# Tight complexes are Golod

Kouyemon Iriye  Department of Mathematics, Osaka Metropolitan University,
Sakai, 599-8531, Japan<EMAIL_ADDRESS>and  Daisuke Kishimoto  Faculty of
Mathematics, Kyushu University, Fukuoka 819-0395, Japan
<EMAIL_ADDRESS>
###### Abstract.

Golodness of a simplicial complex is defined algebraically in terms of the
Stanley-Reisner ring, and it has been a long-standing problem to find its
combinatorial characterization. Tightness of a simplicial complex is a
combinatorial analogue of a tight embedding of a manifold into Euclidean
space, and has been studied in connection to minimal manifold triangulations.
In this paper, by using an idea from toric topology, we prove that tight
complexes are Golod, and as a corollary, we obtain that a triangulation of a
closed connected orientable manifold is Golod if and only if it is tight,
which is a combinatorial characterization of Golodness for manifold
triangulations.

###### Key words and phrases:

Golod complex, tight complex, manifold triangulation, Massey product, higher
prism operator

###### 2020 Mathematics Subject Classification:

13F55, 55S30, 57Q15

##  1\. Introduction

Let $K$ be a simplicial complex, and let $\mathbb{F}[K]$ denote the Stanley-
Reisner ring of $K$ over a field $\mathbb{F}$. The following problem is of
particular interest.

######  Problem 1.1.

Given an algebraic property of $\mathbb{F}[K]$, find a combinatorial
characterization of it.

There are some cases where Problem 1.1 is resolved. A typical example is
Cohen-Macaulayness which is characterized by Reisner’s criterion. In this
paper, we consider Golodness in Problem 1.1. Let $\mathbb{F}$ be a field, and
let $S=\mathbb{F}[x_{1},\ldots,x_{n}]$ where we set each $x_{i}$ to be of
degree 2. Serre [32] proved that for a noetherian ring $R=S/I$ where $I$ is a
homogeneous ideal of $S$ such that $I\subset(x_{1},\ldots,x_{n})^{2}$, there
is a coefficient-wise inequality

$P(\mathrm{Tor}^{R}(\mathbb{F},\mathbb{F});t)\leq\frac{(1+t^{2})^{n}}{1-t(P(\mathrm{Tor}^{S}(R,\mathbb{F});t)-1)}$

where $P(V;t)$ denotes the Poincaré series of a graded vector space $V$ in the
indeterminate $t$. If equality holds, then $R$ is called _Golod_. Later, it
was Golod who proved that $R$ is Golod if and only if all products and
(higher) Massey products in the Koszul homology of $R$ (with respect to
$x_{1},\ldots,x_{n}$) are trivial. Therefore Golodness is a fairly complex
property.

We say that $K$ is _$\mathbb{F}$ -Golod_ if $\mathbb{F}[K]$ is Golod. As
Golodness is quite complicated as mentioned above, Problem 1.1 for Golodness
has been widely open for a long time. Recently, toric topology provided new
techniques and viewpoints for attacking Problem 1.1 for Golodness. More
precisely, as in [6], a space called the moment-angle complex for $K$, or a
more general polyhedral product, realizes the Koszul complex of
$\mathbb{F}[K]$ as its cellular cochain complex, up to homotopy, and so
topological techniques apply to produce new results on Golodness of a
simplicial complex as in [3, 9, 8, 10, 12, 13, 14, 15, 17, 20, 29]. Using this
connection of topology to combinatorics, other results in combinatorics and
topology were obtained as in [1, 11, 18, 16, 19, 24, 22, 27]. See the survey
[2] and references therein for more details and results. Remarkably, we can
get a new point of view that $\mathbb{F}$-Golod manifold triangulations might
be connected to the minimality of triangulations as follows. As in [9, 17], a
triangulation of a circle is $\mathbb{F}$-Golod if and only if it is a
triangle which is the minimal triangulation of a circle, and as in [13], a
triangulation of a connected closed $\mathbb{F}$-orientable surface is
$\mathbb{F}$-Golod if and only if it is neighborly, i.e. every two vertices
are joined by an edge, where most neighborly triangulations are minimal as in
[30].

We give the definition of a tight complex. For a non-empty subset $I$ of the
vertex set of a simplicial complex $K$, the full subcomplex of $K$ over $I$ is
defined by

$K_{I}=\\{\sigma\in K\mid\sigma\subset I\\}.$

A simplicial complex $K$ is called _$\mathbb{F}$ -tight_ if the natural map

$H_{*}(K_{I};\mathbb{F})\to H_{*}(K;\mathbb{F})$

is injective for each non-empty subset $I$ of the vertex set of $K$. Tightness
of a simplicial complex was introduced by Kühnel [28] as a combinatorial
analogue of a tight embedding of a manifold into the Euclidean space, and
since then, it has been intensely studied. In particular, Kühnel and Lutz [27]
conjectured that tight manifold triangulations are strongly minimal, that is,
the number of simplices are minimal in each dimension, and this conjecture was
verified in dimension $\leq 3$ [5, 28, 27]. Then since we may expect a
connection of Golodness of manifold triangulations and the minimality of
triangulations, we may also expect a connection between Golodness and
tightness for manifold triangulations, which is of particular interest because
the two notions from different contexts might be equivalent. By using the
moment-angle complex, the authors [18] proved topologically that if $K$ is a
triangulation of a closed connected $\mathbb{F}$-orientable 3-manifold, then
$K$ is $\mathbb{F}$-Golod if and only if it is $\mathbb{F}$-tight. Moreover,
the authors [18] also proved that if a simplicial complex, not necessarily a
manifold triangulation, is $\mathbb{F}$-tight, then it is weakly
$\mathbb{F}$-Golod, that is, all products in the Koszul homology of
$\mathbb{F}[K]$ are trivial.

In this paper, we will prove:

######  Theorem 1.2.

Every $\mathbb{F}$-tight complex is $\mathbb{F}$-Golod.

The converse of Theorem 1.2 does not hold, in general. For example, let $K$ be
a simplicial complex consisting of two vertices. Then the Koszul homology of
$\mathbb{F}[K]$ is the exterior algebra generated by a single element of
degree 3, and so $K$ is $\mathbb{F}$-Golod. But $K$ is not $\mathbb{F}$-tight
because $\mathbb{F}$-tight complexes must be neighborly.

Now we restrict simplicial complexes, and consider the converse of Theorem
1.2. We say that a simplicial complex $K$ is _weakly $\mathbb{F}$-Golod_ if
all products, not necessarily all (higher) Massey products, in the Koszul
homology of $\mathbb{F}[K]$ are trivial. Clearly, if a simplicial complex is
$\mathbb{F}$-Golod, then it is weakly $\mathbb{F}$-Golod. It is proved in [18]
that if a triangulation of a closed connected $\mathbb{F}$-orientable manifold
is weakly $\mathbb{F}$-Golod, then it is $\mathbb{F}$-tight. So by Theorem
1.2, we get:

######  Corollary 1.3.

A triangulation of a closed connected $\mathbb{F}$-orientable manifold is
$\mathbb{F}$-Golod if and only if it is $\mathbb{F}$-tight.

By definition, it is hard to directly check Golodness. But by Corollary 1.3,
we get a criterion for Golodness, which is easy to check. Let $K$ be a
triangulation of a connected manifold of dimension $d\geq 3$. We say that $K$
is _tight-neighborly_ if

$\binom{m-d-1}{2}=\binom{d+2}{2}\dim H_{1}(K;\mathbb{Q})$

where $m$ is the number of vertices of $K$. It is easy to check tight-
neighborliness. As in [5, 7], if $K$ is tight-neighborly, then it is
$\mathbb{F}$-tight for any field $\mathbb{F}$, where the converse holds in
dimension $3$. Thus by Corollary 1.3, if $K$ is tight-neighborly, then it is
$\mathbb{F}$-Golod for any field $\mathbb{F}$.

It is quite hard to show whether or not a given weakly $\mathbb{F}$-Golod is
$\mathbb{F}$-Golod because we need to check the triviality of all (higher)
Massey products. But there are classes of simplicial complexes where
$\mathbb{F}$-Golodness and weak $\mathbb{F}$-Golodness are equivalent [13, 14,
17, 18, 21]. (Berglund and Jöllenbeck [4] claimed that it is always the case
for every simplicial complex, but Katthän [21] proved that this claim is
false.) By the result on weak Golodness and tightness for manifold
triangulations mentioned above [18] together with Theorem 1.2, we can find an
interesting such class.

######  Corollary 1.4.

A triangulation of a closed connected $\mathbb{F}$-orientable manifold is
$\mathbb{F}$-Golod if and only if it is weakly $\mathbb{F}$-Golod.

We briefly explain the outline of the proof of Theorem 1.2. In Section 3, we
will introduce the differential graded algebra (dga, for short)
$\mathcal{C}^{*}(K)$ which is homotopy equivalent to the Koszul complex of
$\mathbb{F}[K]$, whereas the dga $\mathcal{C}^{*}(K)$ is isomorphic with the
cellular cochain complex of the moment-angle complex for $K$. Then we need to
prove that all products and (higher) Massey products in
$H^{*}(\mathcal{C}^{*}(K))$ are trivial whenever $K$ is $\mathbb{F}$-tight.
Every possibly non-trivial product of $\mathcal{C}^{*}(K)$ is given by the map

$K_{I\sqcup J}\to K_{I}\star K_{J},\quad\sigma\mapsto(\sigma\cap
I)\sqcup(\sigma\cap J)$

in cohomology for disjoint non-empty subsets $I,J$ of the vertex set of $K$,
where $K_{1}\star K_{2}$ denotes the join of simplicial complexes $K_{1}$ and
$K_{2}$. The composite of it with the inclusion $K_{I}\star K_{J}\to K\star K$
is null-homotopic because it is homotopic to the inclusion into the first
factor $K\subset K\star K$, which is null-homotopic. If $K$ is
$\mathbb{F}$-tight, then the triviality of the composite $K_{I\cup J}\to
K\star K$ in cohomology implies the triviality of the map $K_{I\sqcup J}\to
K_{I}\star K_{J}$ in cohomology, implying the triviality of all products in
$H^{*}(\mathcal{C}^{*}(K))$. We will generalize this argument to (higher)
Massey products by considering higher homotopies. Specifically, the map

$K_{I_{0}\sqcup\cdots\sqcup I_{q}}\to
K^{\star(q+1)},\quad\sigma\mapsto(\sigma\cap
I_{0})\sqcup\cdots\sqcup(\sigma\cap I_{q})$

for pairwise disjoint non-empty subsets $I_{0},\ldots,I_{q}$ of the vertex set
of $K$ can be deformed into $K^{\star(q+1)}\supset K^{\star
q}\supset\cdots\supset K^{\star 2}\supset K$ in order by a higher homotopy
that we will construct in Section 2. If $K$ is $\mathbb{F}$-tight, then we
will see that the triviality of $(q+1)$-fold Massey products can be deduced by
considering the higher prism operator corresponding to the above higher
homotopy, which will be constructed in Section 2 and applied in Section 4.

### Acknowledgements

The authors were supported by JSPS KAKENHI Grant Numbers JP19K03473 and
JP17K05248.

##  2\. Higher prism operator

In this section, we define higher prism operators, by which we will control
(higher) Massey products. Hereafter, algebras and modules will be considered
over a field $\mathbb{F}$. We set notation that we are going to use throughout
the paper. Let $K$ and $L$ be simplicial complexes. We define the join of $K$
and $L$ by

$K\star L=\\{\sigma\sqcup\tau\mid\sigma\in K,\,\tau\in L\\}.$

Let $K^{\star n}$ denote the join of $n$ copies of $K$. Let $C^{*}(K)$ denote
the cochain complex of a simplicial complex $K$. Then for $a\in C^{p}(K)$ and
$b\in C^{q}(L)$, we can define $a\star b\in C^{p+q+1}(K\star L)$ by $(a\star
b)(\sigma\sqcup\tau)=a(\sigma)b(\tau)$ for $\sigma\in K$ and $\tau\in L$, and
so we get a homomorphism

$\star\colon C^{p}(K)\otimes C^{q}(L)\to C^{p+q+1}(K\star L),\quad a\otimes
b\mapsto a\star b.$

Let $V(K)$ denote the vertex set of a simplicial complex $K$. For
$\emptyset\neq I\subset V(K)$, let

$j_{I}\colon K_{I}\to K$

denote the inclusion. For $\emptyset\neq I,J\subset V(K)$ with $I\cap
J=\emptyset$, we define an injective simplicial map

$\iota_{I,J}\colon K_{I\cup J}\to K_{I}\star
K_{J},\quad\sigma\mapsto(\sigma\cap I)\sqcup(\sigma\cap J).$

To define a prism operator, we need shuffles. Recall that a $(p,q)$-shuffle is
a sequence $(s_{0},s_{1},\ldots,s_{q+1})$ satisfying $0=s_{0}\leq
s_{1}\leq\cdots\leq s_{q}\leq s_{q+1}=p$. Let $\mathcal{S}(p,q)$ denote the
set of all $(p,q)$-shuffles. It is useful to think of a $(p,q)$-shuffle
$(s_{0},s_{1},\ldots,s_{q+1})$ as the following lattice path in
$\\{0,1,\ldots,p\\}\times\\{0,1,\ldots,q\\}$.

