11institutetext: Qure.ai
11email: {ujjwal.upadhyay,mukul.ranjan,satish.golla,swetha.tanamala,
preetham.sreeniva<EMAIL_ADDRESS>
22institutetext: Christian Medical College Ludhiana, India
22email<EMAIL_ADDRESS>33institutetext: Pacific Neuroscience
Institute, USA
33email<EMAIL_ADDRESS>
# Deep-ASPECTS: A Segmentation-Assisted Model for Stroke Severity Measurement

Ujjwal Upadhyay 11 Mukul Ranjan 11 Satish Golla 11 Swetha Tanamala 11 Preetham
Sreenivas 11 Sasank Chilamkurthy 11 Jeyaraj Pandian 22
Jason Tarpley 33

###### Abstract

A stroke occurs when an artery in the brain ruptures and bleeds or when the
blood supply to the brain is cut off. Blood and oxygen cannot reach the
brain’s tissues due to the rupture or obstruction resulting in tissue death.
The Middle cerebral artery (MCA) is the largest cerebral artery and the most
commonly damaged vessel in stroke. The quick onset of a focused neurological
deficit caused by interruption of blood flow in the territory supplied by the
MCA is known as an MCA stroke. Alberta stroke programme early CT score
(ASPECTS) is used to estimate the extent of early ischemic changes in patients
with MCA stroke. This study proposes a deep learning-based method to score the
CT scan for ASPECTS. Our work has three highlights. First, we propose a novel
method for medical image segmentation for stroke detection. Second, we show
the effectiveness of AI solution for fully-automated ASPECT scoring with
reduced diagnosis time for a given non-contrast CT (NCCT) Scan. Our algorithms
show a dice similarity coefficient of 0.64 for the MCA anatomy segmentation
and 0.72 for the infarcts segmentation. Lastly, we show that our model’s
performance is inline with inter-reader variability between radiologists.

###### Keywords:

Stroke Infarct Automated ASPECTS scoring framework.

##  1 Introduction

A stroke is a medical emergency that requires immediate attention. Brain
injury and other consequences can be avoided if intervention is taken early.
There are two main types of stroke. Hemorrhagic Stroke. When a blood vessel
ruptures, it results in hemorrhagic stroke. Aneurysms or arteriovenous
malformations (AVM) are the most common causes of hemorrhagic stroke. Ischemic
Stroke.When the blood flow to a portion of the brain is blocked or diminished,
brain tissue is deprived of oxygen and nutrients, resulting in an ischemic
stroke. Within minutes, brain cells begin to die. Ischemic and hemorrhagic
strokes are managed differently since they have different causes and effects
on the body. Rapid diagnosis is critical for minimising brain damage and
allowing the doctor to treat the stroke with the most appropriate treatment
strategy for the type.

Figure 1: Segmentation output from Deep-ASPECTS. Refer to figure 2 for more
information on how this is generated.

In this paper, we focus on ischemic strokes. The symptoms of such a stroke
vary depending on the brain area affected and the quantity of tissue that has
been damaged. The severity of damage can be assessed by two methods "ASPECTS"
and "Hypodensity of >1/3 MCA Territory" rule. In clinical practice, ASPECTS
detects significant EIC in a higher proportion of the early scans [13].
ASPECTS is a topographic scoring system that applies a quantitative approach
and does not ask physicians to estimate volumes from two-dimensional images.
It is scored out of 10 points. ASPECTS has been one of the recognized scoring
scales that serve as key selection criteria on the management of acute stroke
in the MCA region, where endovascular therapy in patients with baseline
ASPECTS$\geq 6$ is recommended [15, 6, 5]. Variations of the ASPECT scoring
system is used in the posterior circulation and referred to as pc-ASPECTS
[16].

In stroke cases, "time is brain"[18]. The outcomes become progressively worse
with time. In the current clinical practice, radiologists must read the NCCT
scan to report the ASPECT score, which can take time due to the high volume of
cases. We propose an automated ASPECT scoring system, a task currently
affected by high inter-reader variability due to manual selection and
measurement of the relevant MCA regions scored for ASPECTS. We present Deep-
ASPECTS that provides an efficient way to prioritize and detect stroke cases
in less than 1 minute, making it clinically relevant.

The proposed deep learning solution involves segmenting the acute infarcts and
the MCA territory from an NCCT scan. The segmentation step is followed by
overlapping these maps to get the affected region across the slices in basal
ganglia and corona radiata level. We demonstrate the robustness of our
framework by validating both the segmentation maps and the estimated ASPECTS
against ground truth clinical data, which contains 150 scans.

##  2 Related Work

Various models have been developed and studied for the segmentation of NCCT
scan for qualitative estimation of infarcts, haemorrhage and different other
critical findings in brain[11, 14, 10, 3, 21]. These studies have presented
essential findings related to the NCCT. However, to the best of our knowledge,
we could not find any published research which combines the Infarct
segmentation and MCA anatomy segmentation to predict the ASPECT score of the
NCCT scan in an end-to-end fashion.

Deep learning based semantic segmentation models with encoder and decoder
blocks connected through various skip connections have shown to be effective
in the medical imaging domain[2, 17, 23, 1, 7]. Unet[17] presents a simple
encoder-decoder based CNN architecture while TransUnet[2] uses transformer[22,
4] based encoder in which a CNN layer is first used to extract the features
and then patch embedding is applied to $1\times 1$ features map.Linknet[1] is
another variation of Unet[17] which replaces the CNN block of Unet with
Residual connection[8] and instead to stacking the features of encoder blocks
to the decoder block in the skip connections it adds them. UNet++ further
extends UNet by connecting encoder and decoder blocks through a series of
dense Convolutional blocks instead of directly stacking or adding them.

Figure 2: ASPECTS Framework. It has three main components: (1) Infarct
segmentation network (2) MCA anatomy segmentation network. (3) ASPECTS
generation function.

##  3 Dataset

We have built our dataset containing 50000 studies out of which 8000 have
infarcts. Clinicians manually marked pixel-level ground truths for infarcts
and MCA anatomy regions on 1500 training examples. A skilled specialist
double-checked the annotation findings under stringent quality control. The
labelled dataset was separated into three groups at random in training,
validation, and testing at a ratio of 8:1:1. The number of slices in each scan
in our dataset varies from subject to subject. Our collaborating hospitals
have reviewed the data collection process with approval from the local
research ethics committee. Details for data collection process is added to
supplementary.

##  4 Methods

The ASPECT score is calculated by dividing the MCA territory into ten regions:
Caudate, Lentiform Nucleus, Internal Capsule, Insular Cortex, M1, M2, M3 M4
M5, M6.

Figure 3: MCA segmentation model results on our test set. These results are
based on EfficientNet [20], achieving a DSC of 0.64. Masks are shown in color.

M1 to M3 is at the level of the basal ganglia. M4 to M6 are at the level of
the ventricles immediately above the basal ganglia called corona radiata. 1
point is deducted from the initial score of 10 for every region showing early
ischemic signs, such as focal swelling or infarcts. The score was created to
aid in identifying patients who were most likely to benefit clinically from
intravenous thrombolysis.

Figure 2 illustrates the pipeline of the proposed framework. The UNet [17]
segmentation models with EfficientNet backbone are used for both infarct and
anatomy segmentation networks. ASPECTS generation function is used to overlap
segmentation masks from both segmentation networks and report the ASPECT
Score. More qualitative results on segmentation mask can be found in figure 1.
Infarct volume and masked output are also reported to aid the radiologists in
making informed decisions regarding following procedures.

ASPECTS generation function, in figure 2, overlays the masks from 2
segmentation over each other and finds the anatomy being overlapped. These
overlapped regions are considered affected regions and scored for aspects.
This function also finds volume using the ‘pixel spacing‘ attribute, part of
dicom metadata. Pixel spacing represents how much volume is enclosed by a
voxel.

$V(\text{ml})=\sum{\text{infarct}_{mask}}*\mathit{spacing_{x}}*\mathit{spacing_{y}}*\mathit{spacing_{z}}$
(1)

Figure 4: Infarct segmentation model results on our test set. These results
are based on EfficientNet [20], achieving a DSC of 0.72. Infarct masks are
shown in red color.

Model outputs for infarcts and MCA anatomy are visualized in figures 4 and 3.
Models achieve good results even under challenging conditions when infarct is
small and hard to identify. Quantitative results are shown in table 1(b).

All the scans were rescaled to 224x224 and used as input to our framework.
Data augmentation is also applied by introducing random noise, rotation,
shift, and flipping. The models were trained on 1 Nvidia 1080Ti GPU with 12GB
RAM. We used SGD optimizer with a cyclic learning rate scheduler for all
experiments with a learning rate of 5e-3.

$L=\alpha*L_{1}+\beta*L_{2}+\gamma*L_{3}$ (2)

In equation 2, We used 3 loss functions - focal loss [12] ($L_{1}$), boundary
loss [9] ($L_{2}$) and dice loss [19] ($L_{3}$). We found $\alpha=3$,
$\beta=1$ and $\gamma=1$, worked best for both infarct segmentation and MCA
anatomy segmentation task, after grid search on log linear between 1 and 100.
Final evaluation of the acquired data was performed using the dice similarity
coefficient (DSC).

##  5 Results

How does volume of infarct influence the segmentation map?. From table 1(a),
it can be seen that UNet performed better than other models. We also compared
the models based on their ability to segment infarct based on their volume. In
table 1(b), it can be seen that UNet was not the best model for infarct volume
less than 3ml. UNet++ was better by 0.04 DSC. However, for other volume
categories, UNet was still better. Low volume infarct in terms of ASPECTS
translates to a higher ASPECT score i.e. more than 8. In table 3 and 4, it is
evident that the performance is not hampered as even with low dice score.
Though in future work, it gives scope of improvement.

Table 1: Comparison of different models on infarct segmentation.

