S_{v_\pm} \cap \mathbb{R}X_\varepsilon$. By construction $H_\pm$ belongs to $\{H_1(\beta); H_2(\beta)\}$. We have two cases: either $H_+$ equals $H_-$, i.e. $v_+$ equals $v_-$, or $H_+$ is different from $H_-$ and so are $v_+$ and $v_-$. In the first case $H_+$ separates $\beta$ from all the other vertices of $S$ and $v_+ + v_-$ vanishes, i.e. $D^2\varepsilon_{\sigma^{n-1}}$ equals $\rho_{\sigma^{n-1}}(\beta)$. Hence $\mathbb{R}X_\varepsilon$ is not twisted along $\sigma^{n-1}$ in the direction of $\beta$. In the
other case, $\mathbb{R}X_\epsilon$ is twisted along $\sigma^{n-1}$ in the direction of $\beta$, and $H_\pm$ does not separate $\beta$ from $\text{lk}(\sigma^{n-1}; \sigma_\mp^n)$. $\square$

<figure><img src="image_16.png" /><figcaption>(a) The four different intersections of $\mathbb{R}X_\epsilon$ with the lifts of $S$.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(b) The four different intersections of $\mathbb{R}X_\epsilon$ with the lifts of $S$ inside $S$.</figcaption></figure>

Figure 11: The behavior of a curve $\mathbb{R}X_\epsilon$ in the star of an edge along which it is untwisted.

<figure><img src="image_18.png" /><figcaption>(a) The four different intersections of $\mathbb{R}X_\epsilon$ with the lifts of $S$.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(b) The four different intersections of $\mathbb{R}X_\epsilon$ with the lifts of $S$ inside $S$.</figcaption></figure>

Figure 12: The behavior of a curve $\mathbb{R}X_\epsilon$ in the star of an edge along which it is twisted.

**Examples 4.5.** 1. We assume that $n$ equals 2, so that $\mathbb{R}X_\epsilon$ is a curve. We consider an interior edge $\sigma^1$ of $K$. Following Proposition 2.14, the function $\rho_{\sigma^1}$ is constant. Hence $\mathbb{R}X_\epsilon$ can either be twisted along $\sigma^1$ in all the directions or in none. This corresponds to the usual notion of twist$^{25}$.

(a) In Figure 11a we depicted all the intersections of $\mathbb{R}X_\epsilon$ with the star $S$ of $\sigma^1$. In this case the curve $\mathbb{R}X_\epsilon$ is not twisted along $\sigma^1$. In Figure 11b, we represented all these intersections in $S$. The two curves intersecting $\sigma^1$ are represented disjoint, but in reality they would meet “tangentially” at the middle of $\sigma^1$.

(b) Figure 12 is the analogous of Figure 11 for a curve $\mathbb{R}X_\epsilon$, twisted along $\sigma^1$. In Figure 12b, the two curves intersecting $\sigma^1$ meet “transversally” at the middle point of $\sigma^1$.

<footnote>$^{25}$c.f. [RS23], §7. Case of plane curves p.973.</footnote>
2. We assume now that $n$ equals 3, so that $\mathbb{R}X_\epsilon$ is a surface. Let $\sigma^2$ be an interior triangle of $K$. As we discussed earlier, the set of directions in which $\mathbb{R}X_\epsilon$ can be twisted along $\sigma^2$ is determined by the geometry of $K$. If we assume that the function $\rho_{\sigma^2}$ vanishes, then either $\mathbb{R}X_\epsilon$ is twisted along $\sigma^2$ in every direction or in none.

(a) In Figure 13, We depicted the intersections of a surface $\mathbb{R}X_\epsilon$, that is untwisted along $\sigma^2$, with all the lifts of the star $S$ of $\sigma^2$. Figure 14 represents the four pairs of surfaces of $S$ obtained by intersecting its lifts with $\mathbb{R}X_\epsilon$. In Figure 14b, Figure 14c, and Figure 14d, the pairs of surfaces are disjoint. In reality they would meet “tangentially” along their common intersection with $\sigma^2$.

(b) Now we consider a surface $\mathbb{R}X_\epsilon$ that is twisted along $\sigma^2$ in every direction. In this example Figure 15 is the analogous of Figure 13. It represents the different intersections of $\mathbb{R}X_\epsilon$ with the lifts of $S$. Figure 16 represents the four pairs of surfaces of $S$ obtained by intersecting its lifts with $\mathbb{R}X_\epsilon$. In Figure 16b, Figure 16c, and Figure 16d, the pairs of surfaces meet “transversally” along their common intersection with $\sigma^2$.

<figure><img src="image_20.png" /><figcaption>Figure 13: The eight intersections of $\mathbb{R}X_\epsilon$ with the different lifts of $S$. In this example $t(\mathbb{F}_2)$ is $\mathbb{F}_2^3$.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>(a) The parallel pair.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>(b) The pair of the first vertex.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>(c) The pair of the second vertex.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>(d) The pair of the third vertex.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>(a) Quadrant (0,0,0)</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>(b) Quadrant (1,0,0)</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>(c) Quadrant (1,1,0)</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>(d) Quadrant (0,1,0)</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>(e) Quadrant (0,0,1)</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>(f) Quadrant (1,0,1)</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>(g) Quadrant (1,1,1)</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>(h) Quadrant (0,1,1)</figcaption></figure>

<caption>Figure 15: The eight intersections of $\mathbb{R}X_\epsilon$ with the different lifts of $S$.</caption>
<figure><img src="image_33.png" /><figcaption>(a) The Parallel Pair.</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>(b) The pair of the first vertex.</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>(c) The pair of the second vertex.</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>(d) The pair of the third vertex.</figcaption></figure>

Figure 16: A representation of the four pairs of intersections of lifts of S with $\mathbb{R}X_\varepsilon$.

**Maximality of the number of connected component.** We recall from Theorem 1.4 p.947 of [RS23] that the number of connected components of $\mathbb{R}X_\varepsilon$ cannot exceed the number $1+h^{n-1,0}(X)$. When a T-hypersurface achieves this upper bound we will say that it has a maximal number of connected components. In this paragraph we compute the matrix $M_{0,n-2}(\varepsilon)$ which leads us to a characterisation of T-hypersurfaces achieving this upper bound.

**Notations.** Let W be a p-dimensional sub-space of $t(\mathbb{F}_2)$. We denote by $[W] \in \Lambda^p t(\mathbb{F}_2)$ the generator of the line $\Lambda^p W$.

**Lemma 4.6.** Let $\sigma^p \le \sigma^q$ be a pair of simplices of K with p positive. We have the following alternative:

- Either $\sigma^q$ belongs to the boundary $\partial K$ and $F_{n-1}^X(\sigma^p; \sigma^q)$ vanishes;

- Or $\sigma^q$ is interior and we have the exact sequence:

$$C_2(\sigma^p; \mathbb{F}_2) \xrightarrow{\partial} C_1(\sigma^p; \mathbb{F}_2) \xrightarrow{\Upsilon_{\sigma^p, \sigma^q}} F_{n-1}^X(\sigma^p; \sigma^q) \longrightarrow 0$$

$$c \longmapsto \sum_{\sigma^1 \le \sigma^p} \langle c; \sigma^1 \rangle [N(\sigma^1)]$$

Moreover, if we have $\sigma^p \le \sigma^r \le \sigma^q$ with $\sigma^q$ interior, and $c \in C_1(\sigma^p; \mathbb{F}_2)$ then $\Upsilon_{\sigma^r, \sigma^q}(c)$ equals $\Upsilon_{\sigma^p, \sigma^q}(c)$.
**Proof.** If $\sigma^q$ belongs to $\partial K$, then $s := \dim \mathrm{Sed}(\sigma^q)$ is positive. If $s$ is greater than 1, the vector space $\Lambda^{n-1} t(\mathbb{F}_2)/\mathrm{Sed}(\sigma^q)$ vanishes. This latter vector space equals $F_{n-1}^P(\sigma^p; \sigma^q)$. It contains $F_{n-1}^X(\sigma^p; \sigma^q)$ which must vanish as well. If $s$ equals 1, $\Lambda^{n-1} t(\mathbb{F}_2)/\mathrm{Sed}(\sigma^q)$ is a line. Since $\omega(\sigma^p)$ divides the generator of the dual line, the contraction against it is injective. Thus $F_{n-1}^X(\sigma^p; \sigma^q)$ is also trivial. We assume now that $\sigma^q$ is interior. For all $\sigma^1 \le \sigma^p$, the $(n-1)$-vector $[N(\sigma^1)]$ is the generator of $F_{n-1}^X(\sigma^1; \sigma^q)$. Hence by the fourth point of Definition 1.11, the morphism $\Upsilon_{\sigma^p, \sigma^q}$ is surjective. Let $\sigma^1$ be an edge of $\sigma^p$. The vector $[N(\sigma^1)]$ equals $\omega(\sigma^1) \cdot [t(\mathbb{F}_2)]$. However, the cochain $[\sigma^1 \mapsto \omega(\sigma^1)]$ is closed, c.f. Proposition 2.9. Therefore, $[\sigma^1 \mapsto [N(\sigma^1)]]$ is also closed, and the image of $\partial$ is contained in the kernel of $\Upsilon_{\sigma^p, \sigma^q}$. It follows from Proposition 2.11, that for all vertices $\beta$ of $\sigma^p$, the set $\{[N(\sigma^1)] : \sigma^p \ge \sigma^1 \ge \beta\}$ is linearly independent. Hence, it is a basis of $F_{n-1}^X(\sigma^p; \sigma^q)$, and this space has dimension $p$. The rank of the morphism $\partial$ equals the codimension of the kernel of $d : C^1(\sigma^p; \mathbb{F}_2) \to C^2(\sigma^p; \mathbb{F}_2)$. However $\mathrm{ker}(d)$ has dimension $(p+1) - 1 = p$. Therefore, the image of $\partial$ has codimension $p$ and the sequence is exact. If we have $\sigma^p \le \sigma^r \le \sigma^q$ with $\sigma^q$ interior, $F_{n-1}^X(\sigma^p; \sigma^q)$ is included in $F_{n-1}^X(\sigma^r; \sigma^q)$. The last statement is then a direct consequence of the definition. $\square$

**Lemma 4.7.** Let $\gamma$ be a cochain in $C^0(K; \partial K; \mathbb{F}_2)$ and $[\sum_{\sigma^n \in K} w(\sigma^n) \otimes [\sigma^n; \sigma^n]]$ be a homology class in $H_{n-1,0}(X; \mathbb{F}_2)$. We have the formula:

$$g^0(\gamma) \cdot \left[ \sum_{\sigma^n} w(\sigma^n) \otimes [\sigma^n; \sigma^n] \right] = \sum_{\sigma^n \in K} \left( \sum_{\beta * \sigma^{n-1} = \sigma^n} \gamma(\beta) \omega(\sigma^{n-1}) \right) (w(\sigma^n)).$$