$(0,0)=(s_{0},0)$$(s_{0}+1,0)$$(s_{1},0)$$(s_{1},1)$$(s_{1}+1,1)$$(s_{2},1)$$(s_{2},2)$$(s_{q},q-1)$$(s_{q},q)$$(s_{q}+1,q)$$(s_{q+1},q)=(p,q)$
Figure 1.

For example, the latice path corresponding to the $(6,3)$-shuffle
$(0,2,3,3,6)$ is given by the following figure.

$(0,0)$$(1,0)$$(2,0)$$(2,1)$$(3,1)$$(3,2)$$(3,3)$$(4,3)$$(5,3)$$(6,3)$ Figure
2.

Let $s=(s_{0},\ldots,s_{q+1})\in\mathcal{S}(p,q)$. We define

$\displaystyle s(\rightarrow)$ $\displaystyle=\\{0<i<p+q\mid
s_{k}+k<i<s_{k+1}+k\text{ for some }0\leq k\leq q\\}\cup A\cup B$
$\displaystyle s(\uparrow)$ $\displaystyle=\\{0\leq i<p+q\mid i=s_{k}+k\text{
and }s_{k}=s_{k+1}\text{ for some }0\leq k\leq q\\}\cup C$ $\displaystyle
s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})$ $\displaystyle=\\{0<i<p+q\mid
i=s_{k}+k-1\text{ and }s_{k-1}<s_{k}\text{ for some }0\leq k\leq q+1\\}$
$\displaystyle s(\Rsh)$ $\displaystyle=\\{0<i<p+q\mid i=s_{k}+k\text{ and
}s_{k}<s_{k+1}\text{ for some }0\leq k\leq q\\}$

where we set

$A=\begin{cases}\\{0\\}&s_{0}<s_{1}\\\ \emptyset&s_{0}=s_{1}\end{cases}\quad
B=\begin{cases}\\{p+q\\}&s_{q}<s_{q+1}\\\
\emptyset&s_{q}=s_{q+1}\end{cases}\quad
C=\begin{cases}\emptyset&s_{q}<s_{q+1}\\\ \\{p+q\\}&s_{q}=s_{q+1}.\end{cases}$

Then each $0\leq i\leq p+q$ belongs to one of these sets, and for $0<i<p+q$,
elements of these sets are the $i$-th vertex of the lattice path of Figure 1
depicted as the black vertex in the following figure.

$s(\rightarrow)$$s(\uparrow)$$s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})$$s(\Rsh)$

Moreover, $0$ and $p+q$ belong to either $s(\rightarrow)$ or $s(\uparrow)$ if
they are the black vertices in the following figure.

$(0,0)$$(1,0)$$s(\rightarrow)$$s_{0}<s_{1}$$(0,0)$$(0,1)$$s(\uparrow)$$s_{0}=s_{1}$$(p-1,q)$$(p,q)$$s(\rightarrow)$$s_{q}<s_{q+1}$$(p,q-1)$$(p,q)$$s(\uparrow)$$s_{q}=s_{q+1}$

Then we have

$\\{0,1,\ldots,p+q\\}=s(\rightarrow)\sqcup s(\uparrow)\sqcup
s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})\sqcup s(\Rsh)$

where this partition characterizes the shuffle $s$. For example, for a
$(6,3)$-shuffle $s=(0,2,3,3,6)$, we have

$s(\rightarrow)=\\{0,1,7,8,9\\},\quad s(\uparrow)=\\{5\\},\quad
s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})=\\{2,4\\},\quad s(\Rsh)=\\{3,6\\}.$

where the corresponding lattice path is as in Figure 2. For $i\in
s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})$ with $i=s_{k}+k-1$, we define a
$(p,q)$-shuffle

$\alpha_{i}(s)=(s_{0},\ldots,s_{k-1},s_{k}-1,s_{k+1},\ldots,s_{q+1}).$

For $j\in s(\Rsh)$ with $j=s_{l}+l$, we also define a $(p,q)$-shuffle

$\beta_{j}(s)=(s_{0},\ldots,s_{l-1},s_{l}+1,s_{l+1},\ldots,s_{q+1}).$

Then the lattice path corresponding to $\alpha_{i}(s)$ (resp. $\beta_{j}(s)$)
is obtained by flipping the corner at the $i$-th (resp. $j$-th) vertex in the
lattice path corresponding to the $(p,q)$-shuffle $s$.

######  Lemma 2.1.

The map

$\coprod_{s\in\mathcal{S}(p,q)}s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})\times
s\to\coprod_{s\in\mathcal{S}(p,q)}s(\Rsh)\times
s,\quad(i,s)\mapsto(i,\alpha_{i}(s))$

is bijective.

###### Proof.

By definition, for $i\in s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})$ and $j\in
s(\Rsh)$, we have $i\in\alpha_{i}(s)(\Rsh)$ and
$j\in\beta_{j}(s)(\rotatebox[origin={c}]{-90.0}{$\Lsh$})$. Moreover, we have

$\beta_{i}(\alpha_{i}(s))=s\quad\text{and}\quad\alpha_{j}(\beta_{j}(s))=s.$

Then the map

$\coprod_{s\in\mathcal{S}(p,q)}s(\Rsh)\times
s\to\coprod_{s\in\mathcal{S}(p,q)}s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})\times
s,\quad(i,s)\mapsto(i,\beta_{i}(s))$

is the inverse of the map in the statement. ∎

For $k=0,1,\ldots,q$, we define maps

$\lambda_{k}\colon\mathcal{S}(p-1,q)\to\mathcal{S}(p,q),\quad(s_{0},\ldots,s_{q+1})\mapsto(s_{0},\ldots,s_{k},s_{k+1}+1,\ldots,s_{q+1}+1)$

and

$\nu_{k}\colon\mathcal{S}(p,q-1)\to\mathcal{S}(p,q),\quad(s_{0},\ldots,s_{q})\mapsto(s_{0},\ldots,s_{k-1},s_{k},s_{k},s_{k+1},\ldots,s_{q}).$

Clearly, the maps $\lambda_{k}$ and $\nu_{k}$ are injective. For
$s=(s_{0},\ldots,s_{q})\in\mathcal{S}(p,q)$, we set

$\operatorname{sgn}(s)=(-1)^{s_{1}+\cdots+s_{q}}.$

Then for $s=(s_{0},\ldots,s_{q+1})\in\mathcal{S}(p-1,q)$ and
$t=(t_{0},\ldots,t_{q})\in\mathcal{S}(p,q-1)$, we have

(2.1)
$\operatorname{sgn}(\lambda_{k}(s))=(-1)^{q-k}\operatorname{sgn}(s)\quad\text{and}\quad\operatorname{sgn}(\nu_{k}(t))=(-1)^{t_{k}}\operatorname{sgn}(t).$

We say that a simplicial complex $K$ is ordered if $V(K)$ is equipped with a
total order. Note that every simplicial complex can be an ordered simplicial
complex, where any ordering will do for our purpose. Let $K$ and $L$ be
ordered simplicial complexes. Then we equip $V(K)\times V(L)$ with the product
order such that $(v_{1},w_{1})\leq(v_{2},w_{2})\in V(K)\times V(L)$ if
$v_{1}\leq v_{2}$ and $w_{1}\leq w_{2}$. We define the product of $K$ and $L$,
denoted by $K\otimes L$, to be the simplicial complex with vertex set
$V(K)\times V(L)$, whose simplices are chains
$(v_{0},w_{0})<(v_{1},w_{1})<\cdots<(v_{p},w_{p})$ such that
$\\{v_{0},\ldots,v_{p}\\}\in K$ and $\\{w_{0},\ldots,w_{p}\\}\in L$.

We will always assume that a $q$-simplex $\Delta^{q}$ is ordered in such a way
that its vertex set is $\\{0,1,\ldots,q\\}$ with the usual ordering. Let $K$
be an ordered simplicial complex. For a $(p,q)$-shuffle
$s=(s_{0},\ldots,s_{q+1})$ and a $p$-simplex $\sigma=\\{v_{0}<\cdots<v_{p}\\}$
of $K$, we define a $(p+q)$-simplex of $K\otimes\Delta^{q}$ by

$\sigma\odot
s=[(v_{s_{0}},0),\ldots,(v_{s_{1}},0),(v_{s_{1}},1),\ldots,(v_{s_{2}},1),\ldots,(v_{s_{q}},q),\ldots,(v_{s_{q+1}},q)]$

which is quite similar to thinking of shuffles as lattice paths of Figure 1.
Let $L$ be a simplicial complex. Now, for a simplicial map $H\colon
K\otimes\Delta^{q}\to L$, we define a higher prism operator $P_{H}\colon
C_{*}(K)\to C_{*+q}(L)$ by

$P_{H}(\sigma)=\sum_{s\in\mathcal{S}(p,q)}\operatorname{sgn}(s)H_{*}(\sigma\odot
s).$

Observe that for $q=0$, $P_{H}=H_{*}$ and for $q=1$, $P_{H}$ is the usual
prism operator. We prove a coherent property among higher prism operators.

######  Lemma 2.2.

For a simplicial map $H\colon K\otimes\Delta^{q}\to L$, there is an identity

$\partial
P_{H}+(-1)^{q-1}P_{H}\partial=\sum_{i=0}^{q}(-1)^{i}P_{H\circ(1\otimes
d^{i})}$

where $d^{i}\colon\Delta^{q-1}\to\Delta^{q}$ denotes the $i$-th coface
operator.

###### Proof.

Let $s=(s_{0},\ldots,s_{q+1})$ be a $(p,q)$-shuffle. Then

$\displaystyle\partial P_{H}(\sigma)$
$\displaystyle=\sum_{j=0}^{p+q}(-1)^{j}\partial_{j}\left(\sum_{s\in\mathcal{S}(p,q)}\operatorname{sgn}(s)H_{*}(\sigma\odot
s)\right)$
$\displaystyle=\sum_{j=0}^{p+q}(-1)^{j}\sum_{s\in\mathcal{S}(p,q)}\operatorname{sgn}(s)H_{*}(\partial_{j}(\sigma\odot
s))$ $\displaystyle=\sum_{s\in\mathcal{S}(p,q)}\left(\sum_{j\in
s(\rightarrow)}(-1)^{j}\operatorname{sgn}(s)H_{*}(\partial_{j}(\sigma\odot
s))\right.+\sum_{j\in
s(\uparrow)}(-1)^{j}\operatorname{sgn}(s)H_{*}(\partial_{j}(\sigma\odot s))$
$\displaystyle\quad\left.+\sum_{j\in
s(\Rsh)}(-1)^{j}\operatorname{sgn}(s)H_{*}(\partial_{j}(\sigma\odot
s))+\sum_{j\in
s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})}(-1)^{j}\operatorname{sgn}(s)H_{*}(\partial_{j}(\sigma\odot
s))\right).$

Let $j\in s(\rightarrow)$. If $0<j<p+q$, then
$s_{k_{j}}+k_{j}<j<s_{k_{j}+1}+k_{j}$ for some $0\leq k_{j}\leq q$. For
$j=0,p+q$, we put $k_{j}=0,q$, respectively. Then we have

$\displaystyle\partial_{j}(\sigma\odot s)$
$\displaystyle=[(v_{0},0),\dots,(v_{j-k_{j}-1},k_{j}),(v_{j-k_{j}+1},k_{j}),\dots,(v_{p},q)]$
$\displaystyle=(\partial_{j-k_{j}}\sigma)\odot\lambda_{k_{j}}^{-1}(s).$

Let $j\in s(\uparrow)$. If $0<j<p+q$, then $j=s_{k_{j}}+k_{j}$ and
$s_{k_{j}}=s_{k_{j}+1}$ for some $0\leq k_{j}\leq q$. For $j=0,p+q$, we put
$k_{j}=0,q$, respectively. Then we have