(a) Model Evaluation on infarct segmentation task. Methods | DSC | Sensitivity | Specificity
---|---|---|---
UNet | 0.72 | 0.77 | 0.99
TransUNet | 0.57 | 0.61 | 0.98
UNet++ | 0.67 | 0.76 | 0.99
LinkNet | 0.54 | 0.57 | 0.96

(b) Infarct volume used to characterise the performance of the models based on DSC. Methods | Infarct Volume
---|---
<3ml | 3-16ml | 16-66ml | >66ml
UNet | 0.45 | 0.72 | 0.78 | 0.91
TransUNet | 0.41 | 0.55 | 0.61 | 0.72
UNet++ | 0.49 | 0.62 | 0.71 | 0.86
LinkNet | 0.31 | 0.54 | 0.52 | 0.79

How do different models segment MCA territory? Table 2 shows that UNet
performed better than other models when we look at the overall DSC of the MCA
territory. Though the smaller regions like Lentiforum nucleus, Internal
Capsule, and Insular Ribbon UNet is not the best model, we can see that the
variation is not very large between UNet and the best models.

Table 2: Comparison of different models on MCA territory segmentation task.The
table contains DSC for different regions and across all regions.

Methods | Overall | Caudate | Lentiform Nucleus | Internal Capsule | Insular Ribbon | M1, M4 | M2, M5 | M3, M6
---|---|---|---|---|---|---|---|---
UNet | 0.64 | 0.70 | 0.60 | 0.59 | 0.52 | 0.72 | 0.77 | 0.54
TransUNet | 0.59 | 0.63 | 0.59 | 0.57 | 0.43 | 0.73 | 0.71 | 0.49
UNet++ | 0.62 | 0.69 | 0.61 | 0.58 | 0.50 | 0.69 | 0.75 | 0.54
LinkNet | 0.63 | 0.70 | 0.61 | 0.61 | 0.53 | 0.71 | 0.77 | 0.49

Model performance for different ASPECT score. In table 3, we can see the
performance of the best performing model from infarct segmentation and anatomy
segmentation tasks. This performance is measured against two readers.

From table 3, it can be observed that our model has low specificity for some
ASPECT scores. However, it is important to notice that specificity is low even
between the score of 2 readers. This is attributed to the subjective nature of
ASPECTS. Therefore, it is best to judge the performance of model-based on
binned ASPECTS, in table 4. The score was binned based on the treatment
outcome given the score. If the score is between 10-8, the treatment outcome
is usually favourable and less complex. The ASPECTS cutoff value determined
for the prediction of unfavourable outcomes was equal to 7 [6]. Patients with
ASPECTS < 4 has even less chance of good functional outcome. The improvement
in specificity is evident from table 4. Here we observe that our agreement
with Reader B is more than Reader A. The measure of variability between these
two readers is discussed in section 6.

Table 3: Performance of trained model, Deep-ASPECTS against 2 readers and
inter-reader difference for each ASPECTS.

ASPECTS | Model vs Reader A | Model vs Reader B | Reader A vs Reader B
---|---|---|---
Sensitivity | Specificity | Sensitivity | Specificity | Sensitivity | Specificity
0 | 0.99 | 0.17 | 0.98 | 0.67 | 0.96 | 0.50
1 | 1.00 | 0.50 | 1.00 | 0.50 | 0.99 | 0.50
2 | 1.00 | 0.20 | 0.99 | 0.20 | 0.99 | 1.00
3 | 0.94 | 0.38 | 0.99 | 0.50 | 0.99 | 0.27
4 | 0.99 | 0.25 | 0.98 | 0.25 | 0.97 | 0.00
5 | 0.97 | 0.33 | 0.96 | 0.83 | 0.94 | 0.33
6 | 0.95 | 0.22 | 0.97 | 0.22 | 0.96 | 0.27
7 | 0.92 | 0.40 | 0.89 | 0.30 | 0.92 | 0.47
8 | 0.90 | 0.20 | 0.92 | 0.30 | 0.92 | 0.25
9 | 0.81 | 0.30 | 0.69 | 0.40 | 0.73 | 0.52
10 | 0.85 | 0.59 | 0.93 | 0.35 | 0.92 | 0.41

Table 4: Performance comparison across binned ASPECT score.

ASPECTS | Model vs Reader A | Model vs Reader B | Reader A vs Reader B
---|---|---|---
Sensitivity | Specificity | Sensitivity | Specificity | Sensitivity | Specificity
A (0-3) | 0.96 | 0.48 | 0.99 | 0.65 | 0.95 | 0.56
B (4-7) | 0.87 | 0.55 | 0.85 | 0.66 | 0.84 | 0.66
C (8-10) | 0.74 | 0.93 | 0.74 | 0.86 | 0.84 | 0.85

Number of parameters for models and their computation time. Table 5 shows that
other that TransUNet, all of the other three models UNet, UNet++ and LinkNet
have a comparable number of parameters since they have the same backbone of
EfficientNet[20]. Despite the fact that UNet has 0.9M more parameters compared
to LinkNet, its performance on infarct and MCA anatomy segmentation was better
than LinkNet. Computational time was also comparable for both models.
Therefore we found UNet to be a better option than LinkNet. This evaluation is
done on 1 Nvidia 1080Ti GPU with 12GB RAM.

Table 5: Total number of parameters for each model and time taken to process each scan. Method | Parameters | Time (seconds)
---|---|---
UNet | 42.1M | 28-30
TransUNet | 105M | 50-63
UNet++ | 42.9M | 28-32
LinkNet | 41.2M | 27-29

##  6 Inter-Reader Variability

Out of 1500 scans, 150 scans were read by two radiologists. This set is the
same as the test set to learn what the acceptable ASPECT score might be for a
scan. In Tables 3 and 4, we compare how much disagreement there is between 2
readers with regards to ASPECTS from 10 and a binned version of score.

ASPECTS is a very subjective score, and it varies from reader to reader. We
found the inter-reader agreement on our data to be 39.45% when we expect an
exact score match. However, the agreement increases to 76.87% when the
difference of 2 points is allowed. Pearson correlation between the reads of 2
readers came out to be 73.17%.

The ASPECTS reported by our model have an agreement of 42.17% and 36.73% with
readers A and B, respectively. If the agreement score is relaxed by 2 points,
then the agreement increases to 69.38% and 76.19%. These numbers suggest that
our deep learning model performed closely with radiologists maintaining
comparable agreement scores.

##  7 Conclusion

We proposed a novel end-to-end system for automated ASPECT scoring.
Additionally, a pilot study was conducted, showing the effectiveness of such
an AI-based model in getting ASPECTS promptly. Deploying a system that can
score an NCCT in less than a minute with reasonable accuracy can aid
preliminary diagnosis for a suspected stroke case. Deep-ASPECTS can save
precious time from the diagnosis phase and prevent further deterioration in
patients’ conditions.

Deep-ASPECTS got an agreement of 76.19% with reader B (radiologists), which is
the same as reader A (refer section 6). The proposed method consistently
performs as good as radiologists. In sum, the method shows a high potential to
improve the clinical success rate by alerting the radiologist or neurologists
about potential stroke cases with their severity reported as ASPECTS.
Furthermore, a more realistic study should be conducted with a bigger sample
size. In future, we intend to do a more thorough architecture search for the
encoders, investigate more anatomical priors, and improve the model
performance even further.

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# Golod and tight 3-manifolds

Kouyemon Iriye  Department of Mathematical Sciences, Osaka Prefecture
University, Sakai, 599-8531, Japan<EMAIL_ADDRESS>and  Daisuke
Kishimoto  Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan
<EMAIL_ADDRESS>
###### Abstract.

The notions Golodness and tightness for simplicial complexes come from algebra
and geometry, respectively. We prove these two notions are equivalent for
3-manifold triangulations, through a topological characterization of a
polyhedral product for a tight-neighborly manifold triangulation of dimension
$\geq 3$.

###### Key words and phrases:

Golod complex, tight triangulation, polyhedral product, fat-wedge filtration

###### 2010 Mathematics Subject Classification:

Primary 57Q15, Secondary 13F55, 55U10

##  1\. Introduction

Let $\mathbb{F}$ be a field, and let $S=\mathbb{F}[x_{1},\ldots,x_{m}]$, where
we assume each $x_{i}$ is of degree 2. Serre [26] proved that for $R=S/I$
where $I$ is a homogeneous ideal of $S$, there is a coefficientwise inequality

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

where $P(V;t)$ denotes the Poincaré series of a graded vector space $V$. In
the extreme case that the equality holds, $R$ is called _Golod_. It was Golod
who proved that $R$ is Golod if and only if all products and (higher) Massey
products in the Koszul homology of $R$ vanish, where the Koszul homology of
$R$ is isomorphic with $\operatorname{Tor}^{S}(R,\mathbb{F})$ as a vector
space.

Let $K$ be a simplicial complex with vertex set $[m]=\\{1,2,\ldots,m\\}$. Let
$\mathbb{F}[K]$ denote the Stanley-Reisner ring of $K$ over $\mathbb{F}$,
where we assume generators of $\mathbb{F}[K]$ are of degree 2. Then
$\mathbb{F}[K]$ expresses combinatorial properties of $K$, and conversely, it
is of particular interest to translate a given algebraic property of the
Stanley-Reisner ring $\mathbb{F}[K]$ into a combinatorial property of $K$. We
say that $K$ is _$\mathbb{F}$ -Golod_ if $\mathbb{F}[K]$ is Golod. We aim to
characterize Golod complexes combinatorially.

Recently, a new approach to a combinatorial characterization of Golod
complexes is taken. We can construct a space $Z_{K}$, called the _moment-angle
complex_ for $K$, in accordance with the combinatorial information of $K$.
Then combinatorial properties are encoded on the topology of $Z_{K}$, and in
particular, Golodness can be read from a homotopical property of $Z_{K}$ as
follows. Baskakov, Buchstaber and Panov [6] proved that the cohomology of
$Z_{K}$ with coefficients in $\mathbb{F}$ is isomorphic with the Koszul
homology of $\mathbb{F}[K]$, where the isomorphism respects products and
(higher) Massey products. Then it follows that $K$ is Golod over any field
whenever $Z_{K}$ is a suspension, and so Golod complexes have been studied
also in connection with desuspension of $Z_{K}$ and a more general _polyhedral
product_ [10, 11, 12, 14, 15, 16, 17, 19, 18]. See a survey [4] for more
information about moment-angle complexes and polyhedral products. Here we
remark that there is a Golod complex $K$ such that $Z_{K}$ is not a suspension
as shown by Yano and the first author [20].

In [15, 17, 19], the authors characterized Golod complexes of dimension one
and two in terms of both combinatorial properties of $K$ and desuspension of
$Z_{K}$. Here we recall a characterization of Golodness of a closed connected
surface triangulation, proved in [15]. Recall that a simplicial complex is
called _neighborly_ if every pair of vertices forms an edge.

