A note on Lefschetz spheres and their relatives

TL;DR

This paper explores specific classes of PL-spheres, particularly those that are both vertex-decomposable and strongly edge-decomposable, linking them to recent proof methods related to the $g$-conjecture.

Contribution

It identifies a class of spheres where vertex-decomposability and strong edge-decomposability intersect, enabling the application of Adiprasito's proof techniques.

Findings

Identified the intersection of vertex-decomposable and strongly edge-decomposable spheres.

Linked this class to the applicability of Adiprasito's proof method.

Provided insights into the structure of spheres related to the $g$-conjecture.

Abstract

Inspired by the works of Adiprasito, Babson, Nevo, and Murai on the $g$-conjecture, we consider different classes of PL-spheres and the relations between them. We focus on a certain class of spheres that is in the intersection of vertex-decomposable spheres (a concept due to Provan and Billera) and strongly edge-decomposable spheres (a concept due to Eran Nevo). The spheres in this class are exactly those vertex-decomposable ones for which Adiprasito's recent proof method works.

Equations

$$h_j=\sum _{i=0}^j(-1{)}^{j-i}(\genfrac{}{}{0ex}{}{d-i}{j-i})f_{i-1}\text{ for all }0\le j\le d.$$

$${\omega }^{d-2j}:(A/\Theta {)}_j\to (A/\Theta {)}_{d-j}$$

$$\mathcal{C}(e,\Sigma )=\{\sigma \in \Sigma |u\notin \sigma \}\cup \{(\sigma \setminus \{u\})\cup \{v\}|u\in \sigma \in \Sigma \}.$$

$$D_j=(\Gamma \setminus \{v_0\})\setminus \{v_1\}\setminus \dots \setminus \{v_j\}=(\Sigma \setminus \{v_1\})\setminus \dots \setminus \{v_j\}$$

$$\bigcup _{2\le i\le j}\mathrm{st}(v_i,\mathcal{C}(e,\Sigma ))$$

Full text

Introduction

Let $\Sigma$ be a $(d-1)$-dimensional simplicial complex. The $f$-vector $f(\Sigma)=(f_{-1},f_{0},\ldots,f_{d-1})$ of $\Sigma$ lists the number of various dimensional faces in $\Sigma$. The $h$-vector $h(\Sigma)=(h_{0},h_{1},\ldots,h_{d})$ of $\Sigma$ is defined via the following equations

[TABLE]

It is well-known and easy to see that the $f$-vector and the $h$-vector contain the same information. However, many interesting relations can be stated more clearly in terms of the $h$-numbers. Among them is the Dehn-Sommerville relations $h_{j}=h_{d-j}$, $0\leq j\leq d$ for simplicial (homology) spheres [Sta96, p. 67]. In particular, these relations show that the face numbers of a (homology) sphere $\Sigma$ are completely determined from the first “half” of the $h$-vector, or equivalently from the $g$-vector $g(\Sigma)=(g_{0},g_{1},\ldots,g_{\lfloor d/2\rfloor})$ of $\Sigma$ where $g_{0}=1$ and $g_{j}=h_{j}-h_{j-1}$ for all $1\leq j\leq\lfloor d/2\rfloor$.

One of the major open problems on face numbers of simplicial complexes is McMullen’s $g$-conjecture. It states that a necessary and sufficient condition for a vector $\mathbf{g}=(g_{0},g_{1},\ldots,g_{\lfloor d/2\rfloor})$ to be the $g$-vector of a $(d-1)$-dimensional homology sphere is that there exists a set $\mathbf{M}$ of monomials closed under taking divisors, such that for all $1\leq j\leq\lfloor d/2\rfloor$ the number of monomials of degree $j$ in $\mathbf{M}$ is $g_{j}$.

Billera and Lee [BL80] proved the sufficiency of McMullen’s $g$-conjecture. The necessity part would follow from an algebraic approach that we briefly describe now. We refer to the book [Sta96] by Stanley for undefined algebraic terminology and to the survey article [Swa14] for a more detailed description.