$\displaystyle\partial_{j}(\sigma\odot s)$
$\displaystyle=[(v_{0},0),\dots,(v_{s_{k_{j}}},k_{j}-1),(v_{s_{k_{j}+1}},k_{j}+1),\dots,(v_{p},q)]$
$\displaystyle=(1\otimes d^{k_{j}})(\sigma\odot\nu_{k_{j}}^{-1}(s))$

where $\lambda_{k}^{-1}(s)$ and $\nu_{k}^{-1}(s)$ make sense because
$\lambda_{k}$ and $\nu_{k}$ are injective. Thus by (2.1), we get

$\displaystyle\sum_{s\in\mathcal{S}(p,q)}\left(\sum_{j\in
s(\rightarrow)}(-1)^{j}\operatorname{sgn}(s)H_{*}(\partial_{j}(\sigma\odot
s))+\sum_{j\in
s(\uparrow)}(-1)^{j}\operatorname{sgn}(s)H_{*}(\partial_{j}(\sigma\odot
s))\right)$ $\displaystyle=\sum_{s\in\mathcal{S}(p,q)}\left(\sum_{j\in
s(\rightarrow)}(-1)^{j}\operatorname{sgn}(s)H_{*}((\partial_{j-k_{j}}\sigma)\odot\lambda_{k_{j}}^{-1}(s))\right.$
$\displaystyle\quad\left.+\sum_{j\in
s(\uparrow)}(-1)^{j}\operatorname{sgn}(s)H_{*}((1\otimes
d^{k_{j}})(\sigma\odot\nu_{k_{j}}^{-1}(s)))\right)$
$\displaystyle=(-1)^{q}\sum_{i=0}^{p}\sum_{s\in\mathcal{S}(p-1,q)}(-1)^{i}\operatorname{sgn}(s)H_{*}((\partial_{i}\sigma)\odot
s)$
$\displaystyle\quad+\sum_{i=0}^{q}\sum_{s\in\mathcal{S}(p,q-1)}(-1)^{i}\operatorname{sgn}(s)H_{*}\circ(1\otimes
d^{i})(\sigma\odot s)$
$\displaystyle=(-1)^{q}P_{H}\partial(\sigma)+\sum_{i=0}^{q}(-1)^{i}P_{H\circ(1\otimes
d^{i})}(\sigma).$

For $j\in s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})$, we have

$\partial_{j}(\sigma\odot
s)=\partial_{j}(\sigma\odot\alpha_{j}(s))\quad\text{and}\quad\operatorname{sgn}(s)=-\operatorname{sgn}(\alpha_{j}(s)).$

Then by Lemma 2.1, we get

$\displaystyle\sum_{s\in\mathcal{S}(p,q)}\sum_{j\in
s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})}(-1)^{j}\operatorname{sgn}(s)H_{*}(\partial_{j}(\sigma\odot
s))$ $\displaystyle=-\sum_{s\in\mathcal{S}(p,q)}\sum_{j\in
s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})}(-1)^{j}\operatorname{sgn}(\alpha_{j}(s))H_{*}(\partial_{j}(\sigma\odot\alpha_{j}(s)))$
$\displaystyle=-\sum_{s\in\mathcal{S}(p,q)}\sum_{j\in
s(\Rsh)}(-1)^{j}\operatorname{sgn}(s)H_{*}(\partial_{j}(\sigma\odot s))$

implying

$\sum_{s\in\mathcal{S}(p,q)}\left(\sum_{j\in
s(\Rsh)}(-1)^{j}\operatorname{sgn}(s)H_{*}(\partial_{j}(\sigma\odot
s))+\sum_{j\in
s(\rotatebox[origin={c}]{-90.0}{$\Lsh$})}(-1)^{j}\operatorname{sgn}(s)H_{*}(\partial_{j}(\sigma\odot
s))\right)=0.$

Thus we obtain

$\partial
P_{H}(\sigma)=(-1)^{q}P_{H}\partial(\sigma)+\sum_{i=0}^{q}(-1)^{i}P_{H\circ(1\otimes
d^{i})}(\sigma)$

which is the desired identity. ∎

We define a higher homotopy between simplicial complexes that we are going to
use. Let $K$ be an ordered simplicial complex, and let $I$ denote a partition
$V(K)=I_{0}\sqcup I_{1}\sqcup\cdots\sqcup I_{q}$ such that
$I_{0},\ldots,I_{q}$ are non-empty and pairwise disjoint. For
$i=0,1,\ldots,q$, we define a simplicial map $h_{I}^{i}\colon K\to
K^{\star(q+1)}$ by

$h_{I}^{i}(\sigma)=(\sigma\cap I_{0})\sqcup\cdots\sqcup(\sigma\cap
I_{i-1})\sqcup(\sigma\cap(I_{i}\sqcup\cdots\sqcup
I_{q}))\sqcup\underbrace{\emptyset\sqcup\cdots\sqcup\emptyset}_{q-i}$

for a simplex $\sigma$ of $K$. Then we have:

######  Lemma 2.3.

For $i=0,1,\ldots,q-1$, the image of $h_{I}^{i}$ is in $K^{\star q}\subset
K^{\star q}\star K=K^{\star(q+1)}$.

By definition, the map $h_{I}^{q}$ is concerned with the $(q+1)$-fold products
in $\mathcal{C}^{*}(K)$, and for $i=0,1,\ldots,q-1$, we may think of
$h_{I}^{i}$ as a deformation of $h_{I}^{q}$ into the first $(i+1)$-th join
$K^{\star(i+1)}\subset K^{\star(i+1)}\star K^{\star(q-i)}=K^{\star(q+1)}$. We
define a higher homotopy among these deformations.

######  Lemma 2.4.

The map

$V(K)\times\\{0,1,\ldots,q\\}\to\underbrace{V(K)\sqcup\cdots\sqcup
V(K)}_{q+1},\quad(v,i)\mapsto h_{I}^{i}(v).$

defines a simplicial map $H_{I}\colon K\otimes\Delta^{q}\to K^{\star(q+1)}$.

###### Proof.

Let $\sigma=\\{(v_{0},w_{0}),\ldots,(v_{p},w_{p})\\}$ be a simplex of
$K\otimes\Delta^{q}$. Then $\\{v_{0},\ldots,v_{p}\\}$ is a simplex of $K$,
hence its subsets are simplices of $K$. By definition, we have

$H_{I}(\sigma)=\sigma_{0}\sqcup\cdots\sqcup\sigma_{q}.$

where each $\sigma_{i}$ is a subset of $\\{v_{0},\ldots,v_{p}\\}$. Then each
$\sigma_{i}$ is a simplex of $K$, and so $H_{I}(\sigma)$ is a simplex of
$K^{\star(q+1)}$, completing the proof. ∎

We denote the higher prism operator $P_{H_{I}}$ by $P(I)$. As mentioned above,
$P(I)$ is the identity map of $K$ for $q=0$. For $i=0,1,\ldots q-1$, let
$I(i)$ be the partition $V(K)=J_{0}\sqcup\cdots\sqcup J_{q-1}$ such that

$J_{k}=\begin{cases}I_{k}&0\leq k\leq i-1\\\ I_{i}\sqcup I_{i+1}&k=i\\\
I_{k+1}&i+1\leq k\leq q-1.\end{cases}$

For $i=0,1,\ldots,q-1$, we set

$\mu_{i}=(j_{I_{0}\sqcup\cdots\sqcup I_{i}}\star j_{I_{i+1}\sqcup\cdots\sqcup
I_{q}})\circ\iota_{I_{0}\sqcup\cdots\sqcup I_{i},I_{i+1}\sqcup\cdots\sqcup
I_{q}}\colon K\to K\star K.$

By definition, the partition $I$ defines $\mu_{i}$, and we write $\mu_{i}^{I}$
when we specify the defining partition $I$. We specialize Lemma 2.2 to the
higher prism operators $P(I)$.

######  Lemma 2.5.

There is an identity

$\partial P(I)+(-1)^{q-1}P(I)\partial\\\ =\sum_{i=0}^{q-1}(-1)^{i}(1_{K^{\star
i}}\star\mu_{i}\star 1_{K^{\star(q-i-1)}})_{*}\circ
P(I(i))+(-1)^{q}P_{H_{I}\circ(1_{K}\otimes d^{q})}.$

###### Proof.

Let $\sigma$ be a simplex of $K$. For $0\leq j<i<q$, we have

$\displaystyle(1_{K^{\star i}}\star\mu_{i}\star 1_{K^{\star(q-i-1)}})\circ
h_{I(i)}^{j}(\sigma)$ $\displaystyle=(1_{K^{\star i}}\star\mu_{i}\star
1_{K^{\star(q-i-1)}})((\sigma\cap I_{0})\sqcup\cdots\sqcup(\sigma\cap
I_{j-1})$ $\displaystyle\quad\sqcup(\sigma\cap(I_{j}\sqcup\cdots\sqcup
I_{q}))\sqcup\underbrace{\emptyset\sqcup\cdots\sqcup\emptyset}_{q-j-1})$
$\displaystyle=(\sigma\cap I_{0})\sqcup\cdots\sqcup(\sigma\cap
I_{j-1})\sqcup(\sigma\cap(I_{j}\sqcup\cdots\sqcup
I_{q}))\sqcup\underbrace{\emptyset\sqcup\cdots\sqcup\emptyset}_{q-j}$
$\displaystyle=h_{I}^{j}(\sigma).$

For $0\leq i=j<q$, we have

$\displaystyle(1_{K^{\star i}}\star\mu_{i}\star 1_{K^{\star(q-i-1)}})\circ
h_{I(i)}^{i}(\sigma)$ $\displaystyle=(1_{K^{\star i}}\star\mu_{i}\star
1_{K^{\star(q-i-1)}})((\sigma\cap I_{0})\sqcup\cdots\sqcup(\sigma\cap
I_{i-1})$ $\displaystyle\quad\sqcup(\sigma\cap(I_{i}\sqcup\cdots\sqcup
I_{q}))\sqcup\underbrace{\emptyset\sqcup\cdots\sqcup\emptyset}_{q-i-1})$
$\displaystyle=(\sigma\cap I_{0})\sqcup\cdots\sqcup(\sigma\cap
I_{i})\sqcup(\sigma\cap(I_{i+1}\sqcup\cdots\sqcup
I_{q}))\sqcup\underbrace{\emptyset\sqcup\cdots\sqcup\emptyset}_{q-i-1}$
$\displaystyle=h_{I}^{i+1}(\sigma)=h_{I}^{j+1}(\sigma),$

and for $0\leq i<j<q$, we have

$\displaystyle(1_{K^{\star i}}\star\mu_{i}\star 1_{K^{\star(q-i-1)}})\circ
h_{I(i)}^{j}(\sigma)$ $\displaystyle=(1_{K^{\star i}}\star\mu_{i}\star
1_{K^{\star(q-i-1)}})((\sigma\cap I_{0})\sqcup\cdots\sqcup(\sigma\cap
I_{i-1})\sqcup(\sigma\cap(I_{i}\sqcup I_{i+1}))$
$\displaystyle\quad\sqcup(\sigma\cap I_{i+2})\sqcup\cdots\sqcup(\sigma\cap
I_{j})\sqcup(\sigma\cap(I_{j+1}\sqcup\cdots\sqcup
I_{q}))\sqcup\underbrace{\emptyset\sqcup\cdots\sqcup\emptyset}_{q-j-1})$
$\displaystyle=(\sigma\cap I_{1})\sqcup\cdots\sqcup(\sigma\cap
I_{j})\sqcup(\sigma\cap(I_{j+1}\sqcup\cdots\sqcup
I_{q}))\sqcup\underbrace{\emptyset\sqcup\cdots\sqcup\emptyset}_{q-j-1}$
$\displaystyle=h_{I}^{j+1}(\sigma).$

Then the statement follows from Lemma 2.2. ∎

######  Lemma 2.6.

Let $a_{i}\in C^{*}(K)$ for $i=0,1,\ldots,q$. Then we have

$(P(I)^{*}\delta+(-1)^{q-1}\delta P(I)^{*})(a_{0}\star\cdots\star a_{q})\\\
=\sum_{i=0}^{q-1}(-1)^{i}P(I(i))^{*}(1_{K^{\star i}}\star\mu_{i}\star
1_{K^{\star(q-i-1)}})^{*}(a_{0}\star\cdots\star a_{q}).$

###### Proof.

By Lemma 2.3 and the fact that the inclusion $K^{\star q}\to K^{\star q}\star
K=K^{\star(q+1)}$ is trivial in cohomology, we have $(H_{I}\circ(1\otimes
d^{q}))^{*}(a_{0}\star\cdots\star a_{q})=0$, implying $P_{H_{I}\circ(1\otimes
d^{q})}^{*}(a_{0}\star\cdots\star a_{q})=0$. Then by Lemma 2.5, the proof is
finished. ∎

##  3\. The dga $\mathcal{C}^{*}(K)$

Let $K$ be an (ordered) simplicial complex. In this section, we introduce the
dga $\mathcal{C}^{*}(K)$ which is homotopy equivalent to the Koszul complex of
the Stanley-Reisner ring $\mathbb{F}[K]$, and show its properties concerning
(higher) Massey products.