######  Theorem 1.1.

Let $S$ be a triangulation of a closed connected $\mathbb{F}$-orientable
surface. Then the following statements are equivalent:

  1. (1)

$S$ is $\mathbb{F}$-Golod;

  2. (2)

$S$ is neighborly;

  3. (3)

$Z_{S}$ is a suspension.

We introduce another notion of a simplicial complex coming from geometry.
S.-S. Chern and R.K. Lashof proved that the total absolute curvature of an
immersion $f\colon M\to\mathbb{R}^{n}$ of a compact manifold $M$ is bounded
below by the Morse number of some Morse function on $M$. On the other hand,
the Morse number is bounded below by the Betti number. Tightness of an
immersion $f$ is defined by the equality between the total absolute curvature
of an immersion $f$ and the Betti number of $M$, which is the case that the
total absolute curvature is minimal and attained by the Betti number. See [22,
23]. It is known that an immersion $f$ is tight if and only if for almost
every closed half-space $H$, the inclusion $f(M)\cap H\to f(M)$ is injective
in homology.

Tightness of a simplicial complex is defined as a combinatorial analog of
tightness of an immersion. See [22] for details. Let $K$ be a simplicial
complex with vertex set $[m]$. For $\emptyset\neq I\subset[m]$, the full
subcomplex of $K$ over $I$ is defined by

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

######  Definition 1.2.

Let $K$ be a connected simplicial complex with vertex set $[m]$. We say that
$K$ is _$\mathbb{F}$ -tight_ if the natural map $H_{*}(K_{I};\mathbb{F})\to
H_{*}(K;\mathbb{F})$ is injective for each $\emptyset\neq I\subset[m]$.

Golodness and tightness have origins in different fields of mathematics,
algebra and geometry, respectively. The aim of this paper is to prove the
seemingly irrelevant these two notions are equivalent for 3-manifold
triangulations through the topology of $Z_{K}$ or a more general _polyhedral
product_ (see Section 5). Now we state the main theorem.

######  Theorem 1.3.

Let $M$ be a triangulation of a closed connected $\mathbb{F}$-orientable
3-manifold. Then the following statements are equivalent:

  1. (1)

$M$ is $\mathbb{F}$-Golod;

  2. (2)

$M$ is $\mathbb{F}$-tight;

  3. (3)

$Z_{M}$ is a suspension.

Recall that a $d$-manifold triangulation is called _stacked_ if it is the
boundary of a $(d+1)$-manifold triangulation whose interior simplices are of
dimension $\geq d$. Stacked manifold triangulations have been studied in
several directions, and we will use its connection to tightness (Section 2).
See [3, 9, 22] and references therein for more on stacked manifold
triangulations. Bagchi, Datta and Spreer [3] (cf. Theorem 2.3) proved that a
closed connected $\mathbb{F}$-orientable 3-manifold triangulation is
$\mathbb{F}$-tight if and only if it is neighborly and stacked. Then we get
the following corollary of Theorem 1.3, which enables us to compare with the
2-dimensional case Theorem 1.1.

######  Corollary 1.4.

Let $M$ be a triangulation of a closed connected $\mathbb{F}$-orientable
3-manifold. Then the following statements are equivalent:

  1. (1)

$M$ is $\mathbb{F}$-Golod;

  2. (2)

$M$ is neighborly and stacked;

  3. (3)

$Z_{M}$ is a suspension.

We will investigate a relation between Golodness and tightness of $d$-manifold
triangulations for $d\geq 3$, not only for $d=3$, through tight-
neighborliness. We will prove the following theorem, where Theorem 1.3 is its
special case $d=3$.

######  Theorem 1.5.

Let $M$ be a triangulation of a closed connected $\mathbb{F}$-orientable
$d$-manifold for $d\geq 3$, and consider the following conditions:

  1. (1)

$M$ is $\mathbb{F}$-Golod;

  2. (2)

$M$ is $\mathbb{F}$-tight;

  3. (3)

$M$ is tight-neighborly;

  4. (4)

the fat-wedge filtration of $\mathbb{R}Z_{M}$ is trivial.

Then there are implications

$(1)\quad\Longrightarrow\quad(2)\quad\Longleftarrow\quad(3)\quad\Longrightarrow\quad(4)\quad\Longrightarrow\quad(1).$

Moreover, for $d=3$, the implication $(2)\,\Rightarrow\,(3)$ also holds, so
all conditions are equivalent.

Remarks on Theorem 1.5 are in order. Tight-neighborly triangulations of
$d$-manifolds for $d\geq 3$ will be defined in Section 2. The space
$\mathbb{R}Z_{K}$ is the real moment-angle complex, and properties of its fat-
wedge filtration will be given in Section 5. In particular, we will see that
if the fat-wedge filtration of $\mathbb{R}Z_{K}$ is trivial, then $Z_{K}$ is a
suspension. So Theorem 1.3 is the special case of Theorem 1.5 for $d=3$ as
mentioned above. Datta and Murai [9] proved that if $M$ is tight-neighborly
and $d\geq 4$, then it is $\mathbb{F}$-tight and $\beta_{i}(M;\mathbb{F})=0$
for $2\leq i\leq d-2$, where $\beta_{i}(M;\mathbb{F})=\dim
H_{i}(M;\mathbb{F})$ denotes the $i$-th Betti number. So if
$\beta_{i}(M;\mathbb{F})=0$ for $2\leq i\leq d-2$ and $d\geq 4$, then all
conditions in Theorem 1.5 are equivalent, where the triviality of the Betti
numbers is necessary because there is a $\mathbb{F}$-tight 9-vertex
triangulation of $\mathbb{C}P^{2}$ for any field $\mathbb{F}$ as in [2,
Example 3.15] which is not tight-neighborly.

The paper is organized as follows. Section 2 collects properties of tight and
tight-neighborly manifold triangulations that will be needed in later
sections. Section 3 introduces a weak version of Golodness and proves the weak
Golodness implies tightness of orientable manifold triangulations. Section 4
investigates a simplicial complex $F(M)$ constructed from a tight-neighborly
$d$-manifold triangulation $M$ for $d\geq 3$, and Section 5 recalls the fat-
wedge filtration technique for polyhedral products which is the main
ingredient in desuspending $Z_{K}$. Section 6 applies the results in Sections
4 and 5 to prove Theorem 1.5. Finally, Section 7 poses two problems for a
further study of a relationship between Golodness and tightness.

Acknowledgement: The first author was supported by JSPS KAKENHI 26400094, and
the second author was supported by JSPS KAKENHI 17K05248.

##  2\. Tightness

This section collects facts about tight and tight-neighborly manifold
triangulations that we will use. As mentioned in Section 1, tightness of a
simplicial complex is a discrete analog of a tight space studied in
differential geometry with connection to minimality of the total absolute
curvature, and tight complexes have been studied mainly for manifold
triangulations. First, we show:

######  Lemma 2.1.

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

###### Proof.

Let $K$ be an $\mathbb{F}$-tight complex. Then for two vertices $v,w$ of $K$,
the natural map $H_{0}(K_{\\{v,w\\}};\mathbb{F})\to H_{0}(K;\mathbb{F})$ is
injective. Since $K$ is connected, $H_{0}(K;\mathbb{F})\cong\mathbb{F}$, and
so $H_{0}(K_{\\{v,w\\}};\mathbb{F})\cong\mathbb{F}$. Then $v$ and $w$ must be
joined by an edge. ∎

Next, we explain a conjecture on tight manifold triangulations. Let $|K|$
denote the geometric realization of $K$, and let
$f(K)=(f_{0}(K),f_{1}(K),\ldots,f_{\dim K}(K))$ denotes the $f$-vector of $K$.
We say that $K$ is _strongly minimal_ if for any simplicial complex $L$ with
$|K|\cong|L|$, it holds that

$f_{i}(K)\leq f_{i}(L)$

for each $i\geq 0$. Kühnel and Lutz [22] conjectured that every
$\mathbb{F}$-tight triangulation of a closed connected manifold is strongly
minimal. Clearly, the only $\mathbb{F}$-tight closed connected 1-manifold
triangulation is the boundary of a 2-simplex, so the conjecture is true in
dimension 1. Moreover, the 2-dimensional case was verified as mentioned in
[22], and the 3-dimensional case was verified by Bagchi, Datta and Spreer [3].
But the case of dimensions $\geq 4$ is still open.

As for minimality of a manifold triangulation, we have another notion
introduced by Lutz, Sulanke and Swartz [24].

######  Definition 2.2.

A closed connected $d$-manifold triangulation $M$ with vertex set $[m]$ for
$d\geq 3$ is _tight-neighborly_ if

$\binom{m-d-1}{2}=\binom{d+2}{2}\beta_{1}(M;\mathbb{F}).$

Tight-neighborly manifold triangulations are known to be vertex minimal. By
definition, tight-neighborliness seems to depend on the ground field
$\mathbb{F}$, but it is actually independent of the ground field $\mathbb{F}$
as tight-neighborly manifold triangulations are neighborly and stacked.
Tightness and tight-neighborliness have the following relation. Let
$S^{1}\widetilde{\times}S^{d-1}$ denote a non-trivial $S^{d-1}$-bundle over
$S^{1}$.

######  Theorem 2.3.

Let $M$ be a closed connected $\mathbb{F}$-orientable $d$-manifold
triangulation for $d\geq 3$, and consider the following conditions:

  1. (1)

$M$ is $\mathbb{F}$-tight;

  2. (2)

$M$ is tight-neighborly;

  3. (3)

$M$ is neighborly and stacked;

  4. (4)

$M$ has the topological type of either

$S^{d},\quad(S^{1}\widetilde{\times}S^{d-1})^{\\#k},\quad(S^{1}\times
S^{d-1})^{\\#k}.$

Then there are implications

$(1)\quad\Longleftarrow\quad(2)\quad\Longleftrightarrow\quad(3)\quad\Longrightarrow\quad(4).$

Moreover, the implication $(1)\,\Rightarrow\,(2)$ also holds for $d=3$.