Let $\Sigma$ be a $(d-1)$-dimensional simplicial complex on the vertex set $V=V(\Sigma)$. Let $\mathbb{F}$ be an infinite field and let $R=\mathbb{F}[x_{v}|v\in V]$ be the polynomial ring with grading induced by setting $\mathrm{deg}\ x_{v}=1$ for all $v\in V$. The Stanley-Reisner ring $A=R/I_{\Sigma}$ of $\Sigma$ over $\mathbb{F}$ is obtained by dividing $R$ by the ideal $I_{\Sigma}$ generated by all square-free monomials whose support is not in $\Sigma$. Let $\Sigma$ be Cohen-Macaulay over $\mathbb{F}$ with a symmetric $h$-vector. We say that $\Sigma$ has the strong Lefschetz property over $\mathbb{F}$ (or $\Sigma$ is a Lefschetz complex over $\mathbb{F}$) if there exist a linear system of parameters $\theta_{1},\ldots,\theta_{d}\in R_{1}$ and a linear form $\omega\in R_{1}$ such that the multiplication map

[TABLE]

is a bijection for all $0\leq j\leq\lfloor d/2\rfloor$, where $\Theta$ is the ideal of $A$ generated by $\theta_{1},\ldots,\theta_{d}$. The algebraic $g$-conjecture states that every homology sphere is a Lefschetz complex over $\mathbb{R}$. The algebraic $g$-conjecture implies the $g$-conjecture.

The prototype of this algebraic approach is Stanley’s paper [Sta80], where by using tools from toric geometry he showed that boundary complexes of simplicial polytopes are Lefschetz over $\mathbb{R}$.

In [Nev07], Nevo introduced the class of strongly edge-decomposable spheres. It was shown by Babson and Nevo [BN10] that strongly edge-decomposable spheres are Lefschetz over any field of characteristic zero. Their result was generalized by Murai, see [Mur10, Corollary 3.5], where he considers a more general class of strongly edge-decomposable complexes without any assumption on the characteristic of the field.

Recently, Adiprasito announced a proof of the algebraic $g$-conjecture for $\mathbb{R}$-homology spheres [Adi18], and in particular for vertex-decomposable spheres. While the reach and validity of his general claim awaits confirmation, in this note we focus on a class of vertex-decomposable spheres considered in his work [Adi18, Theorem 6.3] (see also Section 6 of his most recent manuscript [Adi19]) that we call strongly vertex-decomposable. Adiprasito’s path of argument offers a proof of the algebraic $g$-conjecture that is valid for this class of spheres. We show that strongly vertex-decomposable spheres are in fact strongly edge-decomposable, and thus, the algebraic $g$-conjecture for this class of spheres follows also from the above-mentioned works by Babson and Nevo [BN10] and by Murai [Mur10].

In Figure 1 we illustrate the relations between different classes of spheres and Lefschetz complexes. A solid directed arrow from a class $X$ to the class $Y$ indicates the containment relation $X\subset Y$.

If we disregard the class of Lefschetz complexes in this figure, then $X\subset Y$ if and only if there is a directed path from $X$ to $Y$, except possibly for the case when “$Y$=strongly edge-decomposable” and “$X=$vertex-decomposable” or “$X=$shrinkable” (see Section 4).

Edge Contraction

We begin by establishing the notation. For undefined terminology and the basic facts about piecewise linear (PL) topology, we refer the reader to the chapter [Bjö95] by Björner.

Let $\Sigma$ be a simplicial complex and $\sigma$ be a face of $\Sigma$. The star $\mathrm{st}(\sigma,\Sigma)$ of $\sigma$ in $\Sigma$ is the set of all faces $\tau\in\Sigma$ such that $\sigma\cup\tau\in\Sigma$. The antistar $\mathrm{ast}(\sigma,\Sigma)$ of $\sigma$ in $\Sigma$ is the set of all faces $\tau\in\Sigma$ such that $\sigma\cap\tau=\emptyset$. The link of $\sigma$ in $\Sigma$ is defined by $\mathrm{lk}(\sigma,\Sigma):=\mathrm{st}(\sigma,\Sigma)\cap\mathrm{ast}(\sigma,\Sigma)$. The deletion $\Sigma\setminus\sigma$ of $\sigma$ from $\Sigma$ is the subcomplex of $\Sigma$ consisting of all faces that do not contain $\sigma$.

If $e=uv$ is an edge of $\Sigma$, the contraction of $\Sigma$ with respect to $e$ is the simplicial complex $\mathcal{C}(e,\Sigma)$ obtained from $\Sigma$ by identifying $u$ and $v$. More precisely, $\mathcal{C}(e,\Sigma)$ is a simplicial complex on the vertex set $V(\Sigma)\setminus\{u\}$ with the following set of faces

[TABLE]

Let $\Sigma$ be a PL-sphere and $e=uv$ be an edge of $\Sigma$. We say that $\Sigma$ satisfies the link condition with respect to $e$ if $\mathrm{lk}(v,\Sigma)\cap\mathrm{lk}(u,\Sigma)=\mathrm{lk}(e,\Sigma)$.