We define

$\mathcal{C}^{p}(K)=\bigoplus_{\emptyset\neq I\subset
V(K)}\widetilde{C}^{p-|I|-1}(K_{I})$

for $p>0$ and $\mathcal{C}^{0}(K)=\mathbb{F}$, where $\widetilde{C}^{*}(K)$
denotes the reduced cochain complex of $K$. The differential of
$\mathcal{C}^{*}(K)$ is defined by the coboundary maps $\delta$ of
$\widetilde{C}^{*}(K_{I})$, and so we have

$H^{p}(\mathcal{C}^{*}(K))=\bigoplus_{\emptyset\neq I\subset
V(K)}\widetilde{H}^{p-|I|-1}(K_{I})$

for $p>0$ and $H^{0}(\mathcal{C}^{*}(K))=\mathbb{F}$. The product of
$\mathcal{C}^{*}(K)$ is defined by setting

$\widetilde{C}^{p-|I|-1}(K_{I})\otimes\widetilde{C}^{q-|J|-1}(K_{J})\to\widetilde{C}^{p+q-|I\cup
J|-1}(K_{I\cup J})$

to be trivial for $I\cap J\neq\emptyset$ and to be the composite

$\widetilde{C}^{p-|I|-1}(K_{I})\otimes\widetilde{C}^{q-|J|-1}(K_{J})\xrightarrow{\star}\widetilde{C}^{q-|I\cup
J|-1}(K_{I}\star K_{J})\xrightarrow{\iota_{I,J}^{*}}\widetilde{C}^{p+q-|I\cup
J|-1}(K_{I\cup J})$

for $I\cap J=\emptyset$.

Recall that the Stanley-Reisner ring of a simplicial complex $K$ with vertex
set $\\{1,2,\ldots,n\\}$ is defined by

$\mathbb{F}[K]=\mathbb{F}[v_{1},\ldots,v_{n}]/(v_{i_{1}}\cdots
v_{i_{k}}\mid\\{i_{1},\ldots,i_{k}\\}\not\in K)$

where we set $|v_{i}|=2$. Then the Koszul complex of $\mathbb{F}[K]$ (with
respect to $v_{1},\ldots,v_{n}$) is a dga defined by

$\mathcal{K}(K)=\mathbb{F}[K]\otimes\Lambda(x_{1},\ldots,x_{n}),\quad
dv_{i}=0,\quad dx_{i}=v_{i}$

where $|x_{i}|=1$. Let

$\mathcal{R}(K)=\mathcal{K}(K)/(v_{i}^{2},\,v_{i}x_{i}\mid i=1,2,\ldots,n).$

Then $\mathcal{R}(K)$ is a dga. Baskakov, Buchstaber and Panov [6] studied the
cellular cochain complex of a space called the moment-angle complex for $K$,
which is actually $\mathcal{R}(K)$, and proved that the projection
$\mathcal{K}(K)\to\mathcal{R}(K)$ is a homotopy equivalence. On the other
hand, we can define a map

$\phi\colon\mathcal{C}^{*}(K)\to\mathcal{R}(K)$

by $\phi(\sigma^{*})=v_{i_{1}}\cdots v_{i_{k}}x_{j_{1}}\cdots x_{j_{l}}$ for
$\sigma^{*}\in\widetilde{C}^{*}(K_{I})\subset\mathcal{C}^{*}(K)$, where
$\sigma^{*}$ is the Kronecker dual of a simplex
$\sigma=\\{i_{1}<\cdots<i_{k}\\}\in K_{I}$ and
$I-\sigma=\\{j_{1}<\cdots<j_{l}\\}$. It is easy to see that the map $\phi$ is
an isomorphism of dga’s, and so we get:

######  Proposition 3.1.

There is a homotopy equivalence

$\mathcal{C}^{*}(K)\simeq\mathcal{K}(K).$

We recall the definition of (higher) Massey products. We refer to [25, 31] for
details, where we follow the sign convention of May [31]. Let $A$ be a dga. We
set

$\overline{a}=(-1)^{|a|+1}a$

for $a\in A$, where $|a|$ denotes the degree of $a$. Suppose we are given
cohomology classes $\alpha_{0},\ldots,\alpha_{q}\in H^{*}(A)$ for $q\geq 1$.
We say that a set $\\{a_{i,j}\in A\mid 0\leq i\leq j\leq
q,\,(i,j)\neq(0,q)\\}$ is a defining system for $\alpha_{0},\ldots,\alpha_{q}$
if $a_{i,i}$ represents $\alpha_{i}$ for $i=0,1,\ldots,q$ and the identity

$da_{i,j}=\sum_{k=i}^{j-1}\overline{a}_{i,k}a_{k+1,j}$

holds for $i<j$. If there is a defining system for
$\alpha_{0},\ldots,\alpha_{q}$, then we say that the Massey product
$\langle\alpha_{0},\ldots,\alpha_{q}\rangle$ is defined, which is the set of
all cohomology classes represented by a cocycle
$\sum_{i=0}^{q-1}\overline{a}_{0,i}a_{i+1,q}$, where $\\{a_{i,j}\\}$ ranges
over all defining systems for $\alpha_{0},\ldots,\alpha_{q}$. We say that the
Massey product of $\alpha_{0},\ldots,\alpha_{q}$ is trivial if

$0\in\langle\alpha_{0},\ldots,\alpha_{q}\rangle.$

Observe that for $q=1$, the Massey product of $\alpha_{0},\alpha_{1}$ is
always defined, and
$\langle\alpha_{0},\alpha_{1}\rangle=\\{\alpha_{0}\alpha_{1}\\}$. Then the
triviality of all Massey products implies the triviality of all products, and
so we will not mention the triviality of products when we consider the
triviality of all Massey products.

Observe that Massey products are preserved by a homotopy equivalence. Then by
Proposition 3.1, we get:

######  Proposition 3.2.

A simplicial complex $K$ is $\mathbb{F}$-Golod if and only if all Massey
products in $H^{*}(\mathcal{C}^{*}(K))$ are trivial.

Thus we will consider Massey products in $H^{*}(\mathcal{C}^{*}(K))$ for
Golodness of a simplicial complex $K$, instead of $H^{*}(\mathcal{K}(K))$.

######  Lemma 3.3.

Let $\alpha_{i}\in\widetilde{H}^{*}(K_{I_{i}})\subset
H^{*}(\mathcal{C}^{*}(K))$ for $i=0,1,\ldots,q$ with $q\geq 1$. If $I_{i}\cap
I_{j}\neq\emptyset$ for some $i<j$ and the Massey product of
$\alpha_{0},\ldots,\alpha_{q}$ in $H^{*}(\mathcal{C}^{*}(K))$ is defined, then
$\langle\alpha_{0},\ldots,\alpha_{q}\rangle$ is trivial.

###### Proof.

Take any defining system $\\{a_{k,l}\\}$ for $\alpha_{0},\ldots,\alpha_{q}$.
For $0\leq k\leq l\leq q$ with $(k,l)\neq(0,q)$, we define

$b_{k,l}=\begin{cases}0&k\leq i<j\leq l,\\\
a_{k,l}&\text{otherwise}.\end{cases}$

Since $b_{k,k}=a_{k,k}$ for $k=0,1,\ldots,q$, each $b_{k,k}$ represents
$\alpha_{k}$. If $(k,l)$ does not satisfy $k\leq i<j\leq l$, then we have

$\delta b_{k,l}=\delta
a_{k,l}=\sum_{p=k}^{l-1}\overline{a}_{k,p}a_{p+1,l}=\sum_{p=k}^{l-1}\overline{b}_{k,p}b_{p+1,l}.$

Suppose $k\leq i<j\leq l$. If $p<i$, then $p+1\leq i<j\leq l$, implying
$b_{p+1,l}=0$. If $p\geq j$, then $k\leq i<j\leq p$, implying $b_{k,p}=0$.
Moreover, if $i\leq p<j$, then $\overline{b}_{k,p}b_{p+1,l}=0$ because
$\overline{b}_{k,p}\in\widetilde{C}^{*}(K_{I_{k}\sqcup\cdots\sqcup I_{p}})$
and $b_{p+1,l}\in\widetilde{C}^{*}(K_{I_{p+1}\sqcup\cdots\sqcup I_{l}})$ such
that $(I_{k}\sqcup\cdots\sqcup I_{p})\cap(I_{p+1}\sqcup\cdots\sqcup
I_{l})\neq\emptyset$. Hence we get

$\delta b_{k,l}=0=\sum_{p=k}^{l-1}\overline{b}_{k,p}b_{p+1,l}$

and so $\\{b_{k,j}\mid 0\leq k\leq l\leq q,\,(k,l)\neq(0,q)\\}$ is a defining
system for $\alpha_{0},\ldots,\alpha_{q}$ such that

$\sum_{p=0}^{q-1}\overline{b}_{0,p}b_{p+1,q}=0.$

Thus the Massey product of $\alpha_{0},\ldots,\alpha_{q}$ is trivial,
completing the proof. ∎

Let $\alpha_{i}\in\widetilde{H}^{*}(K_{I_{i}})$ for $i=0,1,\ldots,q$ with
$q\geq 1$. If the Massey product of $\alpha_{0},\ldots,\alpha_{q}$ in
$H^{*}(\mathcal{C}^{*}(K))$ is defined, then we say that its support is
$I_{0}\cup\cdots\cup I_{q}$.

######  Lemma 3.4.

Let $\mathcal{C}$ be a class of simplicial complexes which is closed under
taking full subcomplexes. If for each $K\in\mathcal{C}$, all Massey products
in $H^{*}(\mathcal{C}^{*}(K))$ with support $V(K)$ are trivial, then all
simplicial complexes in $\mathcal{C}$ are $\mathbb{F}$-Golod.

###### Proof.

By definition, $\mathcal{C}^{*}(K)$ is natural with respect to subcomplexes of
$K$, that is, an inclusion $i\colon L\to K$ of a subcomplex induces a dga
homomorphism $i^{*}\colon\mathcal{C}^{*}(K)\to\mathcal{C}^{*}(L)$. Then for
any given Massey product $\langle\alpha_{0},\ldots,\alpha_{q}\rangle$ in
$H^{*}(\mathcal{C}^{*}(K))$ with support $\emptyset\neq I\subset V(K)$, we
have

$j_{I}^{*}(\langle\alpha_{0},\ldots,\alpha_{q}\rangle)\subset\langle
j_{I}^{*}(\alpha_{0}),\ldots,j_{I}^{*}(\alpha_{q})\rangle\subset
H^{*}(\mathcal{C}^{*}(K_{I})).$

Suppose that $K$ belongs to a class $\mathcal{C}$ in the statement. Since the
Massey product $\langle
j_{I}^{*}(\alpha_{0}),\ldots,j_{I}^{*}(\alpha_{q})\rangle$ in
$H^{*}(\mathcal{C}^{*}(K_{I}))$ has support $I=V(K_{I})$, it is trivial by
assumption. Observe that $\mathcal{C}^{*}(K_{I})$ is a retract of
$\mathcal{C}^{*}(K)$. Then we obtain that the Massey product
$\langle\alpha_{0},\ldots,\alpha_{q}\rangle$ itself is trivial, completing the
proof. ∎

######  Lemma 3.5.

Let $\alpha_{i}\in\widetilde{H}^{*}(K_{I_{i}})$ for $i=0,1,\ldots,q$ with
$q\geq 1$ such that $V(K)=I_{0}\sqcup\cdots\sqcup I_{q}$. Suppose that there
are $a_{i,j}\in\widetilde{C}^{*}(K)$ for $0\leq i\leq j\leq q$ such that
$j_{I_{i}}^{*}(a_{i,i})$ represents $\alpha_{i}$ for $i=0,1,\ldots,q$ and

(3.1) $\delta a_{i,j}=\sum_{k=i}^{j-1}\mu_{k}^{*}(\overline{a}_{i,k}\star
a_{k+1,j})$

for $0\leq i<j\leq q$. Then the Massey product of
$\alpha_{0},\ldots,\alpha_{q}$ is defined and trivial.