**Proof.** Let $[w]$ denote the homology class $[\sum_{\sigma^n} w(\sigma^n) \otimes [\sigma^n; \sigma^n]]$. The intersection product $g^0(\gamma) \cdot [w]$ equals $\zeta[w]$ where $\zeta \in H^{n-1,0}(X; \mathbb{F}_2)$ is the Poincaré dual of $g^0(\gamma)$, i.e. $\zeta \cap [X]$ equals $g^0(\gamma)$. Therefore, we will begin by finding a cycle $\zeta \in Z^0\Omega(K; F_X^{n-1})$ whose cohomology class is Poincaré dual to $g^0(\gamma)$. Let $\sigma^p$ be an interior simplex of $K$. We denote by $\zeta(\sigma^p)$ the unique form in $F_X^{n-1}(\sigma^p; \sigma^p)$ satisfying:

$$\Upsilon_{\sigma^p, \sigma^p}^*(\zeta(\sigma^p)) = d\gamma|_{\sigma^p},$$

where $\Upsilon_{\sigma^p, \sigma^p}^*$ is the adjoint morphism of $\Upsilon_{\sigma^p, \sigma^p}$ from Lemma 4.6. For all simplices $\sigma^p$ of the boundary $\partial K$ and all vertices of $K$, we set $\zeta(\sigma^p)$ to be 0. Then we have a well defined 0-cochain $\zeta \in \Omega^0(K; F_X^{n-1})$. Let us show that it is closed. Let $\sigma^p \le \sigma^{p+1}$ be an adjacent pair of simplices of $K$. If $\sigma^p$ is a vertex then $F_X^{n-1}(\sigma^p; \sigma^{p+1})$ vanishes and so does $d\zeta(\sigma^p; \sigma^{p+1})$. We assume now that $p$ is positive. If $\sigma^{p+1}$ belongs to $\partial K$, so does $\sigma^p$, and $d\zeta(\sigma^p; \sigma^{p+1})$ vanishes. If $\sigma^p$ belongs to $\partial K$ but $\sigma^{p+1}$ does not, $d\zeta(\sigma^p; \sigma^{p+1})$ equals the image $\zeta(\sigma^{p+1})'$ of $\zeta(\sigma^{p+1})$ by the restriction morphism $F_X^{n-1}(\sigma^{p+1}; \sigma^{p+1}) \to F_X^{n-1}(\sigma^p; \sigma^{p+1})$. By Lemma 4.6, $\Upsilon_{\sigma^p, \sigma^{p+1}}^*(\zeta(\sigma^{p+1})')$ equals $d\gamma|_{\sigma^p}$. However, the latter cocycle vanishes for $\gamma$ is identically zero on $\partial K$. Since $\Upsilon_{\sigma^p, \sigma^{p+1}}^*$ is injective, it means that $d\zeta(\sigma^p; \sigma^{p+1})$ vanishes. If now both $\sigma^p$ and $\sigma^{p+1}$ are interior simplices, then:

$$\Upsilon_{\sigma^p, \sigma^{p+1}}^* (d\zeta(\sigma^p; \sigma^{p+1})) = d\gamma|_{\sigma^p} + d\gamma|_{\sigma^{p+1}} = 0.$$
The cochain $\zeta$ is closed. We have:

$$
\begin{align*}
\zeta \cap \sum_{\sigma^1 \le \sigma^n} [N(\sigma^1)] \otimes [\sigma^1; \sigma^n] &= \sum_{\sigma^1 \le \sigma^n} \zeta(\sigma^1)[N(\sigma^1)] \otimes [\sigma^1; \sigma^n] \\
&= \sum_{\sigma^1 \le \sigma^n} \Upsilon_{\sigma^1, \sigma^n}^*(\zeta(\sigma^1))(\sigma^1) \otimes [\sigma^1; \sigma^n] \\
&= \sum_{\sigma^1 \le \sigma^n} d\gamma(\sigma^1) \otimes [\sigma^1; \sigma^n].
\end{align*}
$$

The cohomology class of $\zeta$ is the Poincaré dual of $g^0(\gamma)$ for the fundamental class $[X]$ is represented by the cycle $\sum_{\sigma^1 \le \sigma^n} [N(\sigma^1)] \otimes [\sigma^1; \sigma^n]$. Therefore, we have:

$$
g^0(\gamma) \cdot [w] = \sum_{\sigma^n \in K} \zeta(\sigma^n)(w(\sigma^n)).
$$

To complete our proof we need to compute $\zeta(\sigma^n)$ for all simplices $\sigma^n \in K$. Let $\sigma^1$ and $\sigma^{n-1}$ be two simplices of $\sigma^n$. The number $\omega(\sigma^{n-1})[N(\sigma^1)]$ equals the value of $(\omega(\sigma^{n-1}) \wedge \omega(\sigma^1))$ on $[t(\mathbb{F}_2)]$ i.e. 1 if and only if $\sigma^1$ is not contained in $\sigma^{n-1}$. Therefore, for all $\sigma^1 \le \sigma^n$, we have:

$$
\left( \sum_{\beta * \sigma^{n-1} = \sigma^n} \gamma(\beta) \omega(\sigma^{n-1}) \right) [N(\sigma^1)] = d\gamma(\sigma^1),
$$

and the lemma follows.

<figure><img src="image_37.png" /></figure>

**Definition 4.8.** Let $\sigma^{n-1}$ be an interior simplex of $K$, and $\sigma^n \in K$ be adjacent to $\sigma^{n-1}$. For all vertices $\alpha$ of $\sigma^{n-1}$ the vector $e_{\sigma^n}(\alpha) (\mod N(\sigma^{n-1}))$ does not depend on the choice of $\sigma^n$ and is denoted by $e_{\sigma^{n-1}}(\alpha)$.

**Proposition 4.9.** Let $\varepsilon$ be a sign distribution on $K$. Let $\beta$ be an interior vertex of $K$ and $\sigma^{n-2}$ be an interior simplex of $K$. We have the following alternative:

1. The intersection of the open stars of $\beta$ and $\sigma^{n-2}$ is empty and $(\beta; \sigma^{n-2})_\varepsilon$ vanishes;

2. The join of $\beta$ and $\sigma^{n-2}$ is an $(n-1)$-simplex $\sigma^{n-1}$ of $K$ and:

$$
(\beta; \sigma^{n-2})_{\varepsilon} [N(\sigma^{n-1})] = D^2 \varepsilon_{\sigma^{n-1}} + \rho_{\sigma^{n-1}}(\beta),
$$

where $[N(\sigma^{n-1})]$ denotes the generator of the line $N(\sigma^{n-1})$;

3. The vertex $\beta$ belongs to $\sigma^{n-2}$ and:

$$
(\beta; \sigma^{n-2})_{\varepsilon} [N(\sigma^{n-2})] = \sum_{\sigma^{n-1} \ge \sigma^{n-2}} e_{\sigma^{n-1}}(\beta) \wedge \left( D^2 \varepsilon_{\sigma^{n-1}} + \rho_{\sigma^{n-1}}(\text{lk}(\sigma^{n-2}; \sigma^{n-1})) \right),
$$

where $[N(\sigma^{n-2})]$ denotes the generator of the line $\Lambda^2 N(\sigma^{n-2})$.

*Proof.* Let $\sigma^{n-2}$ be an interior simplex of $K$. Following Proposition 3.6, the homology class
$\partial_{n-2,1}^1 g^{n-2}(\chi_{\sigma}^{n-2})$ is represented by the cycle:

$$c := \sum_{\sigma^{n-1} \ge \sigma^{n-2}} w(\sigma^{n-2}) \wedge (\mathrm{D}^2 \varepsilon_{\sigma^{n-1}} + \rho_{\sigma^{n-1}}^{n-2}(\sigma^{n-2})) \otimes [\sigma^{n-1}; \sigma^{n-1}].$$

Since $\sigma^{n-2}$ is interior, its link $L$ in $K$ is homeomorphic to a circle. We denote the vertices of $L$ by $\alpha_i$, for all $i \in \mathbb{Z}/k\mathbb{Z}$, in such a way that $\alpha_i$ and $\alpha_{i+1}$ are the vertices of an edge of $L$. For all $i \in \mathbb{Z}/k\mathbb{Z}$, we denote $\sigma^{n-2} * \alpha_i$ by $\sigma_i^{n-1}$, and $\sigma^{n-2} * \alpha_i * \alpha_{i+1}$ by $\sigma_{i,i+1}^n$. Following Proposition 2.14, $\rho_{\sigma_i^{n-1}}^{n-2}(\sigma^{n-2})$ equals $\rho_{\sigma_i^{n-1}}(\alpha_i)$ for all $i \in \mathbb{Z}/k\mathbb{Z}$. Then we can write:

$$c = \sum_{i \in \mathbb{Z}/k\mathbb{Z}} w(\sigma^{n-2}) \wedge (\mathrm{D}^2 \varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i)) \otimes [\sigma_i^{n-1}; \sigma_i^{n-1}].$$

Let us consider the chain of $\Omega_{n-1,n}(K; F_{n-1}^X)$ given by the formula:

$$b := \sum_{i \in \mathbb{Z}/k\mathbb{Z}} u(\sigma^{n-2}) \wedge (\mathrm{D}^2 \varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i)) \otimes [\sigma_i^{n-1}; \sigma_{i,i+1}^n].$$

By construction, if $\sigma^p$ is a simplex of $K$ that does not contain $\sigma^{n-2}$, then $\langle \partial b; [\sigma^p; \sigma^p] \rangle$ vanishes. For all $i \in \mathbb{Z}/k\mathbb{Z}$, both $\langle \partial b; [\sigma_i^{n-1}; \sigma_i^{n-1}] \rangle$ and $\langle \partial b; [\sigma_{i,i+1}^n; \sigma_{i,i+1}^n] \rangle$ equal $\langle c; [\sigma_i^{n-1}; \sigma_i^{n-1}] \rangle$. Therefore, the cycle $c$ is homologous to the cycle:

$$c' := \sum_{i \in \mathbb{Z}/k\mathbb{Z}} w(\sigma^{n-2}) \wedge (\mathrm{D}^2 \varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i)) \otimes [\sigma_{i,i+1}^n; \sigma_{i,i+1}^n].$$