Theorem 1** (Klee-Kleinschmidt and Nevo).**

Let $\Sigma$ be a $(d-1)$-dimensional PL-sphere and $e=uv$ be an edge of $\Sigma$. Then the following are equivalent:

1. $(i)$

$\mathcal{C}(e,\Sigma)$* is PL-homeomorphic to $\Sigma$,* 2. $(ii)$

$\Sigma$* satisfies the link condition with respect to $e$,* 3. $(iii)$

$(\Sigma\setminus\{u\})\setminus\{v\}$* is a $(d-1)$-dimensional PL-ball,* 4. $(iv)$

$\mathrm{st}(u,\Sigma)\cup\mathrm{st}(v,\Sigma)$* is a $(d-1)$-dimensional PL-ball.*

Proof.

The equivalence between $(i)$, $(ii)$, and $(iii)$ is proved in [KK87, 6.1]. To see the equivalence $(iii)\Leftrightarrow(iv)$, let us recall Newman’s Theorem (see [Bjö95, 12.2], for instance) that: if $\Gamma$ is a $(d-1)$-dimensional PL-ball that is a subcomplex of a $(d-1)$-dimensional PL-sphere $\Sigma$, then the closure of the complement of $||\Gamma||$ in $||\Sigma||$ is a $(d-1)$-dimensional PL-ball (here $||.||$ denotes the geometric realization). Now, letting $\Gamma$ to be either of the complexes $(\Sigma\setminus\{u\})\setminus\{v\}$ or $\mathrm{st}(u,\Sigma)\cup\mathrm{st}(v,\Sigma)$, the closure of the complement of $||\Gamma||$ in $||\Sigma||$ is the geometric realization of other one. ∎

The equivalence between conditions $(i)$ and $(ii)$ of Theorem 1 in the more general setting of PL-manifolds (without boundary) is shown in [Nev07, Theorem 1.4]. One can show that the condition $(iv)$ of Theorem 1 is also equivalent to conditions $(i)$ and $(ii)$ in the more general setting of PL-manifolds.

Classes of PL-spheres

Vertex-Decomposable and Shrinkable Spheres

The concept of vertex-decomposability is a strengthening of the concept of shellability that was introduced by Provan and Billera [PB80] in connection with Hirsch conjecture.

Definition 2**.**

A pure $(d-1)$-dimensional simplicial complex $\Sigma$ is said to be vertex-decomposable if either $\Sigma$ is a $(d-1)$-simplex or there exists a vertex $v$, called a shedding vertex, such that

$\Sigma\setminus\{v\}$ is $(d-1)$-dimensional vertex-decomposable, and 2. 2.

$\mathrm{lk}(v,\Sigma)$ is $(d-2)$-dimensional vertex decomposable.

A shedding order $v_{1},\ldots,v_{k}$ of a $(d-1)$-dimensional vertex-decomposable complex $\Sigma$ is a linear ordering of $k=f_{0}(\Sigma)-d$ of its vertices such that for all $j$ the vertex $v_{j}$ is a shedding vertex for $(\Sigma\setminus\{v_{1}\})\setminus\ldots\setminus\{v_{j-1}\}$. Notice that if $v_{1},\ldots,v_{k}$ is a shedding order, then the rest of vertices of $\Sigma$ forms a $(d-1)$-simplex.

Provan and Billera [PB80, Theorem 3.1.3] proved that $2$-dimensional spheres and balls are vertex-decomposable. It was shown by Klee and Kleinschmidt [KK87, p. 743] that there is a simplicial $4$-polytope with $10$ vertices whose boundary complex is not vertex-decomposable.

The following fact about vertex-decomposable balls is well-known. For the sake of completeness we include a proof.

Lemma 3**.**

Let $\Sigma$ be a $(d-1)$-dimensional vertex-decomposable ball and $v$ be a shedding vertex of $\Sigma$. Then

1. $(i)$

$v$* lies on the boundary of $\Sigma$, and* 2. $(ii)$

$\Sigma\setminus\{v\}$* is a $(d-1)$-dimensional PL-ball.*

Proof.