###### Proof.

For $0\leq i\leq j\leq q$, let $b_{i,j}=j^{*}_{I_{i}\sqcup\cdots\sqcup
I_{j}}(a_{i,j})$. Then for $0\leq i\leq k<j\leq q$,

$\displaystyle j_{I_{i}\sqcup\cdots\sqcup
I_{j}}^{*}\circ\mu_{k}^{*}(\overline{a}_{i,k}\star a_{k+1,j})$
$\displaystyle=((j_{I_{0}\sqcup\cdots\sqcup I_{k}}\star
j_{I_{k+1}\sqcup\cdots\sqcup I_{q}})\circ\iota_{I_{0}\sqcup\cdots\sqcup
I_{k},I_{k+1}\sqcup\cdots\sqcup I_{q}}\circ j_{I_{i}\sqcup\cdots\sqcup
I_{j}})^{*}(\overline{a}_{i,k}\star a_{k+1,j})$
$\displaystyle=((j_{I_{i}\sqcup\cdots\sqcup I_{k}}\star
j_{I_{k+1}\sqcup\cdots\sqcup I_{j}})\circ\iota_{I_{i}\sqcup\cdots\sqcup
I_{k},I_{k+1}\sqcup\cdots\sqcup I_{j}})^{*}(\overline{a}_{i,k}\star
a_{k+1,j})$ $\displaystyle=(\iota_{I_{i}\sqcup\cdots\sqcup
I_{k},I_{k+1}\sqcup\cdots\sqcup I_{j}})^{*}(\overline{b}_{i,k}\star
b_{k+1,j})$ $\displaystyle=\overline{b}_{i,k}b_{k+1,j}$

where the product in the last line is taken in $\mathcal{C}^{*}(K)$. Then for
$0\leq i<j\leq q$, we get an identity in $\mathcal{C}^{*}(K)$

(3.2) $\delta b_{i,j}=\sum_{k=i}^{j-1}\overline{b}_{i,k}b_{k+1,j}.$

By assumption, $b_{i,i}$ represents $\alpha_{i}$ for each $i$, and so by
(3.2), $\\{b_{i,j}\mid 0\leq i\leq j\leq q,\,(i,j)\neq(0,q)\\}$ is a defining
system for $\alpha_{0},\ldots,\alpha_{q}$ in $H^{*}(\mathcal{C}^{*}(K))$. Then
the Massey product $\langle\alpha_{0},\ldots,\alpha_{q}\rangle$ in
$H^{*}(\mathcal{C}^{*}(K))$ is defined, so the RHS of (3.2) for $(i,j)=(0,q)$
represents an element of $\langle\alpha_{0},\ldots,\alpha_{q}\rangle$. Thus
(3.2) for $(i,j)=(0,q)$ shows that
$\langle\alpha_{0},\ldots,\alpha_{q}\rangle$ is trivial, completing the proof.
∎

##  4\. Proof of Theorem 1.2

In this section, we prove Theorem 1.2. Let $K$ be an (ordered) simplicial
complex. To prove Theorem 1.2, we need to show that all Massey products in
$H^{*}(\mathcal{C}^{*}(K))$ are trivial whenever $K$ is $\mathbb{F}$-tight. We
consider the special case that the support of Massey products are $V(K)$. Take
any $\alpha_{i}\in\widetilde{H}^{*}(K_{I_{i}})\subset
H^{*}(\mathcal{C}^{*}(K))$ with $\emptyset\neq I_{i}\subset V(K)$ for
$i=0,1,\ldots,q$ such that $V(K)=I_{0}\cup\cdots\cup I_{q}$. We shall consider
the Massey product of $\alpha_{0},\ldots,\alpha_{q}$. By Lemma 3.3, if
$I_{0},\ldots,I_{q}$ are not pairwise disjoint, then the Massey product of
$\alpha_{0},\ldots,\alpha_{q}$ is trivial whenever it is defined. So we may
suppose that $I_{0},\ldots,I_{q}$ are pairwise disjoint, and by Lemma 3.5, it
is sufficient to construct $a_{ij}\in\widetilde{C}^{*}(K)$ for $0\leq i\leq
j\leq q$ satisfying the condition of Lemma 3.5.

######  Lemma 4.1.

If $K$ is $\mathbb{F}$-tight, then for each $i=0,1,\ldots,q$, there is a
cocycle $a_{i,i}\in\widetilde{C}^{*}(K)$ such that $j_{I_{i}}^{*}(a_{i,i})$
represents $\alpha_{i}$.

###### Proof.

Since $K$ is $\mathbb{F}$-tight, the map $j_{I_{i}}^{*}\colon H^{*}(K)\to
H^{*}(K_{I_{i}})$ is surjective for each $i$. Then the statement follows. ∎

We prove the existence of $a_{i,j}$ for $0\leq i<j\leq q$ by induction on
$j-i$.

######  Lemma 4.2.

If there are cocycles $a_{i,i}\in\widetilde{C}^{*}(K)$ such that each
$j_{I_{i}}^{*}(a_{i,i})$ represents $\alpha_{i}$ for $i=0,1,\ldots,q$, then
for $i=0,1,\ldots,q-1$, there are $a_{i,i+1}\in\widetilde{C}^{*}(K)$
satisfying

$\delta a_{i,i+1}=\mu_{i}^{*}(\overline{a}_{i,i}\star a_{i+1,i+1}).$

###### Proof.

Let $J$ denote the partition $V(K)=J_{0}\sqcup J_{1}$ such that
$J_{0}=I_{0}\sqcup\cdots\sqcup I_{i}$ and $J_{1}=I_{i+1}\sqcup\cdots\sqcup
I_{q}$. By Lemma 2.6, we have

$\displaystyle\mu_{i}^{*}(\overline{a}_{i,i}\star a_{i+1,i+1})$
$\displaystyle=P(J(0))^{*}(\mu_{i}^{*}(\overline{a}_{i,i}\star a_{i+1,i+1}))$
$\displaystyle=(P(J)^{*}\delta+\delta P(J)^{*})(\overline{a}_{i,i}\star
a_{i+1,i+1})$ $\displaystyle=\delta P(J)^{*}(\overline{a}_{i,i}\star
a_{i+1,i+1})$

where $P(J(0))$ is the identity map of $K$ and
$P(J)^{*}\delta(\overline{a}_{i,i}\star a_{i+1,i+1})=0$ as $a_{i,i}$ and
$a_{i+1,i+1}$ are cocycles. Then $a_{i,i+1}=P(J)^{*}(\overline{a}_{i,i}\star
a_{i+1,i+1})$ is the desired element. ∎

Now we assume that for $j-i<n$, we have the desired $a_{i,j}$, and prove the
$j-i=n$ case. Let $s=(s_{0},\ldots,s_{k+1})\in\mathcal{S}(j-i,k)$. We define
the partition $I^{s}$ by $V(K)=J_{0}\sqcup\cdots\sqcup J_{k}$ such that

$J_{p}=\begin{cases}I_{i_{0}}\sqcup\cdots\sqcup I_{i_{1}}&p=0\\\
I_{i_{p}+1}\sqcup\cdots\sqcup I_{i_{p+1}}&p=1,\ldots,k\end{cases}$

where $i_{p}=i+s_{p}$ for $p=0,1,\ldots,k+1$. We set

$\epsilon(s)=\begin{cases}(|a_{i_{1}+1,i_{2}}|+1)+(|a_{i_{3}+1,i_{4}}|+1)+\cdots+(|a_{i_{k-1}+1,i_{k}}|+1)&(k\text{
is even})\\\
(|a_{i_{0},i_{1}}|+1)+(|a_{i_{2}+1,i_{3}}|+1)+\cdots+(|a_{i_{k-1}+1,i_{k}}|+1)&(k\text{
is odd})\end{cases}$

for $k\geq 1$, and

$a^{s}=a_{i_{0},i_{1}}\star a_{i_{1}+1,i_{2}}\star\cdots\star
a_{i_{k}+1,i_{k+1}}.$

We also set

$\epsilon(k)=\begin{cases}0&(k\equiv 1,2\mod 4)\\\ 1&(k\equiv 0,3\mod
4).\end{cases}$

Let $\widehat{\mathcal{S}}(p,q)$ denote the subset of $\mathcal{S}(p,q)$
consisting of $(p,q)$-shuffles $(s_{0},\ldots,s_{q+1})$ satisfying
$0=s_{0}\leq s_{1}<s_{2}<\cdots<s_{q+1}=p$. By definition,
$\widehat{\mathcal{S}}(p,q)=\emptyset$ for $q>p$.

######  Lemma 4.3.

If $a_{i,j}$ exists for $j-i<n$, then for $j-i=n$ and $k=1,2,\ldots,j-i$,

(4.1) $\sum_{p=i}^{j-1}\mu_{p}^{*}(\overline{a}_{i,p}\star
a_{p+1,j})\equiv\sum_{s\in\widehat{\mathcal{S}}(j-i,k)}(-1)^{\epsilon(s)+\epsilon(k)}P(I^{s})^{*}\delta(a^{s})\mod\mathrm{Im}\,\delta.$

###### Proof.

We induct on $k$. By Lemma 2.6, we have

$\displaystyle\mu_{p}^{*}(\overline{a}_{i,p}\star a_{p+1,j})$
$\displaystyle=P(I^{s}(0))^{*}(\mu_{p}^{*}(\overline{a}_{i,p}\star
a_{p+1,j}))$ $\displaystyle=(P(I^{s})^{*}\delta+\delta
P(I^{s})^{*})(\overline{a}_{i,p}\star a_{p+1,j})$
$\displaystyle\equiv(-1)^{\epsilon(s)+\epsilon(1)}P(I^{s})^{*}\delta(a^{s})$
$\displaystyle\mod\mathrm{Im}\,\delta$

where $s=(0,p-i,j-i)\in\widehat{\mathcal{S}}(j-i,1)$ and $P(I^{s}(0))$ is the
identity map of $K$. Then since
$\widehat{\mathcal{S}}(j-i,1)=\\{(0,p-i,j-i)\in\mathcal{S}(j-i,1)\mid i\leq
p<j\\}$, the $k=1$ case is proved. We suppose that (4.1) holds for $k$, and
consider the $k+1$ case. Let
$s=(s_{0},\ldots,s_{k+1})\in\widehat{\mathcal{S}}(j-i,k)$. We can define

$b_{0}=a_{i_{0},i_{1}}\star\overline{a}_{i_{1}+1,i_{2}}\quad\text{and}\quad
b_{p}=a_{i_{p}+1,i_{p+1}}\star\overline{a}_{i_{p+1}+1,i_{p+2}}$

for $p$ even. Clearly, we have

$\overline{b}_{0}=\overline{a}_{i_{0},i_{1}}\star
a_{i_{1}+1,i_{2}}\quad\text{and}\quad\overline{b}_{p}=\overline{a}_{i_{p}+1,i_{p+1}}\star
a_{i_{p+1}+1,i_{p+2}}$

for $p$ even. Then for $k$ even, we have

$\displaystyle(-1)^{\epsilon(s)}\delta(a^{s})$
$\displaystyle=\delta(b_{0}\star b_{2}\star\cdots\star b_{k-2}\star
a_{i_{k}+1,i_{k+1}})$
$\displaystyle=\sum_{p=0}^{\frac{k-2}{2}}\overline{b}_{0}\star\cdots\star\overline{b}_{2p-2}\star(\delta
b_{2p})\star b_{2p+2}\star\cdots\star b_{k-2}\star a_{i_{k}+1,i_{k+1}}$
$\displaystyle\quad+\overline{b}_{0}\star\cdots\star\overline{b}_{k-2}\star(\delta
a_{i_{k}+1,i_{k+1}}).$

For $p=0,1,\ldots,k$, let $\mathcal{S}_{p}(s)$ denote the set of all
$(j-i,k+1)$-shuffles of the form
$(s_{0},\ldots,s_{p},t,s_{p+1},\ldots,s_{k+1})$ satisfying $s_{p}<t<s_{p+1}$.
Then we have

$\displaystyle(\delta b_{0})\star b_{2}\star\cdots\star b_{k-2}\star
a_{i_{k}+1,i_{k+1}}$ $\displaystyle=(\delta
a_{i_{0},i_{1}})\star\overline{a}_{i_{1}+1,i_{2}}\star b_{2}\star\cdots\star
b_{k-2}\star a_{i_{k}+1,i_{k+1}}$
$\displaystyle\quad-\overline{a}_{i_{0},i_{1}}\star\overline{(\delta
a_{i_{1}+1,i_{2}})}\star b_{2}\star\cdots\star b_{k-2}\star
a_{i_{k}+1,i_{k+1}}$
$\displaystyle=\sum_{l=i_{0}}^{i_{1}-1}\mu_{l}^{*}(\overline{a}_{i_{0},l}\star
a_{l+1,i_{1}})\star\overline{a}_{i_{1}+1,i_{2}}\star b_{2}\star\cdots\star
b_{k-2}\star a_{i_{k}+1,i_{k+1}}$
$\displaystyle\quad-\sum_{l=i_{1}+1}^{i_{2}-1}\overline{a}_{i_{0},i_{1}}\star\mu_{l}^{*}(a_{i_{1}+1,l}\star\overline{a}_{l+1,i_{2}})\star
b_{2}\star\cdots\star b_{k-2}\star a_{i_{k}+1,i_{k+1}}$
$\displaystyle=\sum_{t\in\mathcal{S}_{0}(s)}(-1)^{\epsilon(t)}(\mu_{0}^{I^{t}}\star
1_{K^{\star
k}})(a^{t})-\sum_{t\in\mathcal{S}_{1}(s)}(-1)^{\epsilon(t)}(1_{K}\star\mu_{1}^{I^{t}}\star
1_{K^{\star(k-1)}})^{*}(a^{t}).$