Let $\beta$ be an interior vertex of $K$. Following Lemma 4.7, we find that:

$$\begin{aligned} (\beta; \sigma^{n-2})_{\varepsilon} &= g^0(\chi_{\beta}) \cdot [c'] \\ &= \sum_{\substack{i \in \mathbb{Z}/k\mathbb{Z} \\ \beta \in \sigma_{i,i+1}^n}} \omega(\mathrm{lk}(\beta; \sigma_{i,i+1}^n)) (w(\sigma^{n-2}) \wedge (\mathrm{D}^2 \varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i))) \cdot \end{aligned} \quad (12)$$

It implies that $(\beta; \sigma^{n-2})_{\varepsilon}$ vanishes whenever the open stars of $\beta$ and $\sigma^{n-2}$ are disjoint. If now the join of $\beta$ and $\sigma^{n-2}$ is an $(n-1)$-simplex of $K$, then $\beta$ is $\alpha_i$ for a certain $i \in \mathbb{Z}/k\mathbb{Z}$. In this case we have:

$$\begin{aligned} (\beta; \sigma^{n-2})_{\varepsilon} &= \omega(\mathrm{lk}(\alpha_i; \sigma_{i,i+1}^n)) (w(\sigma^{n-2}) \wedge (\mathrm{D}^2 \varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i))) \\ &\quad + \omega(\mathrm{lk}(\alpha_i; \sigma_{i-1,i}^n)) (w(\sigma^{n-2}) \wedge (\mathrm{D}^2 \varepsilon_{\sigma_{i-1}^{n-1}} + \rho_{\sigma_{i-1}^{n-1}}(\alpha_{i-1}))) \\ &= \omega(\sigma_{i+1}^{n-1}) (w(\sigma^{n-2}) \wedge (\mathrm{D}^2 \varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i))) \\ &\quad + \omega(\sigma_{i-1}^{n-1}) (w(\sigma^{n-2}) \wedge (\mathrm{D}^2 \varepsilon_{\sigma_{i-1}^{n-1}} + \rho_{\sigma_{i-1}^{n-1}}(\alpha_{i-1}))) \\ &= |\mathrm{D}^2 \varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i)|, \end{aligned}$$
Because $D^2\varepsilon_{\sigma_{i-1}^{n-1}} + \rho_{\sigma_{i-1}^{n-1}}(\alpha_{i-1})$ belongs to $N(\sigma_{i-1}^{n-1})$, and $\omega(\sigma_{i+1}^{n-1})(w(\sigma^{n-2}) \wedge [N(\sigma_i^{n-1})])$ equals 1. Indeed, if $\alpha$ is a vertex of $\sigma^{n-2}$, we have:

$$ \omega(\sigma_{i+1}^{n-1})(w(\sigma^{n-2}) \wedge [N(\sigma_i^{n-1})]) = \omega(\sigma^{n-2}) \wedge (\alpha_{i+1} - \alpha)(w(\sigma^{n-2}) \wedge [N(\sigma_i^{n-1})]) \\ = (\alpha_{i+1} - \alpha)([N(\sigma_i^{n-1})]) = 1, $$

since the vanishing of $(\alpha_{i+1} - \alpha)([N(\sigma_i^{n-1})])$ would imply the nullity of $[N(\sigma_i^{n-1})]$. If we assume now $\beta$ to be a vertex of $\sigma^{n-2}$, the formula (12) implies that:

$$ (\beta; \sigma^{n-2})_{\varepsilon} = \sum_{i \in \mathbb{Z}/k\mathbb{Z}} \omega(\mathrm{lk}(\beta; \sigma_{i,i+1}^n))(w(\sigma^{n-2}) \wedge (D^2\varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i))). $$

The contraction $\omega(\mathrm{lk}(\beta; \sigma_{i,i+1}^n)) \cdot [\mathbf{t}(\mathbb{F}_2)]$ equals the vector $e_{\sigma_{i,i+1}^n}(\beta)$. Therefore, we have:

$$ (\beta; \sigma^{n-2})_{\varepsilon}[\mathbf{t}(\mathbb{F}_2)] = w(\sigma^{n-2}) \wedge \sum_{i \in \mathbb{Z}/k\mathbb{Z}} e_{\sigma_{i,i+1}^n}(\beta) \wedge (D^2\varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i)). $$

If $n$ equals 2, then $w(\sigma^{n-2})$ equals 1 by definition and we have established the last formula. Now we assume that $n$ is at least equal to 3. Let $\sigma^1$ be an edge of $\sigma^{n-2}$. The number $\omega(\sigma^1)(e_{\sigma_{i,i+1}^n}(\beta))$ equals 1 if and only if $\beta$ belongs to $\sigma^1$, c.f. Proposition 2.11. In particular it does not depend on $i$. The number $\omega(\sigma^1)(D^2\varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i))$ is 0 for all $i \in \mathbb{Z}/k\mathbb{Z}$ for $\sigma^1$ is contained in $\sigma_i^{n-1}$ and $D^2\varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i)$ belongs to $N(\sigma_i^{n-1})$. Using these computations we find that:

$$ \omega(\sigma^1) \cdot \sum_{i \in \mathbb{Z}/k\mathbb{Z}} e_{\sigma_{i,i+1}^n}(\beta) \wedge (D^2\varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i)) = d\chi_{\beta}(\sigma^1) \sum_{i \in \mathbb{Z}/k\mathbb{Z}} D^2\varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i) = 0, $$

and that the vector $\sum_{i \in \mathbb{Z}/k\mathbb{Z}} e_{\sigma_{i,i+1}^n}(\beta) \wedge (D^2\varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i))$ belongs to the line $\Lambda^2 N(\sigma^{n-2})$. Since $u(\sigma^{n-2})$ is the volume element of a supplementary sub-space of $N(\sigma^{n-2})$ we finally find the formula:

$$ (\beta; \sigma^{n-2})_{\varepsilon}[N(\sigma^{n-2})] = \sum_{i \in \mathbb{Z}/k\mathbb{Z}} e_{\sigma_{i,i+1}^n}(\beta) \wedge (D^2\varepsilon_{\sigma_i^{n-1}} + \rho_{\sigma_i^{n-1}}(\alpha_i)). $$

<figure><img src="image_38.png" /></figure>

**Proposition 4.10.** For all interior vertices $\beta$ of $K$ and all simplices $\sigma^{n-2} \ge \beta$ we have:

$$ \sum_{\sigma^{n-1} \ge \sigma^{n-2}} e_{\sigma^{n-1}}(\beta) \wedge \rho_{\sigma^{n-1}}(\beta) = \sum_{\sigma^{n-1} \ge \sigma^{n-2}} e_{\sigma^{n-1}}(\beta) \wedge \rho_{\sigma^{n-1}}(\mathrm{lk}(\sigma^{n-2}; \sigma^{n-1})). $$

*Proof.* We will establish this formula by using a particular sign distribution: the characteristic cochain $\chi_{\beta}$ of the vertex $\beta$. On one hand we have:

$$ D^2\chi_{\beta} = \sum_{\beta \in \sigma^{n-1}} \rho_{\sigma^{n-1}}(\beta) \otimes \sigma^{n-1}. $$
Using the formulæ of Proposition 4.9, we have for all $\sigma^{n-2}$ containing $\beta$:

$$ (\beta; \sigma^{n-2})_{\chi\beta}[N(\sigma^{n-2})] = \sum_{\sigma^{n-1} \ge \sigma^{n-2}} e_{\sigma^{n-1}}(\beta) \wedge (\rho_{\sigma^{n-1}}(\beta) + \rho_{\sigma^{n-1}}(\mathrm{lk}(\sigma^{n-2}; \sigma^{n-1}))) . $$

On the other hand, we have:

$$ (\beta; \sigma^{n-2})_{\chi\beta} = g^{n-2}(\chi_{\sigma^{n-2}}) \cdot \partial_{0,n-1}^{1} g^{0}(\chi_{\beta}), $$

However the tropical homology class $g^0(\chi_\beta)$ can be lifted as a $(n-1)$-homology class of $\mathbb{R}X_{\chi_\beta}$: the fundamental class of the small sphere surrounding $\beta$ in the positive quadrant of $\mathbb{R}K$. This small sphere is isotopic to the link of $\beta$ in $K$. As a consequence $\partial_{0,n-1}^1 g^0(\chi_\beta)$ vanishes. Therefore, we have:

$$ \sum_{\sigma^{n-1} \ge \sigma^{n-2}} e_{\sigma^{n-1}}(\beta) \wedge (\rho_{\sigma^{n-1}}(\beta) + \rho_{\sigma^{n-1}}(\mathrm{lk}(\sigma^{n-2}; \sigma^{n-1}))) = 0. $$

□

**Proposition 4.11.** For all sign distributions $\varepsilon$ on $K$, all interior vertices $\beta$ of $K$, and all simplices $\sigma^{n-2} \ge \beta$ we have:

$$ (\beta; \sigma^{n-2})_{\varepsilon}[N(\sigma^{n-2})] = \sum_{\sigma^{n-1} \ge \sigma^{n-2}} e_{\sigma^{n-1}}(\beta) \wedge (\mathrm{D}^2 \varepsilon_{\sigma^{n-1}} + \rho_{\sigma^{n-1}}(\beta)). $$

*Proof.* This is a direct consequence of Proposition 4.9 and Proposition 4.10. □

**Proposition 4.12.** We assume $n$ to be equal to 2. For all interior vertices $\beta$ of $K$, the number $(\beta; \beta)_\varepsilon$ equals the parity of the number of edges of $K$, adjacent to $\beta$, along which $\mathbb{R}X_\varepsilon$ is twisted. In other words, we have:

$$ (\beta; \beta)_\varepsilon = \sum_{\sigma^1 \ge \beta} |\mathrm{D}^2 \varepsilon_{\sigma^1} + \rho_{\sigma^1}|. $$

*Proof.* Let $\sigma^2$ be a triangle of $K$ and $\sigma^1$ be an edge of $\sigma^2$. For all vertices $\beta$ of $\sigma^2$, the set $\{e_{\sigma^2}(\beta); [N(\sigma^1)]\}$ forms a basis of $\mathfrak{t}(\mathbb{F}_2)$. Hence $e_{\sigma^2}(\beta) \wedge [N(\sigma^1)]$ equals $[\mathfrak{t}(\mathbb{F}_2)]$. Moreover for all interior vertex $\beta$ of $K$, $[N(\beta)]$ equals $[\mathfrak{t}(\mathbb{F}_2)]$. Therefore, Proposition 4.11 implies the formula of the proposition. □

**Remark 4.13.** When $n$ is greater than 3 the formula of Proposition 4.11, can also be interpreted as a number of twists in the direction of $\beta$ along the $(n-1)$-simplices of its star. However, this time some twists must be counted and some must not. For instance, if $\beta$ is an interior vertex and $\sigma^{n-2}$ contains $\beta$, $(\beta; \sigma^{n-2})_\varepsilon$ equals the number of twists of $\mathbb{R}X_\varepsilon$ in the direction of $\beta$ along the $(n-1)$-simplices of $K$ for which $w(\sigma^{n-2}) \wedge e_{\sigma^{n-1}}(\beta) \wedge [N(\sigma^{n-1})]$ equals $[\mathfrak{t}(\mathbb{F}_2)]$. The set of twists we are counting may depend on the choice of the decomposed vectors $w(\sigma^{n-2})$. However, the count itself, will not.