###### Proof.

The implications are shown in [9] for $d\geq 4$ and [3] for $d=3$. ∎

######  Remark 2.4.

The integer $k$ in Theorem 2.3 for $d=3$ is known to satisfy $80k+1$ is a
perfect square. For $k=1,30,99,208,357,546$, tight-neighborly triangulations
of $(S^{1}\widetilde{\times}S^{2})^{\\#k}$ are constructed in [8], but no
tight-neighborly triangulation of $(S^{1}\times S^{2})^{\\#k}$ is not known.

##  3\. Weak Golodness

This section introduces weak Golodness and study it for manifold
triangulations. Let $K$ be a simplicial complex with vertex set $[m]$, and let
$\mathcal{H}_{*}(\mathbb{F}[K])$ denote the Koszul homology of the Stanley-
Reisner ring $\mathbb{F}[K]$. As mentioned in Section 1, $K$ is
$\mathbb{F}$-Golod if and only if all products and (higher) Massey products in
$\mathcal{H}_{*}(\mathbb{F}[K])$ vanish. Now we define weak Golodness.

######  Definition 3.1.

A simplicial complex $K$ is _weakly $\mathbb{F}$-Golod_ if all products in
$\mathcal{H}_{*}(\mathbb{F}[K])$ vanish.

Clearly, $K$ is weakly $\mathbb{F}$-Golod whenever it is $\mathbb{F}$-Golod.
Berglund and Jöllenbeck [7] stated that Golodness and weak Golodness of every
simplicial complex are equivalent, but this was disproved by Katthän [21].
Then defining weak Golodness makes sense.

We recall a combinatorial description of the multiplication of
$\mathcal{H}_{*}(\mathbb{F}[K])$. For disjoint non-empty subsets
$I,J\subset[m]$, there is an inclusion

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

Baskakov, Buchstaber and Panov [6] proved:

######  Lemma 3.2.

There is an isomorphism of vector spaces

$\mathcal{H}_{i}(\mathbb{F}[K])\cong\bigoplus_{\emptyset\neq
I\subset[m]}\widetilde{H}^{i-|I|-1}(K_{I};\mathbb{F})$

for $i>0$ such that for non-empty subsets $I,J\subset[m]$ the multiplication

$\widetilde{H}^{i-|I|-1}(K_{I};\mathbb{F})\otimes\widetilde{H}^{i-|J|-1}(K_{J};\mathbb{F})\to\widetilde{H}^{i+j-|I\cup
J|-1}(K_{I\cup J};\mathbb{F})$

is trivial for $I\cap J\neq\emptyset$ and the induced map of $\iota_{I,J}$ for
$I\cap J=\emptyset$.

Let $M$ be a triangulation of a closed connected $\mathbb{F}$-oriented
$d$-manifold with vertex set $[m]$. We consider a relation between the
inclusion $\iota_{I,J}$ and the Poincaré duality. For any subset
$I\subset[m]$, the Poincaré duality [13, Proposition 3.46] holds such that the
map

$H^{i}(|M_{I}|;\mathbb{F})\to
H_{d-i}(|M|,|M|-|M_{I}|;\mathbb{F}),\quad\alpha\mapsto\alpha\frown[M]$

is an isomorphism, where $[M]$ denotes the fundamental class of $M$. By [25,
Lemma 70.1], $|M|-|M_{I}|\simeq|M_{J}|$ for $J=[m]-I$. Then there is an
isomorphism

$D_{I,J}\colon
H^{i}(M_{I};\mathbb{F})\xrightarrow{\cong}H_{d-i}(M,M_{J};\mathbb{F}).$

Let $\partial\colon H_{*}(M,M_{J};\mathbb{F})\to H_{*-1}(M_{J};\mathbb{F})$
denote the boundary map of the long exact sequence

$\cdots\to H_{*}(M_{J};\mathbb{F})\to H_{*}(M;\mathbb{F})\to
H_{*}(M,M_{J};\mathbb{F})\xrightarrow{\partial}H_{*-1}(M_{J};\mathbb{F})\to\cdots.$

######  Lemma 3.3.

Let $M$ be a triangulation of a closed connected $\mathbb{F}$-oriented
$d$-manifold with vertex set $[m]$. For any partition $[m]=I\sqcup J$ and
$\alpha\in H^{i}(M_{I};\mathbb{F})$,

$(\partial\circ D_{I,J})(\alpha)=(-1)^{i+1}(\alpha\otimes
1)((\iota_{I,J})_{*}([M]))\in H_{d-i-1}(K_{J};\mathbb{F})$

where we regard $(\iota_{I,J})_{*}([M])$ as an element of
$\bigoplus_{i+j=d-1}H_{i}(M_{I};\mathbb{F})\otimes
H_{j}(M_{J};\mathbb{F})\cong H_{d}(M_{I}*M_{J};\mathbb{F})$.

###### Proof.

Let $\varphi\in C^{i}(M_{I};\mathbb{F})$ be a representative of $\alpha$. We
define $\overline{\varphi}\in C^{i}(M;\mathbb{F})$ by

$\overline{\varphi}(\sigma)=\begin{cases}\varphi(\sigma)&\sigma\in M_{I}\\\
0&\text{otherwise.}\end{cases}$

Then $\alpha\frown[M]$ is represented by $\overline{\varphi}\frown\mu$ where
$\mu$ represents $[M]$. Let $[v_{0},\ldots,v_{i}]$ denote an oriented
$i$-simplex with vertices $v_{0},\ldots,v_{i}$. We may set

$\mu=\sum_{k}a_{k}[v_{0}^{k},v_{1}^{k},\ldots,v_{d}^{k}]\in
C_{d}(M;\mathbb{F})$

for $a_{k}\in\mathbb{F}$, where $v_{0}^{k},\ldots,v_{n_{k}}^{k}\in I$ and
$v_{n_{k}+1}^{k},\ldots,v_{d}^{k}\in J$ for some $n_{k}$. Then $(\partial\circ
D_{I,J})(\alpha)$ is represented by

$\partial(\overline{\varphi}\frown\mu)=(\overline{\varphi}\circ\partial)\frown\mu=[(\overline{\varphi}\circ\partial)\frown\mu]=\sum_{k}a_{k}\overline{\varphi}(\partial[v_{0}^{k},\ldots,v_{i+1}^{k}])[v_{i+1}^{k},\ldots,v_{d}^{k}].$

Since
$(\overline{\varphi}\circ\partial)|_{C_{i+1}(M_{I};\mathbb{F})}=\varphi\circ\partial=0$,
$\overline{\varphi}(\partial[v_{0}^{k},\ldots,v_{i+1}^{k}])\neq 0$ only when
$n_{k}=i$. Then $(\partial\circ D_{I,J})(\alpha)$ is represented by

$\sum_{n_{k}=i}a_{k}\overline{\varphi}(\partial[v_{0}^{k},\ldots,v_{i+1}^{k}])[v_{i+1}^{k},\ldots,v_{d}^{k}]=(-1)^{i+1}\sum_{n_{k}=i}a_{k}\varphi([v_{0}^{k},\ldots,v_{i}^{k},\widehat{v_{i+1}^{k}}])[v_{i+1}^{k},\ldots,v_{d}^{k}]$

On the other hand, since the $C_{i}(M_{I};\mathbb{F})\otimes
C_{d-i-1}(M_{J};\mathbb{F})$ part of $\mu$ is
$\sum_{n_{k}=i}a_{k}[v_{0}^{k},\ldots,v_{d}^{k}]$, $(\iota_{I,J})_{*}([M])$ is
represented by

$\sum_{n_{k}=i}a_{k}[v_{0}^{k},\ldots,v_{i}^{k}]\otimes[v_{i+1}^{k},\ldots,v_{d}^{k}].$

Thus the proof is complete. ∎

Now we are ready to prove:

######  Theorem 3.4.

If a triangulation of a closed connected $\mathbb{F}$-orientable $d$-manifold
is weakly $\mathbb{F}$-Golod, then it is $\mathbb{F}$-tight.

###### Proof.

Let $M$ be a triangulation of a closed connected $\mathbb{F}$-oriented
$d$-manifold with vertex set $[m]$. Let $[m]=I\sqcup J$ be a partition.
Suppose that the map $\iota_{I,J}$ is trivial in cohomology with coefficients
in $\mathbb{F}$. Then by the universal coefficient theorem, $\iota_{I,J}$ is
trivial in homology with coefficients in $\mathbb{F}$ too. Thus by Lemma 3.3
the boundary map $\partial\colon H_{*}(M,M_{J};\mathbb{F})\to
H_{*-1}(M_{J};\mathbb{F})$ is trivial, and so the natural map
$H_{*}(M_{J};\mathbb{F})\to H_{*}(M;\mathbb{F})$ is injective, completing the
proof. ∎

##  4\. The complex $F(M)$

Throughout this section, let $M$ be a closed connected tight-neighborly
$d$-manifold triangulation for $d\geq 3$ with vertex set $[m]$. Let $K$ be a
simplicial complex with vertex set $[m]$. A subset $I\subset[m]$ is a _minimal
non-face_ of $K$ if every proper subset of $I$ is a simplex of $K$ and $I$
itself is not a simplex of $K$. Define a simplicial complex $F(M)$ by filling
all minimal non-faces of cardinality $d+1$ into $M$. This section investigates
the complex $F(M)$.

We set notation. Let $K$ be a simplicial complex with vertex set $[m]$. The
link of a vertex $v$ in a simplicial complex $K$ is defined by

$\operatorname{lk}_{K}(v)=\\{\sigma\in K\mid v\not\in\sigma\text{ and
}\sigma\sqcup v\in K\\}.$

For a finite set $S$, let $\Delta(S)$ denote the simplex with vertex set $S$.
Then $I\subset[m]$ is a minimal non-face of $K$ if and only if
$K_{I}=\partial\Delta(I)$. Let $K_{1},K_{2}$ be simplicial complexes of
dimension $d$ such that $K_{1}\cap K_{2}$ is a single $d$-simplex $\sigma$.
Then we write

$K_{1}\\#K_{2}=K_{1}\cup K_{2}-\sigma\quad\text{and}\quad K_{1}\circ
K_{2}=K_{1}\cup K_{2}.$

######  Lemma 4.1.

For each $v\in[m]$, there are $V(v,1),\ldots,V(v,n_{v})\subset[m]$ such that
$|V(v,k)|=d+1$ for $1\leq k\leq n_{v}$ and

$\operatorname{lk}_{M}(v)=\partial\Delta(V(v,1))\\#\cdots\\#\partial\Delta(V(v,n_{v})).$

###### Proof.

The case $d=3$ is proved [3, Proof of Theorem 1.2]. For $d\geq 4$, tight-
neighborliness implies local stackedness, that is, every vertex link is a
stacked sphere, as in [9]. Moreover, stacked spheres are characterized by
Bagchi and Datta [1] such that every stacked $(d-1)$-sphere is of the form
$\partial\Delta^{d}\\#\cdots\\#\partial\Delta^{d}$. Then we obtain the result
for $d\geq 4$. ∎

Generalizing neighborliness, we say that a simplicial complex is _$k$
-neighborly_ if every $k+1$ vertices form a simplex. So 1-neighborliness is
precisely neighborliness.