First notice that for a vertex $u\in\Sigma$ lies on the boundary of $\Sigma$ if and only if $\mathrm{lk}(u,\Sigma)$ is $(d-2)$-dimensional ball and $u\notin\partial\Sigma$ if and only if $\mathrm{lk}(u,\Sigma)$ is $(d-2)$-dimensional sphere. So, in order to prove $(i)$, it suffices to show that for a shedding vertex $v$ one has $\widetilde{H}_{d-2}(\mathrm{lk}(v,\Sigma),\mathbb{Z})=0$. Now, since $\Sigma$ is acyclic and $\Sigma\setminus\{v\}$ is $(d-1)$-connected (see [Bjö95, 11.2]), the Mayer-Vietoris sequence for the pair $\mathrm{st}(v,\Sigma)$ and $\Sigma\setminus\{v\}$ shows that $\widetilde{H}_{d-2}(\mathrm{lk}(v,\Sigma),\mathbb{Z})\cong\widetilde{H}_{d-2}(\Sigma\setminus\{v\},\mathbb{Z})=0$. This complete the argument for part $(i)$.

In order to prove $(ii)$, we use induction on $d$. Note that for $d=2$ the statement clearly holds. For a non-empty face $\sigma\in\Sigma\setminus\{v\}$ we observe that if $v\notin\mathrm{lk}(\sigma,\Sigma)$, then $\mathrm{lk}(\sigma,\Sigma)=\mathrm{lk}(\sigma,\Sigma\setminus\{v\})$ is either a PL-ball or a PL-sphere. On the other hand, if $v\in\mathrm{lk}(\sigma,\Sigma)$, then it is not difficult to show that $v$ is a shedding vertex for $\mathrm{lk}(\sigma,\Sigma)$ (see the proof of Proposition 2.3 in [PB80], for instance). Therefore, $\mathrm{lk}(\sigma,\Sigma)\setminus\{v\}=\mathrm{lk}(\sigma,\Sigma\setminus\{v\})$ must be a PL-ball. In particular, $\Sigma\setminus\{v\}$ is a vertex-decomposable PL-manifold (with non-empty boundary) and hence a PL-ball[Bjö95, Theorem 11.4].

∎

While any vertex-decomposable sphere contain an edge whose contraction results in a PL-sphere (see [KK87, 6.2], for instance), the class of vertex-decomposable spheres is not closed under edge contractions. By focusing on some essential properties of vertex-decomposable spheres, we extend this family to a family of spheres that behave more nicely under edge contraction.

Definition 4**.**

A $(d-1)$-dimensional PL-sphere $\Sigma$ is said to be shrinkable if either $\Sigma$ is the boundary of the $d$-simplex or there exists a facet $\sigma$ of $\Sigma$ so that $V(\Sigma)\setminus V(\sigma)$ can be linearly ordered $v_{1},v_{2},\ldots,v_{k}$ (called a shrinking order) so that for any $1\leq j\leq k$

the subgraph $G_{j}$ of the $1$-skeleton of $\Sigma$ induced by $v_{1},\ldots,v_{j}$ is connected, 2. 2.

The complex $\mathcal{N}_{j}(\Sigma):=\bigcup_{i\leq j}\mathrm{st}(v_{i},\Sigma)$ is a $(d-1)$-dimensional PL-ball.

Theorem 5**.**

Let $\Sigma$ be a $(d-1)$-dimensional vertex-decomposable ball with a shedding order $v_{1},\ldots,v_{k}$. Let $v_{0}$ be a vertex outside $\Sigma$ and let $\Gamma=(v_{0}\ast\partial\Sigma)\cup\Sigma$ be the sphere obtined from taking a cone with apex $v_{0}$ over the boundary of $\Sigma$. Then $\Gamma$ is shrinkable with a shrinking order given by $v_{0},v_{1},\ldots,v_{k}$. In particular, every vertex-decomposable sphere is shrinkable and for all $d\geq 5$ there exist $(d-1)$-dimensional shrinkable but not vertex-decomposable spheres.

Proof.

First we observe that part $(ii)$ of Lemma 3 implies that for any $0\leq j\leq k$, the complex

[TABLE]

is a $(d-1)$-dimensional vertex-decomposable ball. By Newman’s Theorem the closure $X$ of the complement of $||D_{j}||$ in $||\Gamma||$ is a $(d-1)$-dimensional ball. However, the complex $\mathcal{N}_{j}(\Gamma)=\bigcup_{0\leq i\leq j}\mathrm{st}(v_{i},\Gamma)$ is a triangulation of $X$. Now, since $v_{j+1}$ is a shedding vertex of $D_{j}$, it follows from Lemma 3 that $v_{j+1}$ has to lie on the boundary of $D_{j}$ which is also the boundary of $N_{j}(\Gamma)$. This shows that the subgraph $G_{j}$ of the $1$-skeleton of $\Sigma$ induced by $v_{1},\ldots,v_{j}$ is connected. Thus $v_{0},v_{1},\ldots,v_{k}$ is a shrinking order for $\Gamma$.