Quite similarly, we have

$\displaystyle\overline{b}_{0}\star\cdots\star\overline{b}_{2p-2}\star(\delta
b_{2p})\star b_{2p+2}\star\cdots\star b_{k-2}\star a_{i_{k}+1,i_{k+1}}$
$\displaystyle=\sum_{t\in\mathcal{S}_{2p}(s)}(-1)^{\epsilon(t)}(1_{K^{\star(2p)}}\star\mu_{2p}^{I^{t}}\star
1_{K^{\star(k-2p)}})^{*}(a^{t})$
$\displaystyle\quad-\sum_{t\in\mathcal{S}_{2p+1}(s)}(-1)^{\epsilon(t)}(1_{K^{\star(2p+1)}}\star\mu_{2p+1}^{I^{t}}\star
1_{K^{\star(k-2p-1)}})^{*}(a^{t})$

for $p=1,2,\ldots,\frac{k-2}{2}$ and

$\displaystyle\overline{b}_{0}\star\cdots\star\overline{b}_{k-2}\star(\delta
a_{i_{k}+1,i_{k+1}})=\sum_{t\in\mathcal{S}_{k}(s)}(-1)^{\epsilon(t)}(1_{K^{\star
k}}\star\mu_{k}^{I^{t}})^{*}(a^{t}).$

Then we get

$(-1)^{\epsilon(s)}\delta(a^{s})=\sum_{p=0}^{k}(-1)^{p}\sum_{t\in\mathcal{S}_{p}(s)}(-1)^{\epsilon(t)}(1_{K^{\star
p}}\star\mu_{p}^{I^{t}}\star 1_{K^{\star(k-p)}})^{*}(a^{t}).$

For $k$ odd, we can similarly get

$\displaystyle(-1)^{\epsilon(s)}\delta(a^{s})$
$\displaystyle=\delta(\overline{b}_{0}\star\overline{b}_{2}\star\cdots\star\overline{b}_{k-1})$
$\displaystyle=\sum_{p=0}^{\frac{k-1}{2}}b_{0}\star\cdots\star
b_{2p-2}\star(\delta\overline{b}_{2p})\star\overline{b}_{2p+2}\star\cdots\star\overline{b}_{k-1}$
$\displaystyle=-\overline{(\delta a_{i_{0},i_{1}})}\star
a_{i_{1}+1,i_{2}}\star\overline{b}_{2}\star\cdots\star\overline{b}_{k-1}+\overline{a}_{i_{0},i_{1}}\star(\delta
a_{i_{1}+1,i_{2}})\star\overline{b}_{2}\star\cdots\star\overline{b}_{k-1}$
$\displaystyle\quad+\sum_{p=1}^{\frac{k-1}{2}}(-b_{0}\star\cdots\star
b_{2p-2}\star\overline{(\delta a_{i_{2p}+1,i_{2p+1}})}\star
a_{i_{2p+1}+1,i_{2p+2}}\star\overline{b}_{2p+2}\star\cdots\star\overline{b}_{k-1}$
$\displaystyle\quad+b_{0}\star\cdots\star b_{2p-2}\star
a_{i_{2p}+1,i_{2p+1}}\star(\delta
a_{i_{2p+1}+1,i_{2p+2}})\star\overline{b}_{2p+2}\star\cdots\star\overline{b}_{k-1})$
$\displaystyle=-\sum_{p=0}^{k}(-1)^{p}\sum_{t\in\mathcal{S}_{p}(s)}(-1)^{\epsilon(t)}(1_{K^{\star
p}}\star\mu_{p}^{I^{t}}\star 1_{K^{\star(k-p)}})^{*}(a^{t}).$

Observe that the map

$\widehat{\mathcal{S}}(j-i,k+1)\times\\{0,1,\ldots,k\\}\to\coprod_{s\in\widehat{\mathcal{S}}(j-i,k)}(\mathcal{S}_{0}(s)\sqcup\mathcal{S}_{1}(s)\sqcup\cdots\sqcup\mathcal{S}_{k}(s))$

sending $(t,p)\in\widehat{\mathcal{S}}(j-i,k+1)\times\\{0,1,\ldots,k\\}$ to
$t\in\mathcal{S}_{p}(s)$ is bijective, where
$s=(t_{0},\ldots,t_{p},t_{p+2},\ldots,t_{k+2})\in\widehat{\mathcal{S}}(j-i,k)$
for $t=(t_{0},\ldots,t_{k+2})\in\widehat{\mathcal{S}}(j-i,k+1)$. Then we get

$\displaystyle\sum_{p=i}^{j-1}\mu_{p}^{*}(\overline{a}_{i,p}\star a_{p+1,j})$
$\displaystyle\equiv\sum_{s\in\widehat{\mathcal{S}}(j-i,k)}(-1)^{\epsilon(s)+\epsilon(k)}P(I^{s})^{*}\delta(a^{s})$
$\displaystyle\equiv\sum_{s\in\widehat{\mathcal{S}}(j-i,k)}\sum_{p=0}^{k}(-1)^{p}\sum_{t\in\mathcal{S}_{p}(s)}(-1)^{\epsilon(t)+\epsilon(k+1)}P(I^{t}(p))^{*}(1_{K^{\star
p}}\star\mu_{p}^{I^{t}}\star 1_{K^{\star(k-p)}})^{*}(a^{t})$
$\displaystyle\equiv\sum_{t\in\widehat{\mathcal{S}}(j-i,k+1)}(-1)^{\epsilon(t)+\epsilon(k+1)}\sum_{p=0}^{k}(-1)^{p}P(I^{t}(p))^{*}(1_{K^{\star
p}}\star\mu_{p}^{I^{t}}\star 1_{K^{\star(k-p)}})^{*}(a^{t})$

modulo $\mathrm{Im}\,\delta$. By Lemma 2.6, we also get

$\displaystyle\sum_{t\in\widehat{\mathcal{S}}(j-i,k+1)}(-1)^{\epsilon(t)+\epsilon(k+1)}\sum_{p=0}^{k}(-1)^{p}P(I^{t}(p))^{*}(1_{K^{\star
p}}\star\mu_{p}^{I^{t}}\star 1_{K^{\star(k-p)}})^{*}(a^{t})$
$\displaystyle\equiv\sum_{t\in\widehat{\mathcal{S}}(j-i,k+1)}(-1)^{\epsilon(t)+\epsilon(k+1)}(P(I^{t})^{*}\delta+(-1)^{k}\delta
P(I^{t})^{*})(a^{t})$
$\displaystyle\equiv\sum_{t\in\widehat{\mathcal{S}}(j-i,k+1)}(-1)^{\epsilon(t)+\epsilon(k+1)}P(I^{t})^{*}\delta(a^{t})$

modulo $\mathrm{Im}\,\delta$. Thus the induction on $k$ proceeds, completing
the proof. ∎

Consider the identity (4.1) for $k=j-i$. Observe that
$\widehat{\mathcal{S}}(j-i,j-i)$ consists of a single element
$(0,0,1,\ldots,j-i)$, and $a^{(0,0,1,\ldots,j-i)}=a_{i,i}\star\cdots\star
a_{j,j}$ which is a cocycle because each $a_{k,k}$ is a cocycle. Then we
obtain

$\displaystyle\sum_{k=i}^{j-1}\mu_{k}^{*}(\overline{a}_{i,k}\star a_{k+1,j})$
$\displaystyle\equiv(-1)^{\epsilon((0,0,1,\ldots,j-i))+\epsilon(j-i+1)}P(I^{(0,0,1,\ldots,j-i)})\delta(a_{i,i}\star\cdots\star
a_{j,j})$ $\displaystyle\equiv 0$

modulo $\mathrm{Im}\,\delta$. Hence there is $a_{i,j}\in\widetilde{C}^{*}(K)$
satisfying (3.1), and so the induction on $j-i$ proceeds by Lemmas 4.2 and
4.3. Thus by Lemma 4.1, we get:

######  Lemma 4.4.

If $K$ is $\mathbb{F}$-tight, then there are $a_{ij}\in\widetilde{C}^{*}(K)$
for $0\leq i\leq j\leq q$ satisfying the condition of Lemma 3.5.

Finally, by Lemma 3.5, we obtain:

######  Proposition 4.5.

If $K$ is an $\mathbb{F}$-tight complex, then every Massey product in
$H^{*}(\mathcal{C}^{*}(K))$ with support $V(K)$ is trivial whenever it is
defined.

Now we are ready to prove Theorem 1.2.

###### Proof of Theorem 1.2.

Induct on the cardinality $|V(K)|$. For $|V(K)|=1$, then $K$ is obviously
$\mathbb{F}$-Golod. We assume that all Massey products in
$H^{*}(\mathcal{C}^{*}(K))$ are trivial for $|V(K)|<m$, and we consider the
$|V(K)|=m$ case. By definition, the class of $\mathbb{F}$-tight complexes are
closed under taking full subcomplexes. Then by Lemma 3.4 and the induction
hypothesis, it is sufficient to show that all Massey products in
$H^{*}(\mathcal{C}^{*}(K))$ with support $V(K)$ are trivial. Thus by
Proposition 4.5, the proof is finished. ∎

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# Some applications of the Shapiro time delay

A. Ghasemi Azar1, H. Rezaei2, H<EMAIL_ADDRESS>1
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
University of Maragheh, P.O. Box 55136-553, Maragheh, Iran
2 Department of Mathematics, College of Sciences, Yasouj University,
Yasouj-75914-74831, Iran

###### Abstract

Listening to echoes has long been a way to estimate distances, a technique
whose backbone is the time delay. The gravitational field also creates a time
delay, called Shapiro time delay, that helps us extract some information from
the field and is indeed due to the photon journey through the field. Here, the
ability of the Shapiro effect to distinguish naked singularities from non-
naked ones (black holes) is discussed. It is also inferred that this time
delay may be hired to compare the various types of singularities with
different dimensions. Besides them, the possibility of detecting the rotation
of the assumed objects through surveying the gravitational time delay is also
addressed.

##  I Introduction

Singularity is one of the attractive predictions of general relativity (GR)
Poisson ; HPbook . Although the Cosmic Censorship Hypothesis (CCH) generally
rejects the existence of naked singularity (NS), there is not a common
agreement on CCH, and actually, NS physics has a lot to say HPbook .
Consequently, NS formation has attracted plenty of attempts to itself JNW ;
NS-2 ; NS-1 ; NS0 ; NS1 ; NS2 ; NS3 ; NS4 ; NS5 ; PRL ; NS6 . Subsequently,
distinguishing NS from black holes (non-naked singularities, i.e.
singularities quarantined from the surroundings by the event horizon) appears
as a serious task for physicists, a task that is accomplished by studying
various properties of NS d1 ; d2 ; d3 ; d4 ; d5 ; d6 ; o1 ; o2 ; o3 ; o4 .

Indeed, recent advances in the field of photographing black hole candidates
further encourage us to study NS and the differences with black holes. In this
regard, observational properties of naked singularities have also been studied
o1 ; o2 ; o3 ; o4 . The Sagnac time delay also seems to be able to distinguish
black holes from NS d6 . In this setup, a satellite (as the sender/receiver)
orbiting the object (NS or black hole) is crucial, a serious difficulty for
the idea applicability, as such objects, are very far from us. Therefore,
another type of time delay may be more useful for such studies, i.e. an
experiment that can be done remotely (without any need to send a satellite to
long distances).