**Definition 4.14.** Let $\mathbb{R}X_\varepsilon$ be a T-hypersurface of $\mathbb{R}P$. The rank of $\mathbb{R}X_\varepsilon$, denoted by $\ell(\mathbb{R}X_\varepsilon)$, is the biggest integer $\ell \ge 0$ for which the homological “inclusions” $H_k(\mathbb{R}X_\varepsilon; \mathbb{F}_2) \to H_k(\mathbb{R}P; \mathbb{F}_2)$ are onto for all $0 \le k \le \ell$.
**Theorem 4.15** (Generalised Haas Theorem). Let P be a non-singular polytope of $t^*(\mathbb{R})$ endowed with a primitive triangulation K. For all sign distributions $\epsilon$, the T-hypersurface $\mathbb{R}X_\epsilon$ has a maximal number of connected components if and only if:

1. The triangulation K is $\rho$-uniform;

2. The number $\ell(\mathbb{R}X_\epsilon)$ is at least equal to $\min(n-2; 1)$;

3. The sign distribution $\epsilon$ satisfies:

$$D^2\epsilon = \rho \text{ in } H_{n-1}(K; \mathcal{B}_1K; N). \qquad (13)$$

4. For all simplices $\sigma^{n-2}$ of $\mathcal{B}_1K$ not contained in $\mathcal{B}_0K$, $(\mathrm{ap}(\sigma^{n-2}); \sigma^{n-2})_\epsilon$ vanishes.

*Proof.* Using Theorem 4.26 p.31 in [Che23b], we have the short exact sequence:

$$0 \to E_{0,0}^{\infty}(\mathbb{R}X_{\epsilon}) \to H_0(\mathbb{R}X_{\epsilon}; \mathbb{F}_2) \to E_{n-1,0}^{\infty}(\mathbb{R}X_{\epsilon}) \to 0.$$

Since $E_{0,0}^{\infty}(\mathbb{R}X_\epsilon)$ is isomorphic to $E_{0,0}^1(\mathbb{R}X_\epsilon)$, itself isomorphic to $H_{0,0}(X; \mathbb{F}_2)$ it has dimension 1. Hence $\mathbb{R}X_\epsilon$ has a maximal number of connected components if and only if $E_{n-1,0}^{\infty}(\mathbb{R}X_\epsilon)$ is isomorphic to $E_{n-1,0}^1(\mathbb{R}X_\epsilon)$. We are now going to distinguish three cases: $n$ equals 2, $n$ equals 3, and $n$ is greater than 3. In a first time we assume that $n$ is greater than 3. $E_{n-1,0}^{\infty}(\mathbb{R}X_\epsilon)$ is isomorphic to $E_{n-1,0}^1(\mathbb{R}X_\epsilon)$ if and only if the two following morphisms vanish:

$$\left\{ \begin{array}{l} \partial_{n-2,1}^1 : E_{n-2,1}^1(\mathbb{R}X_\epsilon) \to E_{n-1,0}^1(\mathbb{R}X_\epsilon) \\ \\ \partial_{1,1}^{n-2} : E_{1,1}^{n-2}(\mathbb{R}X_\epsilon) \to E_{n-1,0}^{n-2}(\mathbb{R}X_\epsilon). \end{array} \right.$$

By Theorem 4.30 p.34 in [Che23b] and duality $\partial_{1,1}^{n-2}$ vanishes if and only if the morphism:

$$i^1 : H^1(\mathbb{R}P; \mathbb{F}_2) \to H^1(\mathbb{R}X_\epsilon; \mathbb{F}_2),$$

is injective. Since $i^0 : H^0(\mathbb{R}P; \mathbb{F}_2) \to H^0(\mathbb{R}X_\epsilon; \mathbb{F}_2)$ is always injective, $\partial_{1,1}^{n-2}$ vanishes if and only if $\ell(\mathbb{R}X_\epsilon)$ is at least equal to 1. Since $n$ is greater than 3, Proposition 3.10 ensures us that $\partial_{n-2,1}^1$ vanishes if and only if $M_{0,n-2}(\epsilon)$ does. The second point of Proposition 4.9 implies that for $M_{0,n-2}(\epsilon)$ to vanish we must have:

$$\rho_{\sigma^{n-1}}(\beta) = D^2\epsilon_{\sigma^{n-1}},$$

for all interior simplices $\sigma^{n-1}$ of K, and all vertices $\beta$ of $\sigma^{n-1}$ whose link $\mathrm{lk}(\beta; \sigma^{n-1})$ is not contained in the boundary. In particular, K must be $\rho$-uniform. Reciprocally, if K is $\rho$-uniform, then we must have:

$$D^2\epsilon = \rho \text{ in } H_{n-1}(K; \mathcal{B}_1K; N).$$

Now if we suppose that K is $\rho$-uniform and $\epsilon$ is a solution of (13), $M_{0,n-2}(\epsilon)$ vanishes if and only if $(\beta; \sigma^{n-2})_\epsilon$ vanishes for all pairs of interior simplices $\beta \le \sigma^{n-2}$. We assume that we have such a pair $\beta \le \sigma^{n-2}$ with $\sigma^{n-2}$ containing at least 2 interior vertices, one of them being $\beta$. If $\sigma^{n-1}$ contains $\sigma^{n-2}$ then, by hypothesis, the link of $\beta$ in $\sigma^{n-1}$ contains at least one interior vertex of
K. As a consequence of (13), $D^2\varepsilon_{\sigma^{n-1}}$ equals $\rho_{\sigma^{n-1}}(\beta)$. Hence $(\beta; \sigma^{n-2})_\varepsilon$ vanishes. Therefore, $M_{0,n-2}(\varepsilon)$ vanishes if and only if K is $\rho$-uniform, $\varepsilon$ is a solution of (13), and for all simplices $\sigma^{n-2}$ of $B_1K \setminus B_0K$, $(\mathrm{ap}(\sigma^{n-2}); \sigma^{n-2})_\varepsilon$ vanishes. When $n$ is at most equal to 3, Theorem 4.26 p.31 of [Che23b] implies that the Renaudineau-Shaw spectral sequence of $\mathbb{R}X_\varepsilon$ degenerates at the second page. Hence, in these cases $\mathbb{R}X_\varepsilon$ has a maximal number of connected components if and only if $E_{n-1,0}^2(\mathbb{R}X_\varepsilon)$ is isomorphic to $E_{n-1,0}^1(\mathbb{R}X_\varepsilon)$. This latter condition is equivalent$^{26}$ to the vanishing of $\partial_{n-1,0}^1$. Let $n$ be equal to 2. Following Proposition 3.10, the vanishing of $\partial_{1,0}^1$ is equivalent to the vanishing of $M_{0,0}(\varepsilon)$. We recall that every primitive triangulation of a two dimensional polytope is $\rho$-uniform$^{27}$. As in the preceding case, $M_{0,0}(\varepsilon)$ vanishes if and only if:

$$D^2\varepsilon = \rho \text{ in } H_1(K; B_1K; N),$$

and $(\beta; \beta)_\varepsilon$ vanishes for all $\beta \in B_1K \setminus B_0K$. Finally, the rank $\ell(\mathbb{R}X_\varepsilon)$ being always at least 0, the condition $\ell(\mathbb{R}X_\varepsilon) \ge \min(0,1)$ is always satisfied. Only remains the case $n$ equals 3. Proposition 3.11 provides us with the formula:

$$\dim E_{2,0}^2(\mathbb{R}X_\varepsilon) = \dim E_{2,0}^1(\mathbb{R}X_\varepsilon) - \mathrm{rk} M_{0,1}(\varepsilon) - \delta,$$

where $\delta$ denotes the dimension of the cokernel of $i_1: H_1(\mathbb{R}X_\varepsilon; \mathbb{F}_2) \to H_1(\mathbb{R}P; \mathbb{F}_2)$. Hence, $E_{2,0}^2(\mathbb{R}X_\varepsilon)$ is isomorphic to $E_{2,0}^1(\mathbb{R}X_\varepsilon)$ if and only if both $M_{0,1}(\varepsilon)$ and $\delta$ vanish. The latter condition is equivalent to $\ell(\mathbb{R}X_\varepsilon)$ being at least equal to 1. The former, as in the case $n > 3$ is equivalent to K being $\rho$-uniform and $\varepsilon$ satisfying:

$$D^2\varepsilon = \rho \text{ in } H_2(K; B_1K; N),$$

and $(\mathrm{ap}(\sigma^1); \sigma^1)_\varepsilon = 0$ for all $\sigma^1 \in B_1K \setminus B_0K$. $\square$

**Remark 4.16.** The case $n$ equals 2 of Theorem 4.15 was already known and is due to B. Haas, c.f. [Haa97], Theorem 7.3.0.10 p.70. Our generalisation can also be formulated in terms of twists. Equation (13) forbids $\mathbb{R}X_\varepsilon$ to be twisted in certain directions along the $(n-1)$-simplices of $K \setminus B_1K$ and the fourth condition of Theorem 4.15 imposes that the weighted count of the twists of $\mathbb{R}X_\varepsilon$ around the $(n-2)$-simplices of $B_1K \setminus B_0K$ has to be even. The two first conditions of the theorem are automatically satisfied for T-curves. Because of Proposition 4.39 p.38 of [Che23b], the second condition is automatically satisfied when $\iota[\omega_X] \ge 1$. The latter condition only depends on the “degree” of $X$. For instance odd degree hypersurfaces in projective spaces satisfy $\iota[\omega_X] = n-1$.

In the remaining of this paragraph we tackle the question of knowing if every $\rho$-uniform triangulation admits a sign distribution $\varepsilon$ whose T-hypersurface $\mathbb{R}X_\varepsilon$ has a maximal number of connected components.

**Proposition 4.17.** Let K be a $\rho$-uniform triangulation. Any solution $\varepsilon$ of the following equation:

$$D^2\varepsilon = \rho \text{ in } H_{n-1}(K; B_0K; N), \qquad (14)$$

<footnote>$^{26}$c.f. [Che23b] Theorem 4.27 p.32.</footnote>

<footnote>$^{27}$c.f. Proposition 2.14.</footnote>
yields a T-hypersurface $\mathbb{R}X_{\varepsilon}$ with the maximal number of connected components. Moreover, the T-hypersurfaces constructed from the solutions of (14) are made of:

1. Exactly $h^{0,n-1}(X)$ spheres, bounding disjoint balls. None of them intersect the toric divisors of $\mathbb{R}P$;

2. Another connected component that intersects all of the toric divisors of $\mathbb{R}P$.

When (14) has a solution we say that $K$ is simply integrable.