######  Lemma 4.2.

For each $v\in[m]$ and $1\leq k\leq n_{v}$, $M_{V(v,k)\sqcup v}$ is
$(d-1)$-neighborly.

###### Proof.

By Lemma 4.1, $\operatorname{lk}_{M}(v)_{V(v,k)}$ is $\partial\Delta^{d}$
removed some $(d-1)$-simplices, implying it is $(d-2)$-neighborly. So if $I$
is a subset of $V(v,k)$ with $|I|=d-1$, then $I\sqcup v$ is a simplex of $M$.
It remains to show $M_{V(v,k)}$ is $(d-1)$-neighborly. Let $J$ be any subset
of $V(v,k)$ with $|J|=d$. Then $\partial\Delta(J)$ is a subcomplex of $M$. If
$M_{J}=\partial\Delta(J)$, then $M_{J\sqcup v}=\partial\Delta(J)*v$, which is
contractible. So the inclusion $M_{J}\to M_{J\sqcup v}$ is not injective in
homology with coefficients in $\mathbb{F}$. By Theorem 2.3, $M$ is
$\mathbb{F}$-tight, so we get a contradiction. Thus $J$ must be a simplex of
$M$, completing the proof. ∎

We prove local properties of the complex $F(M)$.

######  Proposition 4.3.

  1. (1)

For each $v\in[m]$,

$\operatorname{lk}_{F(M)}(v)=\partial\Delta(V(v,1))\circ\cdots\circ\partial\Delta(V(v,n_{v})).$

  2. (2)

For each $v\in[m]$ and $1\leq k\leq n_{v}$, $V(v,k)\sqcup v$ is a minimal non-
face of $F(M)$.

###### Proof.

(1) Let $\sigma$ be the $(d-1)$-simplex
$(\partial\Delta(V(v,1))\\#\cdots\\#\partial\Delta(V(v,k)))\cap\partial\Delta(V(v,k+1))$.
Then by Lemma 4.2, $\partial\Delta(\sigma\sqcup v)$ is a subcomplex of $M$,
implying $\sigma\sqcup v$ is a simplex of $F(M)$. Then by induction, we get
$\partial\Delta(V(v,1))\circ\cdots\circ\partial\Delta(V(v,n_{v}))\subset\operatorname{lk}_{F(M)}(v)$.
The reverse inclusion is obvious by the construction of $F(M)$, completing the
proof.

(2) By Lemma 4.2, $V(v,k)$ is a simplex of $F(M)$, so every proper subset $I$
of $V(v,k)\sqcup v$ is a simplex of $F(M)$. By (1), $V(v,k)\sqcup v$ is not a
simplex of $F(M)$. Then the statement is proved. ∎

We compute the homology of $F(M)$. Let

$S(M)=\\{V(v,k)\sqcup v\mid v\in[m]\text{ and }1\leq k\leq n_{v}\\}.$

Then $S(M)$ is the set of all subsets $I\subset[m]$ such that $|I|=d+2$ and
$\operatorname{lk}_{M_{I}}(v)$ is $(d-2)$-neighborly for some $v\in I$.

######  Lemma 4.4.

$F(M)=\bigcup_{I\in S(M)}\partial\Delta(I)$.

###### Proof.

Let $K=\bigcup_{I\in S(M)}\partial\Delta(I)$. By Proposition 4.3, $K\subset
F(M)$. For any $k$-simplex $\sigma$ of $F(M)$ with $0\leq k\leq d-1$ and
$v\in\sigma$, $\sigma-v$ is a simplex of $\operatorname{lk}_{M}(v)$ because
$\sigma$ is a simplex of $M$ too. Then $\sigma-v\subset V(v,l)$ for some
$1\leq l\leq n_{v}$, implying $\sigma$ is a simplex of $K$. Thus the
$(d-1)$-skeleton of $F(M)$ is included in $K$. Take any $d$-simplex $\sigma$
of $F(M)$. Then $\sigma$ is either a simplex or a minimal non-face of $M$. In
both cases, $\partial\Delta(\sigma-v)$ is a subcomplex of
$\operatorname{lk}_{M}(v)$ for $v\in\sigma$. Then $\sigma-v\subset V(v,l)$ for
some $1\leq l\leq n_{v}$, implying $\sigma$ is a simplex of $K$. Thus
$F(M)\subset K$, completing the proof. ∎

We compute the homology of $F(M)$. By Lemma 4.4, there is an inclusion
$g_{I}\colon\partial\Delta(I)\to F(M)$ for each $I\in S(M)$. Let $u_{I}\in
H_{d}(F(M);\mathbb{Z})$ be the Hurewicz image of $g_{I}$.

######  Proposition 4.5.

The integral homology of $F(M)$, except for dimension 1, is given by

$\widetilde{H}_{*}(F(M);\mathbb{Z})=\begin{cases}\mathbb{Z}\langle u_{I}\mid
I\in S(M)\rangle&*=d\\\ 0&*\neq 1,d.\end{cases}$

###### Proof.

Since $F(M)$ is obtained from $M$ by attaching $d$-simplices, we only need to
calculate $H_{d-1}$ and $H_{d}$ by Theorem 2.3. By Lemma 4.1, each component
of $\operatorname{lk}_{M_{I}}(v)$ is $(d-2)$-connected, where
$\operatorname{lk}_{M_{I}}(v)=\operatorname{lk}_{M}(v)_{I-v}$. Then there is
an exact sequence

$0\to\widetilde{H}_{d}(F(M)_{I-v};\mathbb{Z})\to
H_{d}(F(M)_{I};\mathbb{Z})\xrightarrow{\partial}H_{d-1}(\operatorname{lk}_{F(M)_{I}}(v);\mathbb{Z})\\\
\to H_{d-1}(F(M)_{I-v};\mathbb{Z})\to H_{d-1}(F(M)_{I};\mathbb{Z})\to 0.$

By Proposition 4.3, there is an inclusion
$\partial\Delta(V(v,k))\to\operatorname{lk}_{F(M)_{I}}(v)$ for $V(v,k)\sqcup
v\subset I$, and we write the Hurewicz image of this inclusion by
$\bar{u}_{V(v,k)}$. By Proposition 4.3,

$H_{d-1}(\operatorname{lk}_{F(M)_{I}}(v);\mathbb{Z})=\mathbb{Z}\langle\bar{u}_{V(v,k)}\mid
V(v,k)\sqcup v\subset I\rangle$

such that $\partial(u_{V(v,k)\sqcup v})=\bar{u}_{V(v,k)}$. Then $\partial$ is
surjective, so we get an isomorphism

$H_{d-1}(F(M)_{I-v};\mathbb{Z})\cong H_{d-1}(F(M)_{I};\mathbb{Z}).$

Thus we obtain $H_{d-1}(F(M)_{I};\mathbb{Z})=0$ for any $I\subset[m]$ by
induction on $|I|$, where $H_{d-1}(F(M)_{I};\mathbb{Z})=0$ for $|I|=1$. We
also get a split exact sequence

$0\to H_{d}(F(M)_{I-v};\mathbb{Z})\to
H_{d}(F(M)_{I};\mathbb{Z})\xrightarrow{\partial}H_{d-1}(\operatorname{lk}_{F(M)_{I}}(v);\mathbb{Z})\to
0.$

Then by induction on $|I|$, we also obtain

$H_{d}(F(M)_{I};\mathbb{Z})=\mathbb{Z}\langle u_{V(v,k)}\mid V(v,k)\sqcup
v\subset I\rangle.$

Thus the proof is complete. ∎

By Theorem 2.3, $\pi_{1}(|M|)$ is a free group. Since $|F(M)|$ is obtained by
attaching $d$-cells to $|M|$, the inclusion $|M|\to|F(M)|$ is an isomorphism
in $\pi_{1}$, so $\pi_{1}(|F(M)|)$ is a free group too. Then there is a map
$f\colon B\to|F(M)|$ which is an isomorphism in $\pi_{1}$, where $B$ is a
wedge of circles. Let $\widehat{F}(M)$ be the cofiber of $f$. Since there is
an exact sequence

$\cdots\to
H_{*}(B;\mathbb{Z})\xrightarrow{f_{*}}H_{*}(F(M);\mathbb{Z})\to\widetilde{H}_{*}(\widehat{F}(M);\mathbb{Z})\to\cdots$

the natural map $H_{*}(F(M);\mathbb{Z})\to H_{*}(\widehat{F}(M);\mathbb{Z})$
is an isomorphism for $*\neq 1$. Let $\hat{g}_{I}$ be the composite
$|\partial\Delta(I)|\xrightarrow{g_{I}}|F(M)|\to\widehat{F}(M)$ for $I\in
S(M)$, and let $\hat{u}_{I}$ be the Hurewicz image of $\hat{g}_{I}$. By
Proposition 4.5, we get:

######  Corollary 4.6.

The integral homology of $\widehat{F}(M)$ is given by

$\widetilde{H}_{*}(\widehat{F}(M);\mathbb{Z})=\begin{cases}\mathbb{Z}\langle\hat{u}_{I}\mid
I\in S(M)\rangle&*=d\\\ 0&*\neq d.\end{cases}$

Since $\widehat{F}(M)$ is path-connected, there is a map

$g\colon\bigvee_{I\in S(M)}|\partial\Delta(I)|\to\widehat{F}(M)$

such that $g|_{|\partial\Delta(I)|}\simeq\hat{g}_{I}$ for each $I\in S(M)$.
Then by Corollary 4.6 and the J.H.C. Whitehead theorem, we obtain the
following.

######  Corollary 4.7.

The map $g\colon\bigvee_{I\in S(M)}|\partial\Delta(I)|\to\widehat{F}(M)$ is a
homotopy equivalence.