The fourth test of general relativity, established by I. I. Shapiro shap1 , is
based on the time delay of light rays passing through the gravitational field.
In addition to being used to check GR in the Solar system shap2 ; shap3 ;
shap4 , this effect is also helpful in verifying $i$) modified gravity
theories shapmod0 ; shapmod ; ghosh ; Dyadina:2021paa , generalized
uncertainty principle Okcu:2021oke , $ii$) equivalence principle Desai:2016nqu
; Boran:2018ypz ; Minazzoli:2019ugi ; Kahya:2016prx , $iii$) the number of
spacetime dimensions shapdim , and studying the Pulsars Desai:2016nqu ;
shappul0 ; Desai:2015hua ; shappul ; shappul1 ; shappul3 ; shappul2 .
Fortunately, unlike the Sagnac effect, this experiment does not need to send a
satellite and hence is possible remotely. Moreover, although picosecond has
been reported as the accuracy order of Shapiro measurement Will:2014kxa , it
seems that the next generation of gravitational detectors shall significantly
increase the accuracy Ballmer:2009pq ; Sullivan:2020sop . The negative Shapiro
time delay called the gravitational time advancement also happens when the
light rays passing through a weaker gravitational field compared to the field
at the observer’s place, and it is predicted that the modern versions of the
Michelson-Morley experiment shall measure this time advancement Bhadra:2023ptq
.

Indeed, although the Schwarzschild spacetime is the most general spherically
symmetric vacuum solution and a simple solution, it is the backbone of our
understanding of many phenomena like orbits and thermodynamics of black holes
motivating physicists to study this metric and its various generalizations
Wiltshire:2009zza ; inverno ; shapgamma . Employing the Schwarzschild metric
and some generalized forms of this spacetime including the Bardeen, Reissner-
Nordström, and Ayón-Beato-García metric, it seems that the Shapiro time delay
can distinguish these black holes from each other Junior:2023nku . On the
other hand, to find constraints on the deviations from spherical symmetry
(SS), focusing on the $\gamma$-metric (a non-spherically generalization of the
Schwarzschild metric that, depending on the value of $\gamma$, can also
present NS), the Shapiro time delay has been investigated shapgamma . The
obtained time delay is equal to that of Schwarzschild meaning that the Shapiro
time delay cannot be used to constrain $\gamma$, and hence, distinguish
solutions with different $\gamma$ (the criterion of deviation from SS).
Therefore, it seems that it is not possible to distinguish NS from the black
hole (even the simplest black hole solution i.e. the Schwarzschild black hole)
through comparing the corresponding Shapiro time delays.

A long way has been traced to find the rotating version of the Schwarzschild
geometry called Kerr metric which also plays a vital role in discovering the
secrets of various phenomena Wiltshire:2009zza . Since the Schwarzschild
metric is a vacuum solution, it is then obvious to look at all other black
holes as its extensions. The Janis-Newman-Winicour (JNW) metric and its
rotational version are two well-studied generalizations of the Schwarzschild
and Kerr spacetimes, respectively JNW ; d2 . They can also include NS JNW ; d2
, and attract a lot of attention (see Refs. JNW ; d1 ; d2  and their
citations). In five dimensions, the Myers-Perry (MP) metric is an extension of
the Kerr black hole that reduces to the Kerr geometry whenever a
$4$-dimensional spacetime is taken into account. Correspondingly, the $4$ and
$5$-dimensional Schwarzschild metrics are also recoverable if the zero limit
of angular momentums is applied mpmetric . In summary, the background and
importance of these geometries requires that they be studied.

Here, our first aim is to show the power of the Shapiro test in distinguishing
black holes from NSs. To this end, the JNW metric JNW  and its rotating
version d2  are studied in the subsequent sections, respectively. The
possibility of verifying rotation in the fifth dimension shall be studied in
the fourth section by employing the MP metric mpmetric . A summary is also
provided at the end.

##  II Shapiro time delay in JNW spacetime

In the presence of the scalar field
$\Phi(=\frac{q}{2b\sqrt{\pi}}\ln(1-\frac{b}{r}))$ with charge $q$ and mass
$M$, the JNW metric is obtained as

$\displaystyle
ds^{2}=-H^{\nu}dt^{2}+\frac{dr^{2}}{H^{\nu}}+H^{1-\nu}r^{2}(d\theta^{2}+\sin^{2}(\theta)d\phi^{2}),$
(1)

in which $H=1-\frac{b}{r}$, $\nu=\frac{2M}{b}$, and $b=2\sqrt{M^{2}+q^{2}}$
JNW ; d1 . The Schwarzschild spacetime is covered at the appropriate limit of
$q=0$ (or, $\nu=1$). The spacetime represents a singularity at $r=b$ for
$q\neq 0$ and has the photon sphere if $1/2<\nu\leq 1$ JNW ; d1 . Following
the existence and absence of the photon sphere, singularity is called strong
NS if $0\leq\nu\leq 1/2$, and it is called weak NS on the condition that
$1/2<\nu<1$, respectively d1 . Indeed, it is easy to calculate the surface
area at $r=b$ through

$\displaystyle
A=\int_{\theta_{0}=0}^{\theta=\pi}\int_{\phi_{0}=0}^{\phi=2\pi}\big{[}H^{1-\nu}r^{2}\big{]}_{r=b}d\theta
d\phi=0,$ (2)

for $0\leq\nu<1$ meaning that $r=b$ is a NS.

Figure 1: Light ray moves towards $E$ through the path $PE$.

Figure. (1) displays a photon sent from point $P$ towards $E$, while the path
$PE$ has the closest vertical distance $\mu$ from the object $O$ inverno
whose surrounding spacetime is described by the JNW metric. In this manner,
for a small displacement $dx$, we have

$\displaystyle\bigg{\\{}^{dr=\frac{x}{\sqrt{x^{2}+\mu^{2}}}dx}_{d\phi=\frac{\mu}{x^{2}+\mu^{2}}dx}\Rightarrow
d\phi=\frac{\mu}{r\sqrt{r^{2}-\mu^{2}}}dr.$ (3)

Thus, since for a light ray ($ds^{2}=0$) on the $\theta=\frac{\pi}{2}$ plane

$\displaystyle H^{\nu}dt^{2}=\frac{dr^{2}}{H^{\nu}}+H^{1-\nu}r^{2}d\phi^{2},$
(4)

one can easily use Eq. (3) to find

$\displaystyle
dt^{2}=H^{-2\nu}\big{[}1+\frac{\mu^{2}}{r^{2}-\mu^{2}}H\big{]}dr^{2},$ (5)

as the time delay of a photon passing the path of $PE$ in the presence of an
object whose spacetime is described by the JNW metric. At long distances from
$O$, one can approximate

$\displaystyle
dt^{2}\simeq\frac{r^{2}}{r^{2}-\mu^{2}}\big{[}1+\frac{2b}{r}\nu-\frac{b\mu^{2}}{r^{3}}-\frac{2b^{2}\mu^{2}}{r^{4}}\nu\big{]}dr^{2},$
(6)

leading to

$\displaystyle
dt\simeq\frac{rdr}{\sqrt{r^{2}-\mu^{2}}}\bigg{[}1+\frac{2M}{r}-\frac{\mu^{2}}{r^{2}}(\frac{1}{r}+\frac{4M}{r^{2}})\sqrt{M^{2}+q^{2}}\bigg{]},$
(7)

where assuming $\frac{b}{r}\ll 1$, the Taylor expansion has been used (for
example, we have $H^{-2\nu}\simeq 1+\frac{2\nu b}{r}$) and terms including
powers upper than $(\frac{1}{r})^{4}$ are ignored. In fact, unlike Ref.
inverno , here, the $(\frac{1}{r})^{4}$ term is calculated and the
Schwarzschild result is also obtainable by adopting $q=0$ (or equally,
$\nu=1$) inverno . Accordingly, the discrepancy with the Schwarzschild case
emerges in terms including $q$. It is apparent that $dt$ decreases as $q$
increases meaning that, for the same $dr$, NS produces less $dt$ compared to
the Schwarzschild black hole, and accordingly, those NSs that have the photon
sphere, generate more time delay rate compared to those without the photon
sphere.

A Shapiro time delay measurement helps us confine and estimate the value of
$q$. To show it, up to the first order of expansion, one can write Eq. (7) as
$dt\sim dt_{S}-q^{2}\delta t$, where $dt_{S}\equiv dt\big{|}_{q=0}$ (the
Schwarzschild case) and $\delta
t\equiv\frac{\mu^{2}dr}{2Mr^{2}\sqrt{r^{2}-\mu^{2}}}(1+\frac{4M}{r})$. Now,
consider an object with mass $M$ and a measurement with uncertainty $A$
reporting $t=\int dt$ for the time delay. If the object is supposed to be a
Schwarzschild black hole, then mathematical calculations give us $t_{S}=\int
dt_{S}$. In this manner, if $|t-t_{S}|<A$, then it can be said that with the
accuracy $1-A$, the object is a Schwarzschild black hole.

On the other hand, $|t-t_{S}|<A$ can also be used to find an upper bound on
the value of $q$ as $q^{2}<\frac{A}{\int\delta t}$. Therefore, by increasing
the precision of the setup, one can find more accurate upper bound on $q$ (or
equally, one can find the value of $q$ with more certainty). In this line, it
is worthy to mention that although the current detectors have also significant
precision i.e. $A\sim O\big{(}10^{-12}\big{)}$s Will:2014kxa , the next
generation of gravitational wave detectors equips us with more accurate
measurements Ballmer:2009pq ; Sullivan:2020sop .

##  III Rotating $4$-dimensional metric

The geometry of spacetime including a rotating object (the rotational version
of the JNW metric) is described as d2

$\displaystyle
ds^{2}=-h^{1-\nu}\rho(\frac{dr^{2}}{\Delta}+d\theta^{2}+\sin^{2}\theta
d\phi^{2})+$ (8) $\displaystyle h^{\nu}(dt-a\sin^{2}\theta
d\phi)^{2}+2a\sin^{2}\theta(dt-a\sin^{2}\theta d\phi)d\phi,$

where

$\displaystyle h=1-\frac{br}{\rho},$
$\displaystyle\nu=\frac{M}{\sqrt{M^{2}+q^{2}}}=\frac{2M}{b},$
$\displaystyle\Delta=r^{2}+a^{2}-br,$
$\displaystyle\rho=r^{2}+a^{2}\cos^{2}\theta,$ (9)

and the scalar field $\Phi$ takes the form

$\displaystyle\frac{q}{2b}\ln(1-\frac{br}{\rho}).$ (10)

In the above expressions, $q$ and $a=J/M$ are the scalar charge and angular
momentum per mass, respectively, while $J$ denotes the angular momentum and
the JNW case is easily recovered at the $J=0$ limit. It is reduced to the Kerr
and Schwarzschild solutions for $q=0(\equiv\nu=1)$ and $a=q=0$, respectively.
For other values of $\nu$ ($0<\nu<1$), the line element (8) includes a naked
singularity d2 .

Now, considering the approach of the previous section, one reaches

$\displaystyle h^{\nu}dt^{2}+\frac{2\mu
a}{r\sqrt{r^{2}-\mu^{2}}}(1-h^{\nu})drdt+$ (11)
$\displaystyle[\frac{\mu^{2}a^{2}(h^{\nu}-2)}{r^{2}(r^{2}-\mu^{2})}-(\frac{\mu^{2}}{r^{2}(r^{2}-\mu^{2})}+\frac{1}{\Delta})h^{1-\nu}\rho]dr^{2}=0.$

for the photon moving on the plane $\theta=\pi/2$, and easily, one can recover
Eq. (5) at the appropriate limit $a=0$ leading to $h=H$. Of course, whenever
$\theta=\pi/2$, we always have $h=H$ ($\rho=r^{2}$) and thus

$\displaystyle\frac{2\mu
a}{r\sqrt{r^{2}-\mu^{2}}}(h^{-\nu}-1)\simeq\frac{4M\mu
a}{r^{2}\sqrt{r^{2}-\mu^{2}}}\equiv-\Theta,$ $\displaystyle\Delta\simeq
r^{2}+a^{2},$ (12)
$\displaystyle(\frac{\mu^{2}h^{1-2\nu}\rho}{r^{2}(r^{2}-\mu^{2})}+\frac{h^{1-2\nu}\rho}{\Delta})\simeq$
$\displaystyle(1-\frac{b\mu^{2}}{r^{3}}-\frac{(r^{2}-\mu^{2})a^{2}}{r^{2}(r^{2}+a^{2})})\frac{r^{2}h^{-2\nu}}{r^{2}-\mu^{2}},$
$\displaystyle\frac{\mu^{2}a^{2}(h^{\nu}-2)}{h^{\nu}r^{2}(r^{2}-\mu^{2})}\simeq$
$\displaystyle-\frac{\mu^{2}a^{2}}{r^{2}(r^{2}-\mu^{2})}(1+\frac{2\nu
b}{r})\simeq-\frac{\mu^{2}a^{2}}{r^{2}(r^{2}-\mu^{2})}h^{-2\nu},$

whenever $\frac{b}{r}\ll 1$. It finally gives the coefficient of $dr^{2}$ as

$\displaystyle-\frac{h^{-2\nu}r^{2}}{r^{2}-\mu^{2}}\big{(}1-\frac{b\mu^{2}}{r^{3}}-f(a)\big{)}\simeq-\frac{r^{2}}{r^{2}-\mu^{2}}\bigg{[}-\frac{b\mu^{2}}{r^{3}}$
$\displaystyle(1-f(a))(1+\frac{2b}{r}\nu)-\frac{2b^{2}\mu^{2}}{r^{4}}\nu\bigg{]}\equiv-\alpha,$
$\displaystyle
f(a)=\frac{\frac{a^{2}}{r^{2}}[1-\frac{\mu^{2}}{r^{2}}(2+\frac{a^{2}}{r^{2}})]}{1+\frac{a^{2}}{r^{2}}},$
(13)

compared to Eq. (6) to see that, at this level of approximation, the effects
of $a$ are stored into $f(a)$. Therefore, at this limit, Eq. (11) takes the
form $dt^{2}-\Theta drdt-\alpha dr^{2}\simeq 0$ (the alternative of Eq. (6))
that eventually renders