**Proof.** Let $\varepsilon$ be a solution of (14) and $\beta$ be an interior vertex of $K$. For all $(n-1)$-simplices adjacent to $\beta$, $D^2\varepsilon_{\sigma^{n-1}} + \rho_{\sigma^{n-1}}(\beta)$ vanishes. It implies that the vector $v := D\varepsilon_{\sigma^n} + e_{\sigma^n}(\beta)$ does not depend on the choice of the $n$-simplex $\sigma^n \ge \beta$ we used to compute it. For all $\sigma^n \ge \beta$, the cocycle $d\varepsilon|_{\sigma^n} + z_{\sigma^n}^1(v)$ equals $d\chi_{\beta}|_{\sigma^n}$. Therefore, it follows from Proposition 4.1 that the intersection of $\mathbb{R}X_{\varepsilon}$ with the lift of the closed star of $\beta$ corresponding to $v$ consists of a sphere confining the lift of $\beta$ in its interior. Since $\mathbb{R}X_{\varepsilon}$ must intersect all the toric divisors of $\mathbb{R}P$ in a non-empty set, $\mathbb{R}X_{\varepsilon}$ has at least $1 + \text{card}(K^{(0)} \setminus \partial K^{(0)})$ connected component. However, $\text{card}(K^{(0)} \setminus \partial K^{(0)})$ equals $h^{n-1,0}(X)$. Hence, $\mathbb{R}X_{\varepsilon}$ has a maximal number of connected components and satisfies the description of the proposition. $\square$

**Proposition 4.18.** A shellable, $\rho$-uniform triangulation $K$ of a non-singular polytope $P$ is simply integrable if and only if $\partial\rho = 0$ where $\partial$ denotes the connecting morphism:

$$ \partial : H_{n-1}(K; \mathcal{B}_0 K; N) \to H_{n-2}(\mathcal{B}_0 K; \partial K; N), $$

of the long exact sequence associated with the triple $\partial K \subset \mathcal{B}_0 K \subset K$.

**Proof.** Let $K$ be a $\rho$-uniform triangulation. If $K$ is simply integrable, we consider $\varepsilon$ a solution of (14). In this case, the second derivative of $\varepsilon$ belongs to $H_{n-1}(K; \partial K; N)$ and equals $\rho$ modulo $\mathcal{B}_0 K$. Since the kernel of $\partial$ is the image of $H_{n-1}(K; \partial K; N) \to H_{n-1}(K; \mathcal{B}_0 K; N)$, $\partial\rho$ has to vanish. Reciprocally, if we assume $K$ to be a $\rho$-uniform, shellable triangulation, then $\partial\rho = 0$ implies that $\rho$ has a lift $\bar{\rho}$ in $H_{n-1}(K; \partial K; N)$. Since $K$ is shellable, Lemma 2.20 ensures that we can find a sign distribution $\varepsilon$, whose second derivative equals $\bar{\rho}$. By Proposition 4.17, any such sign distribution yields a solution of (14). $\square$

**T-curves.** In what follows we assume $n$ equal to 2. In this special case, Proposition 2.14 implies that there is a canonical class $\bar{\rho} \in H_1(K; \partial K; N)$ that lifts the class $\rho \in H_1(K; \mathcal{B}_0 K; N)$. It is defined as follows:

$$ \bar{\rho} := \sum_{\sigma^1 \notin \partial K} \rho_{\sigma^1} \otimes \sigma^1. $$

**Theorem 4.19.** The solutions of the equation:

$$ D^2\varepsilon = \bar{\rho} \text{ in } H_1(K; \partial K; N), \qquad (15) $$

consist of the images of the polynomial functions of degree 2 in $C^0(K; \mathbb{F}_2)$. As such, every 2-dimensional triangulation, even a non-shellable one, is simply integrable.
Proof. Let $h$ be a quadratic sign distribution. Now let $\beta_0$ and $\beta_1$ be the two vertices of an interior edge of $K$, and $[\beta_0; \beta_1; \alpha_{\pm}]$ the two triangles of $K$ adjacent to $[\beta_0; \beta_1]$. Following Proposition 2.19, affine maps are in the kernel of $D^2$. Since $h$ is quadratic, we can assume, to compute $D^2h_{[\beta_0;\beta_1]}$, that $h$ is given by $[(x; y) \mapsto xy]$ where $(x; y)$ are the affine coordinates in the basis $(\beta_1 + \beta_0; \alpha_+ + \beta_0)$ with origin at $\beta_0$. If $(x_-; y_-)$ are the coordinates of the vector $\alpha_- + \beta_0$, then $y_-$ equals 1 for $(\beta_1 + \beta_0; \alpha_- + \beta_0)$ is also a basis of $t^*(\mathbb{F}_2)$. Hence, we have:

$$ \alpha_+ + \alpha_- = x_-(\beta_1 + \beta_0), $$

and $\rho_{[\beta_0;\beta_1]}$ equals $x_-[N([\beta_0; \beta_1])]$ by Proposition 2.13. Let us write:

$$ D^2h_{[\beta_0;\beta_1]} = |D^2h_{[\beta_0;\beta_1]}| [N([\beta_0; \beta_1])] . $$

Then, we have:

$$ \begin{aligned} |D^2h_{[\beta_0;\beta_1]}| &= x(\alpha_+)y(\alpha_+) + x_-y_- + y_-(x(\beta_0)y(\beta_0) + x(\beta_1)y(\beta_1)) \\ &= 0 \times 1 + x_- \times 1 + x_-(0 \times 0 + 1 \times 0) \\ &= x_- . \end{aligned} $$

And $D^2h_{[\beta_0;\beta_1]}$ equals $\rho_{[\beta_0;\beta_1]}$. Now by Proposition 2.19, the kernel of $D^2$ is exactly the space of affine sign distributions, hence the theorem follows. $\square$

**Remarks 4.20.** 1. Since $t^*(\mathbb{F}_2)$ has dimension 2 there is essentially one quadratic polynomial function $f : t^*(\mathbb{F}_2) \to \mathbb{F}_2$. It is a consequence of Fermat's little theorem. Hence, all the T-curves $\mathbb{R}X_\epsilon$ obtained from a solution $\epsilon$ of (15) are reflections of one another;

2. I. Itenberg and O. Viro proved$^{28}$ that a particular sign distribution always produces a maximal curve of a given topological type for all convex triangulations of the standard triangle of size $d \ge 1$: $\{(x; y) \in \mathbb{R}_+^2 | x + y \le d\}$. They named their sign distribution the Harnack distribution. It is given by the formula $\epsilon(x; y) := 1 + (1 + x)(1 + y) (\text{mod } 2)$, hence by a quadratic polynomial.

**A sharper bound for the number of connected components.** In this paragraph, we provide an estimation of the maximal number of connected components of a T-hypersurface constructed on a non-$\rho$-uniform triangulation.

**Definition 4.21.** Let $K$ be a primitive triangulation of a non-singular polytope. The distance between two simplices is the minimum of the graph length of their vertices in the 1-skeleton of $K$. We denote by $\kappa(K)$ the maximal number of $(n-1)$-simplices of $K$ at which it fails to be $\rho$-uniform and that are all at distance at least 2 of each others.

**Proposition 4.22.** Let $K$ be a primitive triangulation of a non-singular polytope. For all sign distributions $\epsilon \in C^0(K; \mathbb{F}_2)$ the rank of $M_{0,n-2}(\epsilon)$ is always at least equal to $\kappa(K)$. As a consequence we have the following upper bound:

$$ \dim H_0(\mathbb{R}X_\epsilon; \mathbb{F}_2) \le 1 + \dim H_{0,n-1}(X; \mathbb{F}_2) - \kappa(K). \quad (16) $$

<footnote>$^{28}$c.f. [IV96], Proposition p.24.</footnote>
*Proof.* Let $\sigma_1^{n-1}, \dots, \sigma_k^{n-1}$ be simplices of $K$ at which it fails to be $\rho$-uniform and that are all at distance at least 2 of each others. Let $\varepsilon$ be a sign distribution on $K$. Since $K$ fails to be $\rho$-uniform at the simplices $\sigma_1^{n-1}, \dots, \sigma_k^{n-1}$, we can find, for all $1 \le i \le k$, two different interior vertices $\alpha_i, \beta_i$ of $\sigma_i^{n-1}$ for which $\rho_{\sigma_i^{n-1}}(\alpha_i)$ and $\rho_{\sigma_i^{n-1}}(\beta_i)$ are different. Using Proposition 4.9, it follows that either $(\alpha_i; \mathrm{lk}(\alpha_i; \sigma_i^{n-1}))_{\varepsilon}$ or $(\beta_i; \mathrm{lk}(\beta_i; \sigma_i^{n-1}))_{\varepsilon}$ equals 1. We assume that it happens on $\alpha_i$, for all $1 \le i \le k$. The matrix:

$$((\alpha_i; \mathrm{lk}(\alpha_j; \sigma_j^{n-1}))_{\varepsilon})_{1 \le i,j \le k},$$

is a sub-matrix of $M_{0,n-2}(\varepsilon)$ equal to the identity matrix of size $k$. Indeed for all $i$ different from $j$, $\alpha_i$ and $\mathrm{lk}(\alpha_j; \sigma_j^{n-1})$ are at distance at least 2. Therefore, they cannot be both included in the same simplex. Hence, the rank of $M_{0,n-2}(\varepsilon)$ is at least equal to $k$. Consequently, the rank of $M_{0,n-2}(\varepsilon)$ is at least equal to $\kappa(K)$, and the upper bound (16) follows from Proposition 3.11. $\square$

**Asymptotic growth of the expected number of connected components.** We assume that we are given a smooth polytope $P$ of $t^*(\mathbb{R})$, and for every integer $d \ge 1$, a primitive triangulation $K_d$ of its dilate $dP$. We denote by $X(d)$ the dual hypersurface of $K_d$. By Theorem 1.4 p.947 of [RS23], we know that for all $\varepsilon \in C^0(K(d); \mathbb{F}_2)$, the number of connected components $b_0(\mathbb{R}X_{\varepsilon}(d))$ is less than $1 + h^{n-1,0}(X(d))$, where $h^{n-1,0}(X(d))$ is the dimension of $H_{n-1,0}(X(d); \mathbb{F}_2)$. In this paragraph we give an idea of the growth of the number connected components of a typical T-hypersurface as $d$ grows to $+\infty$. If we choose $\varepsilon$ uniformly at random in $C^0(K(d); \mathbb{F}_2)$, the number $b_0(\mathbb{R}X_{\varepsilon}(d))$ becomes a random variable. We denote by $\mathbb{E}[\xi]$ the expected value of a random variable $\xi$.