##  5\. Polyhedral product

Throughout this section, let $K$ be a simplicial complex with vertex set
$[m]$. Let $(\underline{X},\underline{A})=\\{(X_{i},A_{i})\\}_{i=1}^{m}$ be a
collection of pairs of pointed spaces indexed by vertices of $K$. For
$I\subset[m]$, let

$(\underline{X},\underline{A})^{I}=Y_{1}\times\cdots\times Y_{m}$

where $Y_{i}=X_{i}$ for $i\in I$ and $Y_{i}=A_{i}$ for $i\not\in I$. The
_polyhedral product_ of $(\underline{X},\underline{A})$ over $K$ is defined by

$Z_{K}(\underline{X},\underline{A})=\bigcup_{\sigma\in
K}(\underline{X},\underline{A})^{\sigma}.$

For $\emptyset\neq I\subset[m]$, let
$(\underline{X}_{I},\underline{A}_{I})=\\{(X_{i},A_{i})\\}_{i\in I}$. Then we
can define $Z_{K_{I}}(\underline{X}_{I},\underline{A}_{I})$. The following
lemma is immediate from the definition of a polyhedral product.

######  Lemma 5.1.

For each $\emptyset\neq I\subset[m]$,
$Z_{K_{I}}(\underline{X}_{I},\underline{A}_{I})$ is a retract of
$Z_{K}(\underline{X},\underline{A})$.

For a collection of pointed spaces $\underline{X}=\\{X_{i}\\}_{i=1}^{m}$, let
$(C\underline{X},\underline{X})=\\{(CX_{i},X_{i})\\}_{i=1}^{m}$. For $0\leq
i\leq m$, we define a subspace of $Z_{K}(C\underline{X},\underline{X})$ by

$Z_{K}^{i}(C\underline{X},\underline{X})=\\{(x_{1},\ldots,x_{m})\in
Z_{K}(C\underline{X},\underline{X})\mid\text{at least }m-i\text{ of
}x_{1},\ldots,x_{m}\text{ are basepoints}\\}.$

Using the basepoint of each $X_{i}$, we regard
$Z_{K_{I}}(C\underline{X}_{I},\underline{X}_{I})$ as a subspace of
$Z_{K}(C\underline{X},\underline{X})$ so that we can alternatively write

(5.1)
$Z_{K}^{i}(C\underline{X},\underline{X})=\bigcup_{I\subset[m],\,|I|=i}Z_{K_{I}}(C\underline{X}_{I},\underline{X}_{I}).$

There is a filtration

$*=Z_{K}^{0}(C\underline{X},\underline{X})\subset
Z_{K}^{1}(C\underline{X},\underline{X})\subset\cdots\subset
Z_{K}^{m}(C\underline{X},\underline{X})=Z_{K}(C\underline{X},\underline{X})$

which we call the _fat-wedge filtration_ of
$Z_{K}(C\underline{X},\underline{X})$. By [17, Theorem 4.1], we have

$Z_{K}^{i}(C\underline{X},\underline{X})/Z_{K}^{i-1}(C\underline{X},\underline{X})=\bigvee_{I\subset[m],\,|I|=i}|\Sigma
K_{I}|\wedge\widehat{X}^{I}$

where $\widehat{X}^{I}=\bigwedge_{i\in I}X_{i}$. Moreover, it is shown in [17,
Corollary 4.2] that the fat-wedge filtration of
$Z_{K}(C\underline{X},\underline{X})$ splits after a suspension and the
decomposition of Bahri, Bendersky, Cohen and Gitler [5, Theorem 2.2.1] is
reproduced as:

######  Theorem 5.2 (BBCG decomposition).

There is a homotopy equivalence

$\Sigma Z_{K}(C\underline{X},\underline{X})\simeq\Sigma\bigvee_{\emptyset\neq
I\subset[m]}|\Sigma K_{I}|\wedge\widehat{X}^{I}.$

In particular, if the BBCG decomposition desuspends, then
$Z_{K}(C\underline{X},\underline{X})$ itself desuspends. Moreover, if each
$X_{i}$ is a connected CW complex, then the BBCG decomposition desuspends
whenever $Z_{K}(C\underline{X},\underline{X})$ desuspends [17]. Then we aim to
desuspend the BBCG decomposition. Desuspension of the BBCG decomposition was
studied for specific Golod complexes such as shifted complexes [14, 11, 12],
and desuspension for much broader classes of simplicial complexes, including
the previous specific simplicial complexes, can be reproved by using the fat-
wedge filtration technique [17].

The moment-angle complex $Z_{K}$ introduced in Section 1 is the polyhedral
product $Z_{K}(D^{2},S^{1})$. The _real moment-angle complex_
$\mathbb{R}Z_{K}$ is defined to be the polyhedral product
$Z_{K}(D^{1},S^{0})$, and we denote its fat-wedge filtration by

$*=\mathbb{R}Z^{0}_{K}\subset\mathbb{R}Z^{1}_{K}\subset\cdots\subset\mathbb{R}Z^{m}_{K}=\mathbb{R}Z_{K}$

where we choose the basepoint of $S^{0}=\\{-1,+1\\}$ to be $-1$. The fat-wedge
filtration of $\mathbb{R}Z_{K}$ is proved to be a cone decomposition [17,
Theorem 3.1]. For $\emptyset\neq I\subset[m]$, let
$j_{K_{I}}\colon\mathbb{R}Z_{K_{I}}^{|I|-1}\to\mathbb{R}Z_{K}^{|I|-1}$ denote
the inclusion.

######  Theorem 5.3.

For each $\emptyset\neq I\subset[m]$, there is a map
$\varphi_{K_{I}}\colon|K_{I}|\to\mathbb{R}Z^{|I|-1}_{K_{I}}$ such that

$\mathbb{R}Z_{K}^{i}=\mathbb{R}Z_{K}^{i-1}\bigcup_{I\subset[m],\,|I|=i}C|K_{I}|$

where the attaching maps are $j_{K_{I}}\circ\varphi_{K_{I}}$.

We say that the fat-wedge filtration of $\mathbb{R}Z_{K}$ is trivial if
$\varphi_{K_{I}}$ is null homotopic for each $\emptyset\neq I\subset[m]$. We
remark that $\varphi_{K_{I}}$ is null homotopic if and only if so is
$j_{K_{I}}\circ\varphi_{K_{I}}$ because $\mathbb{R}Z_{K_{I}}^{|I|-1}$ is a
retract of $\mathbb{R}Z_{K}^{|I|-1}$. The fat-wedge filtration is useful for
desuspending the BBCG decomposition because we have the following criterion
[17, Theorem 1.2].

######  Theorem 5.4.

If the fat-wedge filtration of $\mathbb{R}Z_{K}$ is trivial then for any
$\underline{X}$,

$Z_{K}(C\underline{X},\underline{X})\simeq\bigvee_{\emptyset\neq
I\subset[m]}|\Sigma K_{I}|\wedge\widehat{X}^{I}.$

For $\emptyset\neq I\subset[m]$, define a map
$\alpha_{I}\colon\mathbb{R}Z_{K_{I}}^{|I|-1}\to\mathbb{R}Z_{K}^{m-1}$ by
$\alpha_{I}(x_{i}\mid i\in I)=(y_{1},\ldots,y_{m})$ such that

$y_{i}=\begin{cases}x_{i}&i\in I\\\ +1&i\not\in I\end{cases}$

for $(x_{i}\mid i\in I)\in\mathbb{R}Z_{K_{I}}^{|I|-1}$. Note that $\alpha_{I}$
is not the natural inclusion because the basepoint of $S^{0}=\\{-1,+1\\}$ is
taken to be $-1$ as mentioned above. For $\emptyset\neq J\subset I\subset[m]$
and $|J|\leq i\leq|I|$, let $\pi$ denote the composite of projections

$\mathbb{R}Z_{K_{I}}^{i}\to\mathbb{R}Z_{K_{J}}\to\mathbb{R}Z_{K_{J}}/\mathbb{R}Z_{K_{J}}^{|J|-1}=|\Sigma
K_{J}|.$

By the construction of $\varphi_{K}$, we have:

######  Lemma 5.5.

For $\emptyset\neq J\subsetneq I\subset[m]$, there is a commutative diagram

$\textstyle{|K_{I}|\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{K_{I}}}$$\textstyle{\mathbb{R}Z_{K_{I}}^{|I|-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\alpha_{I}}$$\textstyle{|\Sigma
K_{J}|\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{|\Sigma
j|}$$\textstyle{|K|\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{K}}$$\textstyle{\mathbb{R}Z_{K}^{m-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{|\Sigma
K_{J\sqcup([m]-I)}|}$

where $j\colon K_{J}\to K_{J\sqcup([m]-I)}$ is the inclusion.

The following two lemmas, proved in [17, Proof of Theorem 7.2] and [17, Lemma
10.1] respectively, are quite useful in detecting the triviality of
$\varphi_{K}$.

######  Lemma 5.6.

Let $\overline{K}$ be a simplicial complex obtained by filling all minimal
non-faces into $K$. Then $\varphi_{K}$ factors through the inclusion
$|K|\to|\overline{K}|$.

######  Lemma 5.7.

If $\varphi_{K_{I}}\simeq*$ for each $\emptyset\neq I\subsetneq[m]$, then the
composite

$|K|\xrightarrow{\varphi_{K}}\mathbb{R}Z_{K}^{m-1}\to\mathbb{R}Z_{K_{J}}\xrightarrow{\pi}|\Sigma
K_{J}|$

is null-homotopic for each $\emptyset\neq J\subsetneq[m]$.

Finally, we estimate the connectivity of $\mathbb{R}Z_{K}$.

######  Lemma 5.8.

If $K$ is $k$-neighborly, then $\mathbb{R}Z_{K}$ is $k$-connected.