Now observe that if $\Gamma$ is a vertex-decomposable sphere with a shedding vertex $v$, then $\Sigma=\Gamma\setminus\{v\}$ is a vertex-decomposable ball and $\Gamma=(v\ast\partial\Sigma)\cup\Sigma$. Hence, every vertex-decomposable sphere is shrinkable.

Finally, let $\Sigma$ be a PL-sphere whose first barycentric subdivision is not shellable. Such spheres exist in all dimensions greater than or equal $3$. By a result of Pachner [Pac91, Theorem 5.8], every PL-sphere can be realized as the boundary complex to a shellable ball. So, we may assume that there exists a shellable ball $\Gamma$ such that $\partial\Gamma=\Sigma$. Let $\mathrm{sd}(\Gamma)$ denote the barycentric subdivision of $\Gamma$. Since the barycentric subdivision of every shellable complex is vertex-decomposable [PB80, Corollary 3.3.2], $\mathrm{sd}(\Gamma)$ is a vertex-decomposable ball and the sphere $\Lambda=(w\ast\mathrm{sd}(\Gamma))\cup\mathrm{sd}(\Gamma)$ is shrinkable. However, $\Lambda$ is not vertex-decomposable. Because $\mathrm{lk}(w,\Lambda)=\mathrm{sd}(\Sigma)$ is not vertex-decomposable while the link of every face in a vertex-decomposable complex is vertex-decomposable [PB80, Proposition 2.3].

∎

Strong Versions

Next, we present the precise definition of the subclass of vertex-decomposable spheres that is considered in Adiprasito’s paper (see the paragraph preceding [Adi18, Theorem 6.3], see also [Adi19, Section 6] for a clarified version).

Definition 6**.**

A $(d-1)$-dimensional vertex-decomposable sphere $\Sigma$ is said to be strongly vertex-decomposable if either $d$ is less than or equal $3$, or there exists a shedding order $v_{1},\ldots,v_{k}$ (that we call a strong shedding order) such that, for any $1\leq j<k$, the complex $\mathrm{lk}(v_{j+1},\partial\mathcal{N}_{j}(\Sigma))$ is strongly vertex-decomposable, where $\mathcal{N}_{j}(\Sigma):=\bigcup_{i\leq j}\mathrm{st}(v_{i},\Sigma)$.

Notice that since every shedding order is a shrinking order, $\mathcal{N}_{j}(\Sigma):=\bigcup_{i\leq j}\mathrm{st}(v_{i},\Sigma)$ is a $(d-1)$-dimensional ball for all $1\leq j<k$ and it makes sense to talk about $\partial\mathcal{N}_{j}(\Sigma)$.

We also consider the strong version of shrinkability. We define the class of strongly shrinkable spheres by replacing “vertex-decomposable” by “shrinkable” in the definition above. Clearly, every strongly vertex-decomposable sphere is strongly shrinkable.

Theorem 7**.**

(Adiprasito [Adi18, Theorem 6.3]). Let $\Sigma$ be a strongly vertex-decomposable sphere and $\mathbb{F}$ be a field of characteristic zero. Then $\Sigma$ is a Lefschetz complex over $\mathbb{F}$.

The following class of PL-spheres is introduced by Nevo [Nev07].

Definition 8**.**

A $(d-1)$-dimensional sphere $\Sigma$ is said to be strongly edge-decomposable if either $\Sigma$ is the boundary of $d$-simplex or else there exists an edge $e\in\Sigma$ such that $\Sigma$ satisfies the link condition with respect to $e$ and $\mathrm{lk}(e,\Sigma)$ and $\mathcal{C}(e,\Sigma)$ are strongly edge-decomposable.

All $2$-dimensional spheres are strongly edge-decomposable. Babson and Nevo [BN10] proved that strongly edge-decomposable spheres are Lefschetz over any field of characteristic zero. This was generalized by Murai [Mur10] where he extended the definition of strong edge-decomposition to complexes that are not necessarily PL-spheres.