**Proposition 4.23.** The asymptotic of $h^{n-1,0}(X(d))$ is the following:

$$h^{n-1,0}(X(d)) = \mathrm{vol}_{\mathbb{Z}}(P)d^n + o(d^{n-1}),$$

where $\mathrm{vol}_{\mathbb{Z}}(P)$ is the integral volume of $P$. That is to say the Lebesgue measure of $P$, normalised so that a parallelogram on a basis of $t^*(\mathbb{Z})$ has volume 1.

*Proof.* The number $h^{n-1,0}(X(d))$ equals the number of lattice points in the relative interior of $P$. By Ehrhart-Macdonald reciprocity$^{29}$ it equals $(-1)^n L(P; -d)$, where $L(P; X)$ is the Ehrhart polynomial$^{30}$ of $P$. The polynomial $L(P; X)$ has degree $n$ and its leading coefficient is $\mathrm{vol}_{\mathbb{Z}}(P)$. $\square$

**Proposition 4.24.** Let $P$ be a $n$-dimensional integral polytope of $t^*(\mathbb{R})$ and $(K_d)_{d \ge 1}$ be a sequence of primitive triangulations of $(dP)_{d \ge 1}$. The function $d \mapsto f_1(K_d)$ is a polynomial of degree $n$.

*Proof.* Let $d \ge 1$ be an integer. The number $f_1(K_d)$ is a linear combination of the $h^*$-vector of $dP$. In particular there is a $t^*(\mathbb{Z})$-translation invariant valuation $\varphi$ on the integral polytopes of $t^*(\mathbb{R})$ such that $f_1(K_d)$ equals $\varphi(dP)$ for all $d \ge 1$. Theorem 5 p.121 of [McM77] implies that $d \mapsto f_1(K_d)$ is given by a polynomial of degree at most $n$. However, the number $f_1(K_d)$

<footnote>$^{29}$c.f. [Mac71], Theorem 4.6 p.192.</footnote>

<footnote>$^{30}$c.f. [Ehr62], pp.616-618.</footnote>
is not smaller than $\frac{1}{2}f_0(K_d)$ since no vertex is isolated. Hence, $d \mapsto f_1(K_d)$ is of degree $n$ for $d \mapsto f_0(K_d) = \text{card}(dP \cap t^*(\mathbb{Z}))$ is polynomial of degree $n$. $\square$

**Proposition 4.25.** For all sequences $(K_d)_{d \ge 1}$ of primitive triangulations of $(dP)_{d \ge 1}$ we have:

$$1 \ge \limsup_{d \to +\infty} \mathbb{E} \left[ \frac{b_0(\mathbb{R}X_\varepsilon(d))}{1 + h^{n-1,0}(X(d))} \right] \ge \liminf_{d \to +\infty} \mathbb{E} \left[ \frac{b_0(\mathbb{R}X_\varepsilon(d))}{1 + h^{n-1,0}(X(d))} \right] > 0.$$

*Proof.* Let $d \ge 1$ be an integer and $\beta \in K_d$ be an interior vertex. We denote by $S_\beta$ the subcomplex of cubical subdivision of $K_d$ made of all the cubes indexed by the pairs of simplices $\sigma^p \le \sigma^q$ with $p \ge 1$ and $\sigma^p$ containing $\beta$. We also denote by $\text{St}_\beta$ the open star of $\beta$, and $\xi_\beta$ the Bernoulli random variable whose value is 1 if and only if $\mathbb{R}X_\varepsilon(d)$ contains a lift of $S_\beta$. By Proposition 4.1 and Proposition 4.17 $\xi_\beta(\varepsilon)$ equals 1 if and only if the image of $\mathbb{D}^2\varepsilon$ in $H_{n-1}(K_d(\text{St}_\beta); \partial K_d(\text{St}_\beta); N)$ equals $\rho(\beta) = \sum_{\sigma^{n-1} \ge \beta} \rho_{\sigma^{n-1}}(\beta) \otimes \sigma^{n-1}$. We have the following commutative diagram with exact rows and columns:

<figure><img src="image_39.png" /></figure>

Hence, we have the following exact sequence:

$$0 \to \begin{array}{c} C^0(K_d; K_d(\text{St}_\beta); \mathbb{F}_2) \\ \oplus \text{Aff}(t^*(\mathbb{F}_2)) \end{array} \to C^0(K_d; \mathbb{F}_2) \xrightarrow{D_\beta^2} H_{n-1}(K_d(\text{St}_\beta); \partial K_d(\text{St}_\beta); N),$$

where $D_\beta^2$ is the composition $\mathbb{D}^2|_{K_d(\text{St}_\beta)}$. Since $D_\beta^2 \chi_\beta$ equals $\rho(\beta)$ there is exactly $2^{n+1+f_0(K_d)-f_0(K_d(\text{St}_\beta))}$ solutions to the equation $D_\beta^2\varepsilon = \rho(\beta)$. Therefore the probability that $\xi_\beta$ equals 1 is $2^{n+1-f_0(K_d(\text{St}_\beta))}$.
We have the following inequality:

$$b_0(\mathbb{R}X_\varepsilon(d)) \ge 1 + \sum_{\beta \in K \setminus \partial K} \xi_\beta(\varepsilon).$$

Which implies the following one:

$$\mathbb{E} \left[ \frac{b_0(\mathbb{R}X_\varepsilon(d))}{1 + h^{n-1,0}(X(d))} \right] \ge \frac{1 + \sum_{\beta \in K \setminus \partial K} 2^{n+1-f_0(K_d(\text{St}_\beta))}}{1 + h^{n-1,0}(X(d))}.$$
The number $f_0(K_d(\mathrm{St}_\beta))$ equals the degree$^{31}$ of the vertex $\beta$ plus 1. Hence, the number $\sum_{\beta \in K \setminus \partial K} 2^{n+1-f_0(K_d(\mathrm{St}_\beta))}$ equals $2^n \sum_{\beta \in K \setminus \partial K} 2^{-\deg \beta}$. By convexity of the function $x \mapsto 2^{-x}$, we have:

$$2^n \sum_{\beta \in K \setminus \partial K} 2^{-\deg \beta} \ge 2^n h^{n-1,0}(X(d)) 2^{-\frac{\sum_{\beta \in K \setminus \partial K} \deg \beta}{h^{n-1,0}(X(d))}} \ge h^{n-1,0}(X(d)) 2^{n-2 \frac{f_1(K_d)}{h^{n-1,0}(X(d))}}.$$

The sequence $(\frac{f_1(K_d)}{h^{n-1,0}(X(d))})_{d \ge 1}$ converges toward a number $c > 0$ as $d$ grows to $+\infty$. Thus, the lim. inf. is at least equal to $2^{n-2c} > 0$. Since $\mathbb{E}[\frac{b_0(\mathbb{R}X_\varepsilon(d))}{1+h^{n-1,0}(X(d))}]$ is at most equal to 1 by A. Renaudineau and K. Shaw's inequality, the proposition follows. $\square$

**A Discrete Dirichlet Problem for T-Curves.** We finally remark that in the special case of curves one can use the symmetry of $M_{0,0}(\varepsilon)$ to define a Laplacian type operator on the space of 0-cochains of $K$. Doing so we can link the number of connected components of $\mathbb{R}X_\varepsilon$ to the dimension of the space of harmonic cochains.

**Definition 4.26.** Let $K$ be a primitive triangulation of the 2-dimensional polytope $P$ and $\varepsilon$ be a sign distribution on $K$. We define the Laplacian $\Delta_\varepsilon$ to be the following endomorphism of $C^0(K; \partial K; \mathbb{F}_2)$:

$$\Delta_\varepsilon : C^0(K; \partial K; \mathbb{F}_2) \longrightarrow C^0(K; \partial K; \mathbb{F}_2) \\ f \longmapsto [\beta \mapsto \sum_\alpha (\beta; \alpha)_\varepsilon f(\alpha)].$$

This operator is symmetric with respect to the canonical scalar product of $C^0(K; \partial K; \mathbb{F}_2)$ since the matrix $M_{0,0}(\varepsilon)$ is symmetric. Also, using Proposition 3.11, the number of connected components of $\mathbb{R}X_\varepsilon$ is equal to 1 plus the dimension of the space of harmonic cochains $\ker(\Delta_\varepsilon)$ that vanish on the boundary.

# 5 Examples and Non-Examples

Let $n$ and $d$ be two positive integers. We recall that we denote the standard $n$-simplex of size $d$ by the symbol $P_d^n$. In this context $t^*(\mathbb{R})$ is $\mathbb{R}^n$, $t^*(\mathbb{Z})$ is $\mathbb{Z}^n$, $e_1, ..., e_n$ is the canonical basis of $\mathbb{R}^n$ and $P_d^n$ is the convex hull of $0, de_1, ..., de_n$.

**Regular Triangulations of F. Knudsen.** In the third chapter of [KKMSD73], F. Knudsen introduces a family of primitive convex triangulations of $(P_d^n)_{n,d \ge 1}$ called *regular triangulations*. For all $n, d \ge 1$, the regular triangulation $R_d^n$ of $P_d^n$ is obtained by cutting $P_d^n$ by the following hyperplanes:

$$H_k^{i,j} := \left\{ \sum_{l=i}^{j} x_l = k \right\},$$

for all $1 \le i \le j \le n$, and all $1 \le k \le d-1$. The Lemma 2.12 p.126 of [KKMSD73] implies that $R_d^n$ is never $\rho$-uniform when $n$ is bigger than 2 and $d$ is big enough.

<footnote>$^{31}$The number of edges adjacent to the vertex.</footnote>
**Viro Triangulations.** In Definition 5.3 p.43 in [Che23b] we described a method of construction of triangulations of $P_{d+1}^{n+1}$ from a triangulation of $P_d^{n+1}$ and of $P_{d+1}^n$. If $K$ is a triangulation of $P_d^{n+1}$ and $L$ is a triangulation of $P_{d+1}^n$, $K+L$ is the unique triangulation of $P_{d+1}^{n+1}$ for which:

1. The restriction of $K+L$ to $P_{d+1}^{n+1} \cap \{x_{n+1} \ge 1\}$ is the translation of $K$;

2. For all integers $0 \le i \le n$, the restriction of the triangulation $K+L$ to the join of simplices $[e_{n+1}, e_{n+1} + de_1, \dots, e_{n+1} + de_i] * [(d+1)e_i, \dots, (d+1)e_n]$, is the join of the restriction of the translation of $K$ to $[e_{n+1}, e_{n+1} + de_1, \dots, e_{n+1} + de_i]$ with the restriction of $L$ to $[(d+1)e_i, \dots, (d+1)e_n]$.