###### Proof.

The proof is done by the same calculation as [17, Proposition 5.3]. The proof
is alternatively done as follows. By definition, $\pi_{*}(\mathbb{R}Z_{K})$ is
isomorphic to $\pi_{*}(\mathbb{R}Z_{K_{k}})$ for $*\leq k$, where $K_{k}$
denotes the $k$-skeleton of $K$. Since $K$ is $k$-neighborly,
$K_{k}=\Delta^{m-1}_{k}$. Since $\Delta^{m-1}_{k}$ is shifted, it follows from
[14] that there is a homotopy equivalence

$\mathbb{R}Z_{\Delta_{k}^{m-1}}\simeq\bigvee_{\emptyset\neq
I\subset[m]}|\Sigma(\Delta^{m-1}_{k})_{I}|.$

Since each $|\Sigma(\Delta^{m-1}_{k})_{I}|$ is $k$-connected, the proof is
done. ∎

##  6\. Proof of Theorem 1.5

Throughout this section, let $M$ be a tight-neighborly triangulation of a
closed connected $\mathbb{F}$-orientable $d$-manifold with vertex set $[m]$
unless otherwise is specified. We aim to prove that the fat-wedge filtration
of $\mathbb{R}Z_{M}$ is trivial. First, we compute the fundamental group of
$|F(M)_{I}|$ for $\emptyset\neq I\subset[m]$.

######  Lemma 6.1.

For each $\emptyset\neq I\subset[m]$, $\pi_{1}(|F(M)_{I}|)$ is a free group.

###### Proof.

Since the fundamental group of a suspension is a free group, we prove
$|F(M)_{I}|$ is a suspension by induction on $I$. For $|I|=1$, $|F(M)_{I}|$ is
obviously a suspension. Suppose that $|F(M)_{I-v}|$ is a suspension for $v\in
I$. Note that

(6.1) $F(M)_{I}=F(M)_{I-v}\cup(\operatorname{lk}_{F(M)_{I}}(v)*v)$

where
$F(M)_{I-v}\cap(\operatorname{lk}_{F(M)_{I}}(v)*v)=\operatorname{lk}_{F(M)_{I}}(v)$.
Since $\operatorname{lk}_{F(M)_{I}}(v)=\operatorname{lk}_{F(M)}(v)_{I-v}$, it
follows from Proposition 4.3 that there are inclusions

$\operatorname{lk}_{F(M)_{I}}(v)\subset(\Delta(V(v,1))\circ\cdots\circ\Delta(V(v,n_{v})))_{I-v}\subset
F(M)_{I-v}.$

Since $M$ is neighborly by Theorem 2.3, so is $M_{I-v}$, implying $F(M)_{I-v}$
is connected. On the other hand, each component of
$(\Delta(V(v,1))\circ\cdots\circ\Delta(V(v,n_{v})))_{I-v}$ is contractible.
Then the inclusion
$|(\Delta(V(v,1))\circ\cdots\circ\Delta(V(v,n_{v})))_{I-v}|\to|F(M)_{I-v}|$ is
null-homotopic, and so the inclusion
$|\operatorname{lk}_{F(M)_{I}}(v)|\to|F(M)_{I-v}|$ is null-homotopic too. Thus
by (6.1), we get a homotopy equivalence

$|F(M)_{I}|\simeq|F(M)_{I-v}|\vee|\Sigma\operatorname{lk}_{F(M)_{I}}(v)|.$

Since $|F(M)_{I-v}|$ is a suspension by the induction hypothesis, $|F(M)_{I}|$
turns out to be a suspension, completing the proof. ∎

Let $\emptyset\neq I\subset[m]$. By Lemma 5.6, the map $\varphi_{M_{I}}$
decomposes as

(6.2) $|M_{I}|\to|F(M)_{I}|\to\mathbb{R}Z_{M_{I}}^{|I|-1}.$

By Lemma 6.1, there is a map $f_{I}\colon B_{I}\to|F(M)_{I}|$, where $B_{I}$
is a wedge of circles, such that $f_{I}$ is an isomorphism in $\pi_{1}$. Let
$\widehat{F}(M)_{I}$ denote the cofiber of $f_{I}$, where
$\widehat{F}(M)_{[m]}$ coincides with $\widehat{F}(M)$ in Section 4. On the
other hand, since $M$ is neighborly by Lemma 2.1, so is $M_{J}$ for any
$\emptyset\neq J\subset[m]$. Then by (5.1) and Lemma 5.8, we can see that
$\mathbb{R}Z_{M_{I}}^{|I|-1}$ is simply-connected. In particular, there is a
commutative diagram

(6.3)
$\textstyle{|F(M)_{I}|\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{F}(M)_{I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{R}Z_{M_{I}}^{|I|-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{R}Z_{M_{I}}^{|I|-1}.}$

Then by combining (6.2) and (6.3), we get:

######  Lemma 6.2.

For each $\emptyset\neq I\subset[m]$, the map $\varphi_{M_{I}}$ factors
through the inclusion $|M_{I}|\to\widehat{F}(M)_{I}$.

######  Proposition 6.3.

For each $\emptyset\neq I\subsetneq[m]$, the map $\varphi_{M_{I}}$ is null-
homotopic.

###### Proof.

As is computed in the proof of Proposition 4.5,
$\widetilde{H}_{*}(F(M)_{I};\mathbb{Z})=0$ unless $*=1,d$. Thus as well as
$\widehat{F}(M)$, we can see that $\widehat{F}(M)_{I}$ is $(d-1)$-connected.
Since $I\neq[m]$, $|M_{I}|$ is homotopy equivalent to a CW complex of
dimension $\leq d-1$. Then we obtain that the inclusion
$|M_{I}|\to\widehat{F}(M)_{I}$ is null-homotopic. Thus by Lemma 6.2, the proof
is complete. ∎

It remains to show $\varphi_{M}$ is null-homotopic. By Lemma 5.5, there is a
commutative diagram

$\textstyle{\bigvee_{I\in
S(M)}|M_{I}|\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigvee_{I\in
S(M)}\varphi_{M_{I}}}$$\textstyle{|M|\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{M}}$$\textstyle{\bigvee_{I\in
S(M)}\mathbb{R}Z_{M_{I}}^{d+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigvee_{I\in
S(M)}\alpha_{I}}$$\textstyle{\mathbb{R}Z_{M}^{m-1}.}$

Then since $F(M)_{I}=\partial\Delta(I)$ for $I\in S(M)$ by Proposition 4.3, we
get a commutative diagram

(6.4) $\textstyle{\bigvee_{I\in
S(M)}|\partial\Delta(I)|\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigvee_{I\in
S(M)}g_{I}}$$\textstyle{|F(M)|\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigvee_{I\in
S(M)}\mathbb{R}Z_{M_{I}}^{d+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigvee_{I\in
S(M)}\alpha_{I}}$$\textstyle{\mathbb{R}Z_{M}^{m-1}.}$

Juxtaposing the commutative diagrams (6.3) and (6.4), we get a commutative
diagram

$\textstyle{\bigvee_{I\in
S(M)}|\partial\Delta(I)|\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\widehat{F}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigvee_{I\in
S(M)}\mathbb{R}Z_{M_{I}}^{d+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigvee_{I\in
S(M)}\alpha_{I}}$$\textstyle{\mathbb{R}Z_{M}^{m-1}}$

and by Corollary 4.7 and Lemma 6.2, we obtain:

######  Lemma 6.4.

The map $\varphi_{M}\colon|M|\to\mathbb{R}Z_{M}^{m-1}$ is homotopic to the
composite

$|M|\to\widehat{F}(M)\xrightarrow{g^{-1}}\bigvee_{I\in
S(M)}|\partial\Delta(I)|\to\bigvee_{I\in
S(M)}\mathbb{R}Z_{M_{I}}^{d+1}\xrightarrow{\bigvee_{I\in
S(M)}\alpha_{I}}\mathbb{R}Z_{M}^{m-1}.$

We will investigate the composite of maps in Lemma 6.4 by identifying a
homotopy set with a homology.

######  Lemma 6.5.

Let $W$ be a finite wedge of $S^{d}$. Then there is an isomorphism of sets

$[|M|,W]\cong H^{d}(M;\mathbb{Z})\otimes H_{d}(W;\mathbb{Z})$

which is natural with respect to maps among finite wedges of $S^{d}$.

###### Proof.

Since $\dim M=d$, the statement follows from the Hopf degree theorem. ∎

######  Lemma 6.6.

For each $v\in I\in S(M)$, the natural map

$H^{d}(M;\mathbb{Z})\otimes H_{d-1}(M_{I-v};\mathbb{Z})\to
H^{d}(M;\mathbb{Z})\otimes H_{d-1}(M_{[m]-v};\mathbb{Z})$

is injective.

###### Proof.

By Lemma 4.2, $|M_{I-v}|$ is contractible or $S^{d-1}$, so in particular,
$H_{d-1}(M_{I-v};\mathbb{Z})$ is a free abelian group, and so there is a
natural isomorphism

(6.5) $H_{d-1}(M_{I-v};\mathbb{F})\cong
H_{d-1}(M_{I-v};\mathbb{Z})\otimes\mathbb{F}.$

By definition, $M_{[m]-v}$ is $M$ removed the open star of $v$, which is
homotopy equivalent to $M$ removed the vertex $v$ by [25, Lemma 70.1]. Then by
Theorem 2.3, $|M_{[m]-v}|$ is homotopy equivalent to a wedge of finitely many,
possibly zero, copies of $S^{1}$ and $S^{d-1}$. Then
$H_{*}(M_{[m]-v};\mathbb{Z})$ is a free abelian group, and so there is a
natural isomorphism

(6.6) $H_{d-1}(M_{[m]-v};\mathbb{F})\cong
H_{d-1}(M_{[m]-v};\mathbb{Z})\otimes\mathbb{F}.$

Since $M$ is $\mathbb{F}$-tight by Theorem 2.3, the natural map
$H_{d-1}(M_{I-v};\mathbb{F})\to H_{d-1}(M_{[m]-v};\mathbb{F})$ is injective.
Then by (6.5) and (6.6), the natural map

$H_{d-1}(M_{I-v};\mathbb{Z})\otimes\mathbb{F}\to
H_{d-1}(M_{[m]-v};\mathbb{Z})\otimes\mathbb{F}$

is injective too. Since both $H_{d-1}(M_{I-v};\mathbb{Z})$ and
$H_{d-1}(M_{[m]-v};\mathbb{Z})$ are free abelian group, the case that $M$ is
orientable is proved because $H^{d}(M;\mathbb{Z})\cong\mathbb{Z}$. If $M$ is
unorientable, then $H^{d}(M;\mathbb{Z})\cong\mathbb{F}_{2}$ and the base field
$\mathbb{F}$ is of characteristic 2, where $\mathbb{F}_{2}$ is the field of
two elements. Thus the case that $M$ is unorientable is proved too, completing
the proof. ∎

######  Proposition 6.7.

The map $\varphi_{M}\colon|M|\to\mathbb{R}Z_{M}^{m-1}$ is null-homotopic.