$\displaystyle
dt\simeq\bigg{[}\sqrt{\alpha}+\frac{\Theta(\frac{\Theta}{4\sqrt{\alpha}}+1)}{2}\bigg{]}dr,$
$\displaystyle\sqrt{\alpha}\simeq\frac{r}{\sqrt{r^{2}-\mu^{2}}}\bigg{[}1-\frac{f(a)}{2}+(1-f(a))\frac{2M}{r}$
$\displaystyle-\frac{\mu^{2}}{r^{2}}(\frac{1}{r}+\frac{4M}{r^{2}})\sqrt{M^{2}+q^{2}}\bigg{]},$
(14)

as the counterpart of Eq. (7) whenever $a\neq 0$ and of course provided that
$f(a)\ll 1$. The plausibility of the latter condition is a reflection of our
great distance from $O$ in agreement with $i)$ the primary assumption $b/r\ll
1$ and also, $ii)$ the fact that such objects are very far from us. Indeed, it
is the only solution of $dt^{2}-\Theta drdt-\alpha dr^{2}\simeq 0$ that
produces Eq. (7) at the limit of $a=0$. Thus, the ability of the Shapiro time
delay in detecting rotation in $4$-dimensional spacetime as well as
distinguishing rotating NS from the rotating black hole is deduced.

##  IV The MP spacetime

One of the Universe’s mysteries is its number of dimensions which seems
searchable using the Shapiro time delay shapdim . Another puzzle is the method
of detecting motions in higher dimensions when we do not have direct access to
the higher dimensions. Here, focusing on the MP metric, we are going to
provide an answer by studying the effects of such a movement on the Shapiro
time delay. The MP geometry is mpmetric

$\displaystyle ds^{2}=-dt^{2}+\frac{M(dt+a\sin^{2}\theta
d\phi+\beta\cos^{2}\theta
d\psi)^{2}}{r^{2}+a^{2}\cos^{2}\theta+\beta^{2}\sin^{2}\theta}$
$\displaystyle+\frac{r^{2}(r^{2}+a^{2}\cos^{2}\theta+\beta^{2}\sin^{2}\theta)}{(r^{2}+a^{2})(r^{2}+\beta^{2})-Mr^{2}}dr^{2}$
(15) $\displaystyle+(r^{2}+a^{2})\sin^{2}\theta
d\phi^{2}+(r^{2}+\beta^{2})\cos^{2}\theta d\psi^{2}$
$\displaystyle+(r^{2}+a^{2}\cos^{2}\theta+\beta^{2}\sin^{2}\theta)d\theta^{2},$

where $\beta$ is related to the angular momentum of fifth dimension.

For $\beta=a=0$, the five-dimensional Schwarzschild solution, i.e.

$\displaystyle
ds^{2}=-(1-\frac{M}{r^{2}})dt^{2}+\frac{dr^{2}}{1-\frac{M}{r^{2}}}$ (16)
$\displaystyle+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}+\cos^{2}\theta
d\psi^{2}),$

is achieved. Following the approach of the previous section, one finds

$\displaystyle
dt^{2}=(1-\frac{M}{r^{2}})^{-2}\big{[}1+\frac{\mu^{2}}{r^{2}-\mu^{2}}(1-\frac{M}{r^{2}})\big{]}dr^{2},$
(17)

as the Shapiro time delay of a $5$-dimensional Schwarzschild black hole
whenever $\theta=\pi/2$ and $d\psi=0$. At long distances from $O$ (or equally,
$M/r\ll 1$), one easily reaches

$\displaystyle
dt^{2}\simeq\frac{r^{2}}{r^{2}-\mu^{2}}[1+\frac{2M}{r^{2}}-\frac{M\mu^{2}}{r^{4}}]dr^{2},$
(18) $\displaystyle\Rightarrow
dt\simeq\frac{rdr}{\sqrt{r^{2}-\mu^{2}}}[1+\frac{M}{r^{2}}-\frac{M\mu^{2}}{2r^{4}}],$

that clearly explains if the Shapiro time delay of a Schwarzschild candidate
obeys this equation instead of the $q=0$ case of Eq. (7), then one can claim
that the Universe has $5$ dimensions. Compared to Eq. (7), it is seen that
$1/r^{3}$ does not appear here, a fact that originated from the $M/r^{2}$ term
in the metric (17) (for the Schwarzschild case, we have $M/r$). Moreover, it
should be noted that the origin of $r^{-4}$ also differs from that of this
term in Eq. (7) as their coefficients differ. Finally and as a preliminary
check, if we only want to keep the first two terms $\big{(}1+\frac{M}{r^{2}}$
in this equation, and $1+\frac{2M}{r}$ in Eq. (7)$\big{)}$, then it is enough
to replace $M/r^{2}$ and $2M/r$ with each other to see that the corresponding
time delays are mutually recovered, a net reflection of the relationship
between the corresponding metrics (Eqs. (5) and (17)).

A three dimensional subspace of metric (IV) with $\theta=\pi/2$ and $\psi=0$
gives the geometry

$\displaystyle
ds^{2}=-(1-\frac{M}{r^{2}+\beta^{2}})dt^{2}+\frac{2Ma}{r^{2}+\beta^{2}}dtd\phi$
(19)
$\displaystyle+\frac{r^{2}(r^{2}+\beta^{2})}{(r^{2}+a^{2})(r^{2}+\beta^{2})-Mr^{2}}dr^{2}$
$\displaystyle+(r^{2}+a^{2}+\frac{Ma^{2}}{r^{2}+\beta^{2}})d\phi^{2},$

leading to

$\displaystyle
0=-(1-\frac{M}{r^{2}+\beta^{2}})dt^{2}+\frac{2Ma\mu}{r(r^{2}+\beta^{2})\sqrt{r^{2}-\mu^{2}}}dtdr$
$\displaystyle+\big{[}(r^{2}+a^{2}+\frac{Ma^{2}}{r^{2}+\beta^{2}})\frac{\mu^{2}}{r^{2}(r^{2}-\mu^{2})}$
$\displaystyle+\frac{r^{2}}{r^{2}+a^{2}-\frac{Mr^{2}}{r^{2}+\beta^{2}}}\big{]}dr^{2},$
(20)

when one uses Eq. (3) and considers a photon ($ds=0$). Clearly, Eq. (17) is
produced for $a=\beta=0$, and additionally, even if the object $M$ does not
rotate in $4$ dimension ($a=0$), then the existence of the fifth dimension
rotation still contributes to the results on the condition that $\beta\neq 0$.
Hence, the detection of rotation in higher dimensions through the Shapiro
effect is possible. When $a=0$ and $\beta\neq 0$, the solution is

$\displaystyle
dt\simeq\frac{rdr}{\sqrt{r^{2}-\mu^{2}}}[1+\frac{M}{r^{2}+\beta^{2}}-\frac{M\mu^{2}}{2r^{2}(r^{2}+\beta^{2})}],$
(21)

where $\frac{M}{r^{2}+\beta^{2}}\ll 1$ has been assumed. Therefore, rotation
in the fifth dimension is verifiable using this effect. Clearly, it also
reduces to Eq. (18) when $\beta=0$. For the $\beta=0$ case (when $a\neq 0$),
as the coefficient of $dtdr$ in Eq. (IV) is not zero, following the approach
that led to Eq. (III), one finally obtains $dt^{2}-\varphi drdt-\epsilon
dr^{2}\simeq 0$ yielding

$\displaystyle
dt\simeq\bigg{[}\sqrt{\epsilon}+\frac{\varphi(1+\frac{\varphi}{4\sqrt{\epsilon}})}{2}\bigg{]}dr,$
(22)
$\displaystyle\sqrt{\epsilon}\simeq\frac{r}{\sqrt{r^{2}-\mu^{2}}}\bigg{[}1-\frac{F(a)}{2}+(1-F(a))\frac{M}{r^{2}}-\frac{M\mu^{2}}{2r^{4}}\bigg{]},$

as the only solution that recovers Eq. (18) at the limit $a=0$. Here,
$\varphi=\frac{2Ma\mu}{r^{3}\sqrt{r^{2}-\mu^{2}}}$ and

$\displaystyle
F(a)=\frac{a^{2}}{r^{2}}\bigg{(}\frac{1-\frac{\mu^{2}}{r^{2}}\big{[}2+\frac{a^{2}}{r^{2}}-\frac{M}{r^{2}}\big{]}}{1+\frac{a^{2}}{r^{2}}-\frac{M}{r^{2}}}\bigg{)},$
(23)

and thus $\varphi=F(a)=0$ for $a=0$.

Generally, when $a,\beta\neq 0$, long calculations lead to $dt^{2}-\psi
drdt-\mathcal{N}dr^{2}\simeq 0$ and thus

$\displaystyle
dt\simeq\bigg{[}\sqrt{\mathcal{N}}+\frac{\psi(1+\frac{\psi}{4\sqrt{\mathcal{N}}})}{2}\bigg{]}dr,$
(24)

where

$\displaystyle\sqrt{\mathcal{N}}$ $\displaystyle\simeq$
$\displaystyle\frac{r}{\sqrt{r^{2}-\mu^{2}}}\bigg{[}1-\frac{\mathcal{F}(a)}{2}+\frac{M(1-\mathcal{F}(a))}{r^{2}+\beta^{2}}$
$\displaystyle-$
$\displaystyle\frac{M\mu^{2}}{2r^{2}(r^{2}+\beta^{2})}\bigg{]},$
$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\frac{2Ma\mu}{r(r^{2}+\beta^{2})\sqrt{r^{2}-\mu^{2}}},$ (25)
$\displaystyle\mathcal{F}(a)$ $\displaystyle=$
$\displaystyle\frac{a^{2}}{r^{2}}\bigg{(}\frac{1-\frac{\mu^{2}}{r^{2}}\big{[}2+\frac{a^{2}}{r^{2}}-\frac{M}{r^{2}+\beta^{2}}\big{]}}{1+\frac{a^{2}}{r^{2}}-\frac{M}{r^{2}+\beta^{2}}}\bigg{)}.$

Clearly, Eqs. (21) and (22) are obtained at the appropriate limits $a=0$ and
$\beta=0$, respectively. Of course, comparing Eqs. (21) and (18), it is
understood that one could have achieved this result by replacing $M/r^{2}$
with $\frac{M}{r^{2}+\beta^{2}}$ in Eqs. (22) and (23).

In summary, while Eq. (21) implies on the implication of the rotation in the
fifth dimension of the Shapiro time delay, difference between $F(a)$ and
$f(a)$ clearly shows that even the existence of the fifth dimension affects
the time delay ($\beta=0$). Indeed, as it is emphasized by the information
stored in $\mathcal{F}(a)$, these effects become more tangible when $\beta\neq
0$. This achievement is strengthened by the point mentioned after Eq. (18),
where the emergence of $M/r^{2}$ instead of $M/r$ in the time delay of a five-
dimensional Schwarzschild spacetime has been argued, a feature again confirmed
by comparing Eqs. (22) and (III) with each other.

##  V Conclusion

Depending on the metric, it is possible to distinguish NSs from black holes
via the gravitational time delay (the Shapiro effect). Moreover, the
possibility of verifying the existence of extra dimensions and detecting
rotation in higher dimensions through the Shapiro time delay has also been
studied. The results imply the ability of this effect in such investigations.
Indeed, comparing Secs. (III) and (IV) shows that this effect can even be used
to compare objects of various dimensions.

Therefore, we can hope that the use of the Shapiro time delay and such
fundamental experiments shall help us analyze the dimensions of the universe
and its contents. Finally, it is worthwhile to mention that challenging the
ideas presented here by different data such as GW could be an interesting
topic for future projects.

## Declarations Conflict of interest

The authors declare no conflict of interest.

## Acknowledgments

The authors would like to appreciate the anonymous referee for the valuable
comments.

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