If both $K$ and $L$ are primitive then so is $K+L$. By Proposition 5.5 p.44 of [Che23b] the same holds for the convexity. Then we recursively defined the Viro triangulations $(V_d^n)_{n,d \ge 1}$ by the formulæ:

$$ \left\{ \begin{array}{ll} V_1^n = P_1^n & \forall n \ge 1 \\ V_{d+1}^{n+1} = V_d^{n+1} + V_{d+1}^n & \forall n, d \ge 1. \end{array} \right. $$

This process can be generalised. Let $L = (L_d)_{d \ge 1}$ be a sequence of primitive triangulations of $(P_d^n)_{d \ge 1}$ we denote by $(V_d(L))_{d \ge 1}$ the sequence of triangulations of $(P_{d+1}^{n+1})_{d \ge 1}$ defined by the formulæ:

$$ \left\{ \begin{array}{ll} V_1(L) = P_1^n & \\ V_{d+1}(L) = V_d(L) + L_{d+1} & \forall d \ge 1. \end{array} \right. $$

**Proposition 5.1.** Let $L = (L_d)_{d \ge 1}$ be a sequence of primitive triangulations of $(P_d^{2n})_{d \ge 1}$. Let $d$ be an even integer. If $L_d$ has a $2n$-simplex that contains only interior vertices, then $V_D(L)$ is not $\rho$-uniform for all $D \ge d+1$.

*Proof.* Let $d$ be such an integer and $\sigma^{2n}$ be a $2n$-simplex of $L_d$ whose vertices are all interior. Using Proposition 2.13, we see that $V_{d+1}(L)$ does not fail to be $\rho$-uniform at $\sigma^{2n}$ if and only if the two apices of the two $(2n+1)$-simplices adjacent to $\sigma^{2n}$ are congruent modulo 2. Indeed, since $\sigma^{2n}$ has an odd number of vertices, and $\sum_{\beta \in \sigma^{2n}} \rho_{\sigma^{2n}}(\beta)$ vanishes, $\rho_{\sigma^{2n}}$ can only be constant to 0. Now the two apices of the two $(2n+1)$-simplices adjacent to $\sigma^{2n}$ are $2e_{2n+1}$ and $(d+1)e_{2n}$. Since $d$ is even they are not congruent and $V_{d+1}(L)$ fails to be $\rho$-uniform at $\sigma^{2n}$. Since all the triangulations $(V_D(L))_{D \ge d+1}$ contain $V_{d+1}(L)$, they all fail to be $\rho$-uniform at $\sigma^{2n}$. $\square$

**Itenberg-Viro Triangulations** In the last example we always perform the first subdivision of the prism $P_{d+1}^{n+1} \cap \{0 \le x_{n+1} \le 1\}$ in the same order. However, we can also reverse the order of the joins.

**Definition 5.2.** If $K$ is a triangulation of $P_d^{n+1}$ and $L$ is a triangulation of $P_{d+1}^n$ we define $K + (-1)L$ as the unique triangulation of $P_{d+1}^{n+1}$ for which:

1. The restriction of $K + (-1)L$ to $P_{d+1}^{n+1} \cap \{x_{n+1} \ge 1\}$ is the translation of $K$;

2. For all integers $0 \le i \le n$, the restriction of the triangulation $K + (-1)L$ to the join of simplices $[e_{n+1} + de_i, \dots, e_{n+1} + de_n] * [0, (d+1)e_1, \dots, (d+1)e_i]$ is the join of the restriction of the translation of $K$ to $[e_{n+1} + de_i, \dots, e_{n+1} + de_n]$ with the restriction of $L$ to $[0, (d+1)e_1, \dots, (d+1)e_i]$.
As a direct consequence of the definition if both K and L are primitive then so is $K + (-1)L$. The same holds for the convexity by a small adaptation of Proposition 5.5 p.44 in [Che23b]. For all $n, d \ge 1$, let $\text{Tr}_d^n$ denote the set of primitive triangulations of $P_d^n$. For all subset $I$ of $\{0, ..., n\}$ of cardinality $(k + 1)$, there is a map $\pi_I : \text{Tr}_d^n \to \text{Tr}_d^k$ that sends a triangulation $K$ to its restriction on the convex hull of $\{de_i : i \in I\}$, where $e_0 = 0$ and the order of the vertices is preserved. Another direct consequence of the definition is that if $K \in \text{Tr}_d^{n+1}$, $L \in \text{Tr}_{d+1}^n$, and $I \subset \{0, ..., n\}$ then $\pi_{I \cup \{n+1\}}(K + (-1)^n L)$ equals $\pi_{I \cup \{n+1\}}(K) + (-1)^n \pi_I(L)$ for all $\eta \in \mathbb{F}_2$. Following [Ite97], we define the *Itenberg-Viro triangulations* $(IV_d^n)_{n,d \ge 1}$ by the formulæ:

$$ 
\left\{ 
\begin{array}{ll}
IV_d^1 & \text{is the only primitive triangulation of } P_d^1 \\
IV_1^n = P_1^n & \forall n \ge 1 \\
IV_{d+1}^{n+1} = IV_d^{n+1} + (-1)^d IV_{d+1}^n & \forall n, d \ge 1.
\end{array}
\right. 
$$

An example of such triangulation is illustrated in Figure 17. As in the case of Viro triangulations this process can be generalised from a sequence of primitive triangulations $(L_d)_{d \ge 1}$ of $(P_d^n)_{d \ge 1}$ to a sequence $(IV_d(L))_{d \ge 1}$ of triangulations of $(P_d^{n+1})_{d \ge 1}$ by the formulæ:

$$ 
\left\{ 
\begin{array}{l}
IV_1(L) = P_1^{n+1} \\
IV_{d+1}(L) = V_d(L) + (-1)^d L_{d+1} \quad \forall d \ge 1.
\end{array}
\right. 
$$

<figure><img src="image_40.png" /><figcaption>Figure 17: The triangulation $IV_4^2$.</figcaption></figure>

**Proposition 5.3.** *The triangulations $L_d$ are $\rho$-uniform for all $d \ge 1$ if and only if the triangulations $IV_d(L)$ are $\rho$-uniform for all $d \ge 1$.*

*Proof.* Let $K$ be a triangulation of $P_d^{n+1}$. We denote, for all $0 \le q \le n+1$, the set of $q$-simplices of $K$ having at least one vertex in the interior $K - \partial K$ by $A_q(K)$. If $L$ is a triangulation of $P_{d+1}^n$, we denote, for all $0 \le q \le n+1$, by $A_q^+(K; L)$ the set of $q$-simplices of $K + e_{n+1}$ that belong to $A_q(K \pm L)$ but not to $A_q(K) + e_{n+1}$. Let us denote by $\underline{K}$ the restriction of $K$ to the bottom face $[0; de_1; ...; de_n]$ of $P_d^{n+1}$. We have:

$$ A_n(K + L) = (A_n(K) + e_{n+1}) \amalg A_n^+(K; L) \amalg ([A_{n-1}(\underline{K}) + e_{n+1}] * (d+1)e_n). $$

Indeed, the $n$-simplices of $A_n(K + L)$ that are neither translates of simplices of $A_n(K)$ nor in $A_n^+(K; L)$ come from joins of simplices of $\underline{K}$ with simplices of $L$. By construction the only such simplices not having all of their vertices in $\partial(K + L)$ belong to the subdivision of the
join $(e_{n+1} + [0; de_1; \dots; de_n]) * (d+1)e_n$. Since $(d+1)e_n$ belongs to $\partial(K+L)$, the remaining $n$-simplices are in the set $([A_{n-1}(K) + e_{n+1}] * (d+1)e_n)$. Likewise we have:

$$A_n(K - L) = (A_n(K) + e_{n+1}) \amalg A_n^-(K; L) \amalg ([A_{n-1}(K) + e_{n+1}] * 0).$$

Now let $L'$ be a triangulation of $P_{d+2}^n$. We have:

$$A_n^-(K + L; L') = (A_n(L) + e_{n+1}) \amalg ([A_{n-1}(L) + e_{n+1}] * 2e_{n+1}),$$

and:

$$A_n^+(K - L; L') = (A_n(L) + e_{n+1}) \amalg ([A_{n-1}(L) + e_{n+1}] * (2e_{n+1} + de_n)).$$

Therefore, we have:

$$\begin{aligned}
A_n(K + L - L') &= (A_n(K + L) + e_{n+1}) \amalg A_n^-(K + L; L') \amalg ([A_{n-1}(L) + e_{n+1}] * 0) \\
&= (A_n(K + L) + e_{n+1}) \amalg (A_n(L) + e_{n+1}) \\
&\qquad \amalg ([A_{n-1}(L) + e_{n+1}] * 2e_{n+1}) \amalg ([A_{n-1}(L) + e_{n+1}] * 0),
\end{aligned}$$

and:

$$\begin{aligned}
A_n(K - L + L') &= (A_n(K - L) + e_{n+1}) \amalg A_n^+(K - L; L') \\
&\qquad \amalg ([A_{n-1}(L) + e_{n+1}] * (d+2)e_n) \\
&= (A_n(K - L) + e_{n+1}) \amalg (A_n(L) + e_{n+1}) \\
&\qquad \amalg ([A_{n-1}(L) + e_{n+1}] * (2e_{n+1} + de_n)) \\
&\qquad \amalg ([A_{n-1}(L) + e_{n+1}] * (d+2)e_n),
\end{aligned}$$

for $K \pm L$ equals $L$. If we assume $K+L$ to be $\rho$-uniform, $K+L-L'$ is $\rho$-uniform as well if and only if the simplices of $(A_n(L) + e_{n+1})$, $([A_{n-1}(L) + e_{n+1}] * 2e_{n+1})$, and $([A_{n-1}(L) + e_{n+1}] * 0)$ satisfy the requirement of Definition 2.22. Let $\sigma^n$ be a simplex of the first set. It bounds the two $(n+1)$-simplices $\sigma^n * 2e_{n+1}$ and $\sigma^n * 0$. By Proposition 2.13 we have:

$$\sum_{\alpha \in \sigma^n} |\rho_{\sigma^n}(\alpha)| \alpha = 2e_{n+1} (\text{mod } 2),$$

and therefore, $\rho_{\sigma^n} = 0$. Now let $\sigma^{n-1}$ be a simplex of $A_{n-1}(L) + e_{n+1}$. It bounds two $n$-simplices of $L + e_{n+1}$, $\sigma_+^n$ and $\sigma_-^n$, so $\sigma^{n-1} * 2e_{n+1}$ bounds the two $(n+1)$-simplices of $K+L-L'$, $\sigma_+^n * 2e_{n+1}$ and $\sigma_-^n * 2e_{n+1}$. Let $\alpha_\pm$ denote the vertex of $\sigma_\pm^n$ opposite to $\sigma^{n-1}$. From Proposition 2.13, we have:

$$\sum_{\beta \in \sigma^n} |\rho_{\sigma^{n-1}*2e_{n+1}}(\beta)| \beta = \alpha_+ - \alpha_- (\text{mod } 2).$$