###### Proof.

Note that $m\geq d+2$. Let $\emptyset\neq J\subset I\in S(M)$. By Lemma 4.2,
$|M_{J}|$ is contractible for $|J|\leq d$, and $|M_{J}|$ is contractible or
$S^{d-1}$ for $|J|=d+1$. Then by Proposition 6.3, there is a homotopy
equivalence

(6.7) $\mathbb{R}Z_{M_{I}}^{d+1}\simeq\bigvee_{v\in I}|\Sigma M_{I-v}|$

where $|\Sigma M_{I-v}|$ is contractible or $S^{d}$ as mentioned above. Let

$A=\bigvee_{I\in S(M)}\bigvee_{v\in I}|\Sigma M_{I-v}|\quad\text{and}\quad
B=\bigvee_{I\in S(M)}\bigvee_{v\in I}|\Sigma M_{[m]-v}|$

where $A\simeq\bigvee_{I\in S(M)}\mathbb{R}Z_{M_{I}}^{d+1}$ by (6.7). Let
$f\colon|M|\to A$ denote the composite of the first three maps in Lemma 6.4.
Then it suffices to show $f$ is null-homotopic. By Lemma 6.5, $f$ is
identified with some element $\phi$ of $H_{d}(M;\mathbb{Z})\otimes
H_{d}(A;\mathbb{Z})$, so that $f$ is null-homotopic if and only if $\phi=0$.

As in the proof of Lemma 6.6, $|\Sigma M_{[m]-v}|$ is a wedge of finitely many
copies of $S^{2}$ and $S^{d}$ for each vertex $v$ of $M$. Let $C_{v}$ denote
the $S^{d}$-wedge part of $|\Sigma M_{[m]-v}|$. Then there is a projection
$q_{v}\colon B\to C_{v}$. By Lemmas 5.5, 5.7 and 6.4, the composite

(6.8) $|M|\xrightarrow{f}A\to|\Sigma M_{I-v}|\to|\Sigma M_{[m]-v}|$

is null homotopic for each $v\in I\in S(M)$. Then by Lemma 6.5, $\phi$ is
mapped to 0 by

$1\otimes(q_{v}\circ j)_{*}\colon H^{d}(M;\mathbb{Z})\otimes
H_{d}(A;\mathbb{Z})\to H^{d}(M;\mathbb{Z})\otimes H_{d}(C_{v};\mathbb{Z})$

for each $v\in I\in S(M)$, where $j\colon A\to B$ denotes the inclusion. Since
the map

$\bigoplus_{v\in I\in S(M)}(q_{v})_{*}\colon
H_{d}(B;\mathbb{Z})\to\bigoplus_{v\in I\in S(M)}H_{d}(C_{v};\mathbb{Z})$

is an isomorphism, we get $(1\otimes j_{*})(\phi)=0$. Thus we obtain $\phi=0$
by Lemma 6.6, completing the proof. ∎

Now we are ready to prove Theorem 1.5:

###### Proof of Theorem 1.5.

The implications (1) $\Rightarrow$ (2) $\Leftarrow$ (3) are proved by Theorems
3.4 and 2.3. The implication (3) $\Rightarrow$ (4) is proved by Propositions
6.3 and 6.7. If (4) holds, then by Theorem 5.4, $Z_{M}$ is a suspension. So by
the fact that $K$ is $\mathbb{F}$-Golod whenever $Z_{K}$ is a suspension, as
mentioned in Section 1, we obtain the implication (4) $\Rightarrow$ (1),
completing the proof. ∎

##  7\. Further problems

This section proposes possible problems for a further study on a relationship
of Golodness and tightness. As proved in Theorem 3.4, weak
$\mathbb{F}$-Golodness implies $\mathbb{F}$-tightness for a closed connected
$\mathbb{F}$-orientable manifold triangulations. But not all weakly
$\mathbb{F}$-Golod complexes are $\mathbb{F}$-tight. For example, if $K$ is
the join of a vertex and the boundary of a simplex, then it is
$\mathbb{F}$-Golod for any field $\mathbb{F}$ as the fat-wedge filtration of
$\mathbb{R}Z_{K}$ is trivial but it is not $\mathbb{F}$-tight as in the proof
of Lemma 4.2. Then it is natural to ask:

######  Problem 7.1.

For which simplicial complexes does weak $\mathbb{F}$-Golodness imply
$\mathbb{F}$-tightness?

Interestingly, the opposite implication always holds.

######  Proposition 7.2.

Let $K$ be a simplicial complex with vertex set $[m]$. If $K$ is
$\mathbb{F}$-tight, then it is weakly $\mathbb{F}$-Golod.

###### Proof.

Take any disjoint subsets $\emptyset\neq I,J\subset[m]$. Then there is a map

$\iota_{I,J}\colon K_{I\sqcup J}\to K_{I}*K_{J}$

as in Section 3. By Lemma 3.2, $K$ is weakly $\mathbb{F}$-Golod if and only if
the map $\iota_{I,J}$ is trivial in homology with coefficients in
$\mathbb{F}$. Now we suppose $K$ is $\mathbb{F}$-tight. Then $K_{I\sqcup J}$
is $\mathbb{F}$-tight too, and so we only need to consider the case $I\sqcup
J=[m]$. By the Künneth theorem, the map

$(j_{I}*j_{J})_{*}\colon\widetilde{H}_{*}(K_{I}*K_{J};\mathbb{F})\to\widetilde{H}_{*}(K*K;\mathbb{F})$

is injective, where $j_{I}\colon K_{I}\to K$ denotes the inclusion. Then it
suffices to show the composite $(j_{I}*j_{J})\circ\iota_{I,J}$ is null-
homotopic.

Now we may assume $|K|\subset\mathbb{R}^{m}$ by identifying a simplex
$\\{i_{1},\ldots,i_{k}\\}\in K$ with

$\\{t_{1}e_{i_{1}}+\cdots+t_{k}e_{i_{k}}\mid t_{1}+\cdots+t_{k}=1,\,t_{1}\geq
0,\,\ldots,t_{k}\geq 0\\}$

where $e_{1},\ldots,e_{m}$ is the standard basis of $\mathbb{R}^{m}$. We may
assume $|K*K|\subset\mathbb{R}^{2m}$ in the same way. Consider a homotopy
$h_{t}^{i}\colon\mathbb{R}^{2m}\times[0,1]\to\mathbb{R}^{2m}$ defined by

$h_{t}^{i}(x_{1},\ldots,x_{m},y_{1},\ldots,y_{m})=(x_{1},\ldots,(1-t)x_{i}+ty_{i},\ldots,x_{m},y_{1},\ldots,tx_{i}+(1-t)y_{i},\ldots,y_{m})$

for $(x_{1},\ldots,x_{m},y_{1},\ldots,y_{m})\in\mathbb{R}^{2m}$. Then
$h_{t}^{i}$ restricts to a homotopy $h_{t}^{i}\colon|K*K|\times[0,1]\to|K*K|$
such that for $i\in I$,

$(j_{I}*j_{J})\circ\iota_{I,J}=h_{0}^{i}\circ(j_{I}*j_{J})\circ\iota_{I,J}\simeq
h_{1}^{i}\circ(j_{I}*j_{J})\circ\iota_{I,J}=(j_{I-i}*j_{J\cup
i})\circ\iota_{I-i,J\cup i}.$

Thus for $v\in[m]$,
$(j_{I}*j_{J})\circ\iota_{I,J}\simeq(j_{v}*j_{[m]-v})\circ\iota_{v,[m]-v}$.
Since $|v*K_{[m]-v}|$ is contractible, we get
$(j_{I}*j_{J})\circ\iota_{I,J}\simeq*$, completing the proof. ∎

In this paper, we have been studying a relationship between Golodness and
tightness through tight-neighborliness which perfectly works in dimension 3.
However, in dimensions $\geq 4$, tight-neighborliness does not work well
because it is not equivalent to tightness as mentioned in Section 1. So we
consider the following problem, the weak version of which is Problem 7.1 for
manifold triangulations of dimensions $\geq 4$.

######  Problem 7.3.

What condition on closed connected $d$-manifold triangulations with $d\geq 4$
guarantees $\mathbb{F}$-Golodness and $\mathbb{F}$-tightness are equivalent?

One approach is to put a topological condition on manifolds. For example, the
condition on the Betti number is stated in Section 1. We also have the
following theorem, in which manifolds triangulations are not tight-neighborly.

######  Theorem 7.4.

Let $M$ be a triangulation of a closed $(d-1)$-connected $2d$-manifold for
$d\geq 2$. Then the following are equivalent:

  1. (1)

$M$ is $\mathbb{F}$-Golod for any field $\mathbb{F}$;

  2. (2)

$M$ is $\mathbb{F}$-tight for any field $\mathbb{F}$;

  3. (3)

$M$ is $d$-neighborly;

  4. (4)

the fat-wedge filtration of $\mathbb{R}Z_{M}$ is trivial.

###### Proof.

The implication (1) $\Rightarrow$ (2) holds by Theorem 3.4 because $M$ is
orientable. Suppose $M$ has a minimal non-face $I$ with $|I|\leq d+1$. Then
$M_{I}=\partial\Delta(I)$, implying $H_{|I|-2}(M_{I};\mathbb{F})\neq 0$. Since
$M$ is $\mathbb{F}$-tight, the natural map $H_{|I|-2}(M_{I};\mathbb{F})\to
H_{|I|-2}(M;\mathbb{F})$ is injective, and since $M$ is $(d-1)$-connected,
$\widetilde{H}_{*}(M;\mathbb{F})=0$ for $*<d$. Then we get a contradiction, so
we obtain the implication (2) $\Rightarrow$ (3). The implication (3)
$\Rightarrow$ (4) follows from [17, Theorem 1.6]. The implication (4)
$\Rightarrow$ (1) holds by the fact that $K$ is $\mathbb{F}$-Golod over any
field $\mathbb{F}$ whenever $Z_{K}$ is a suspension, as mentioned in Section
1. Therefore, the proof is complete. ∎

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