Theorem 9**.**

(Murai [Mur10, Corollary 3.5]). Let $\Sigma$ be a strongly edge-decomposable sphere and $\mathbb{F}$ be an infinite field. Then $\Sigma$ is a Lefschetz complex over $\mathbb{F}$.

Now, we show that strongly vertex-decomposable spheres are strongly edge-decomposable.

Theorem 10**.**

Every strongly shrinkable sphere is strongly edge-decomposable. Consequently, every strongly vertex-decomposable sphere is strongly edge-decomposable.

Proof.

We use induction both on the dimension and the length of the shrinking order. So, let $\Sigma$ be a $(d-1)$-dimensional shrinkable sphere with a shrinking order $v_{1},v_{2},\ldots,v_{k}$. If $d\leq 3$ or $k=1$, then $\Sigma$ is strongly edge-decomposable. Hence, we may assume that $k>1$ and every strongly shrinkable sphere of dimension less than $d-1$ or with a shrinking order of length less than $k$ is strongly edge-decomposable. It follows from the definition that $e=v_{1}v_{2}$ is an edge of $\Sigma$ and $\mathrm{st}(v_{1},\Sigma)\cup\mathrm{st}(v_{2},\Sigma)$ is a $(d-1)$-dimensional ball. Hence, by Theorem 1, one sees that $\Sigma$ satisfies the link condition with respect to $e$. In particular, $\mathcal{C}(e,\Sigma)$ is a PL-sphere. Our desired conclusion follows from the claims below:

Claim 10.1**.**

$\mathcal{C}(e,\Sigma)$* is strongly shrinkable.*

Claim 10.2**.**

$\mathrm{lk}(e,\Sigma)$* is strongly shrinkable.*

Proof of Claim 10.1.

Since for any $2\leq j\leq k$ $(\mathcal{C}(e,\Sigma)\setminus\{v_{2}\})\setminus\ldots\setminus\{v_{j}\}=(\Sigma\setminus\{v_{1}\})\setminus\ldots\setminus\{v_{j}\}$ is a PL-ball, it follows from the Newman’s Theorem that the complex

[TABLE]

is also a PL-ball. ∎

Proof of Claim 10.2.

It follows easily from the definition of strongly shrinkable spheres, the general equation $\mathrm{lk}(e,\Sigma)=\mathrm{lk}(v_{2},\mathrm{lk}(v_{1},\Sigma))$ and the fact that $\mathrm{lk}(v_{1},\Sigma)=\partial\mathcal{N}_{1}(\Sigma)$. ∎

∎

Remarks

Remark 11**.**

In dimensions less than or equal to $4$ every shedding order is also a strong shedding order and, consequently, all vertex-decomposable spheres of dimension less than or equal to $4$ are strongly vertex-decomposable. In all dimensions greater than $4$, using a result of Pachner [Pac91, Theorem 5.8], one can construct examples of shedding orders that are not strong. However, this does not eliminate the possibility of existence of other shedding orders that are strong.

It is interesting to construct a concrete example, if such an example exists at all, of a vertex-decomposable sphere which is not strongly vertex-decomposable. Since, every strongly vertex-decomposable sphere is strongly edge-decomposable, it would be enough to find a vertex-decomposable sphere which is not strongly edge-decomposable.

Problem 12**.**

Find a triangulated sphere which is vertex-decomposable but not strongly edge-decomposable.

Remark 13**.**

For the inductive argument given by Adiprasito in the proof of [Adi18, Theorem 6.3] to work for a PL-sphere $\Sigma$, it is necessary and sufficient that

• (i)

$\Sigma$ is shrinkable with a shrinking order $v_{1},v_{2},\ldots,v_{k}$, and

• (ii)

the complexes $\mathrm{lk}(v_{j+1},\partial\mathcal{N}_{j}(\Sigma))$ are Lefschetz.

In particular, the argument given by Adiprasito shows that:

• (1)

Every strongly shrinkable sphere is a Lefschetz complex.

• (2)

If all PL-spheres of dimension less than or equal to $d$ are Lefschetz complexes, then all shrinkable spheres of dimension less than or equal to $d+2$ are Lefschetz complexes.

See also [Adi19, Section 6] where shrinkable and strongly shrinkable spheres are called $B$-decomposable and $A$-decomposable respectively.

Remark 14**.**

While we were at the final stage of preparing this note, two new papers by Karu [Kar19] and independently by Adiprasito and Steinmeyer [AS19] appeared on arXiv announcing simpler proofs of the $g$-conjecture for PL-spheres.