Since the difference $\alpha_+ - \alpha_-$ belongs to the hyperplane $\{x_{n+1} = 0\}$, the number $|\rho_{\sigma^{n-1}*2e_{n+1}}(2e_{n+1})|$ vanishes and the restriction of $|\rho_{\sigma^{n-1}*2e_{n+1}}|$ to $\sigma^{n-1}$ equals $|\rho_{\sigma^{n-1}}^L|$ (the function $|\rho|$ of the triangulation $L$). The vertex $2e_{n+1}$ belongs to the boundary $\partial(K+L-L')$. It implies that $\rho_{\sigma^{n-1}*2e_{n+1}}$ is constant on the interior vertices $I(\sigma^{n-1} * 2e_{n+1})$ and meets the requirements of
Definition 2.22 if and only if $\rho_{\sigma^{n-1}}^L$ is constant on the interior$^{32}$ vertices $I(\sigma^{n-1})$. The exact same reasoning applies to the simplices of the third set $[A_{n-1}(L) + e_{n+1}] * 0$ and we find that $K + L - L'$ is $\rho$-uniform if and only if $K$ and $L$ are. We can also prove that $K - L + L'$ is $\rho$-uniform if and only if $K$ and $L$ are by the same arguments. The only difference in this case is the two apices with which we suspend the horizontal $n$-simplices $\sigma^n$ of $L$. They are $2e_{n+1} + de_n$ and $(d+2)e_n$. Their difference is also 0 modulo 2 so $\rho_{\sigma^n}$ vanishes as in the case $K + L - L'$. The proposition follows from the construction of the triangulations $(IV_d(L))_{d \ge 1}$. $\square$

**Remark 5.4.** In the last proof we showed that if $\sigma^n$ is a $n$-simplex of the triangulation $IV_d(L)$ of $P_d^{n+1}$ written as a link $\alpha * \sigma^{n-1}$ with $\sigma^{n-1}$ in a floor $L_k$ then the numerical value $|\rho_{\alpha * \sigma^{n-1}}(\beta)|$ is zero for $\beta = \alpha$ and equals $\rho_{\sigma^{n-1}}^{L_k}(\beta)$ otherwise.

**Proposition 5.5.** For all $n, d \ge 1$, for all $\varepsilon \in C^0(IV_d^n; \mathbb{F}_2)$, the number $\ell(\mathbb{R}X_\varepsilon)$ is at least equal to $\lfloor \frac{n-1}{2} \rfloor$.

*Proof.* This is Theorem 5.9 p.45 of [Che23b] for Itenberg-Viro triangulations. The crucial point in the proof of this theorem was Proposition 5.8 of the same article: the fact that the restriction of the Viro triangulation to the face opposite to 0 is still a Viro triangulation. However, the same is true for the Itenberg-Viro triangulation, hence the proposition holds. $\square$

**Theorem 5.6.** For all $n \ge 2$, and all $d \ge 1$, the Itenberg-Viro triangulation $IV_d^n$ is simply integrable. In particular, the Harnack distribution:

$$h^n(x_1, \dots, x_n) := \sum_{i<j} x_i x_j,$$

is a solution of (14).

*Proof.* We will prove, by recursion on $n$, that $h^n$ is a solution of (14) for all $d \ge 1$. Theorem 4.19 ensures that the property is true for $n = 2$. Let us now assume that the property is true for some $n \ge 2$. Let $1 \le d \le D$ be two integers. The $n$-simplices of $IV_D^{n+1} \setminus B_0IV_D^{n+1}$ that have non-empty intersection with the horizontal hyperplane $\{x_{n+1} = D-d\}$ are $n$-simplices of $IV_d^n$ and cones over $(n-1)$-simplices of $IV_d^n \setminus B_0IV_d^n$. If $d$ is even the apices of the cones are $(D-d-1)e_{n+1}$ and $(D-d+1)e_{n+1}$. And if $d$ is odd, they are $(D-d-1)e_{n+1} + (d+1)e_n$ and $(D-d+1)e_{n+1} + (d-1)e_n$. Either way we showed$^{33}$ that $\rho_{\alpha * \sigma^{n-1}}$, with $\sigma^{n-1} \in IV_d^n$, is zero on the apex $\alpha$ of the cone and that the numerical value of $|\rho_{\alpha * \sigma^{n-1}}|$ equals the numerical value $|\rho_{\sigma^{n-1}}^{IV_d^n}|$ of $IV_d^n$. As a consequence, $|D^2h_{\alpha * \sigma^{n-1}}^{n+1}|$ equals $|D^2(h^{n+1}|_{IV_d^n})_{\sigma^{n-1}}|$. However, $h^{n+1}|_{IV_d^n}$ only differs from $h^n$ by an affine function. Hence, by recursion hypothesis $|D^2(h^{n+1}|_{IV_d^n})_{\sigma^{n-1}}|$ equals $|\rho_{\sigma^{n-1}}^{IV_d^n}|$, thus $|D^2h_{\alpha * \sigma^{n-1}}^{n+1}|$ equals $|\rho_{\alpha * \sigma^{n-1}}|$. Now, if $\sigma^n$ belongs to $IV_d^n$, the two $(n+1)$-simplices of $IV_D^{n+1}$ that are adjacent to $\sigma^n$ have their apices congruent modulo 2. It implies that $\rho_{\sigma^n}$ vanishes. These two $(n+1)$-simplices are identical modulo 2, and the value of $h^{n+1}$ is the same on the two apices since they have congruent coordinates. Hence, $D^2h_{\sigma^n}^{n+1}$ vanishes as well. Therefore, we proved that $h^{n+1}$ is a solution of (14). $\square$

**Remark 5.7.** The hypersurface $\mathbb{R}X_h^n$ of $\mathbb{R}P_d^n$ has a maximal number of connected components but also intersects all the toric divisors of $\mathbb{R}P_d^n$ along T-hypersurfaces that have a maximal number of connected components.

<footnote>$^{32}$interior inside $L$ i.e. in $L - \partial L$.</footnote>

<footnote>$^{33}$c.f. Remark 5.4.</footnote>
References

[ARS21] Charles Arnal, Arthur Renaudineau, and Kris Shaw. Lefschetz Section Theorems for Tropical Hypersurfaces. *Annales Henri Lebesgue*, 4:1347–1387, 2021.

[Ber10] Benoit Bertrand. Euler Characteristic of Primitive T-Hypersurfaces and Maximal Surfaces. *Journal of the Institute of Mathematics of Jussieu*, 9(1):1–27, 2010.

[Bjö84] Anders Björner. Posets, Regular CW Complexes and Bruhat Order. *European Journal of Combinatorics*, 5(1):7–16, 1984.

[BLdMR22] Erwan Brugallé, Lucía López de Medrano, and Johannes Rau. Combinatorial Patchworking: Back from Tropical Geometry. To appear in *Transaction of the AMS*, arXiv:2209.14043, 2022.

[Che23a] Jules Chenal. A Poincaré-Lefschetz Theorem for Cellular Cosheaves and an Application to the Tropical Homology of Orbifold Toric Varieties. hal-04164269, jul. 2023.

[Che23b] Jules Chenal. Poincaré Duality, Degeneracy, and Real Lefschetz Property for T-Hypersurfaces. hal-04349997, dec. 2023.

[Ehr62] Eugène Ehrhart. Sur les polyèdres rationnels homothétiques à n dimensions. *Comptes rendus de l'académie des Sciences*, 254(1):616–618, january 1962.

[Ful93] William Fulton. *Introduction to Toric Varieties*. Annals of mathematics studies, no. 131.; William H. Roever lectures in geometry. Princeton University Press, 1993.

[Haa97] Bertrand Haas. *Real Algebraic Curves and Combinatorial Constructions*. PhD thesis, Universität Basel, 1997.

[IKMZ19] Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin, and Ilia Zharkov. Tropical Homology. *Mathematische Annalen*, 374(1):963–1006, 2019.

[Ite97] Ilia Itenberg. Topology of Real Algebraic T-surfaces. *Revista Matemática de la Universidad Complutense de Madrid*, 10(Supl.):131–152, 1997.

[IV96] Ilia Itenberg and Oleg Viro. Patchworking Algebraic Curves Disproves the Ragsdale Conjecture. *The Mathematical Intelligencer*, 18(4):19–28, 1996.

[JRS18] Philipp Jell, Johannes Rau, and Kristin Shaw. Lefschetz (1,1)-theorem in tropical geometry. *Épijournal de Géométrie Algébrique*, Volume 2, 2018.

[KKMSD73] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat. *Toroidal Embeddings 1*. Lecture Notes in Mathematics. Springer Berlin, Heidelberg, 1 edition, 1973.

[KS10] Masaki Kashiwara and Pierre Schapira. *Sheaves on Manifolds*, chapter Constructible sheaves, pages 320–359. Grundlehren der mathematischen Wissenschaften. Springer Berlin, Heidelberg, 1 edition, 12 2010.
[Mac71] I. G. MacDonald. Polynomials Associated with Finite Gell-Complexes. *Journal of the London Mathematical Society*, s2-4(1):181–192, 2023/09/16 1971.

[McM77] P. McMullen. Valuations and Euler-Type Relations on Certain Classes of Convex Polytopes. *Proceedings of the London Mathematical Society*, s3-35(1):113–135, 2023/10/05 1977.

[Ris93] Jean-Jacques Risler. Construction d'hypersurfaces réelles [d'après Viro]. In *Sémi-naire Bourbaki : volume 1992/93, exposés 760-774*, number 216 in Astérisque. Société mathématique de France, 1993. talk:763.

[RS23] Arthur Renaudineau and Kris Shaw. Bounding the Betti Numbers of Real Hypersurfaces Near the Tropical Limit. *Annales Scientifiques de l'École Normale Supérieure*, 56(3):945–980, 2023.

[She85] Allan D. Shepard. *A Cellular Description of the Derived Category of a Stratified Space*. PhD thesis, Brown University, 05 1985.

[Vir06] Oleg Viro. Patchworking Real Algebraic Varieties. arXiv:math/0611382, 2006.

[Zie95] Günter M. Ziegler. *Lectures on Polytopes*. Graduate Texts in Mathematics. Springer New York, NY, 1 edition, 1995.