# ATMOSPHERIC NOISE MEASUREMENTS IN THE GARDEN: DETECTING UNIVERSALITIES IN
INTER-SPHERIC WAITING TIME STATISTICS

István Gere  Department of Physics, Babeş-Bolyai University, Cluj-Napoca,
Romania  email<EMAIL_ADDRESS>
(August 28, 2024)

###### Abstract

Lightning flashes result in an instantaneous emission of electromagnetic (EM)
waves that encompass a broad spectrum of frequencies in the domain of radio
waves. The signature of these impulses in the region of the very low frequency
(VLF) of radio waves and bellow are called spherics. These impulsive signals
traverse great distances in the waveguide between the Earth’s ionosphere and
its surface. Due to the abundance of lightnings the lower end of the EM
spectrum is dominated by the waveforms of these emission. It is called
atmospheric radio noise because of its origin. This article outlines a
straightforward approach for capturing spherics activity within the VLF radio
wave range. Employing data processing techniques, we pinpoint the timestamps
of spherics within time series data. The distribution of inter-spheric times
is analyzed across various detection threshold levels. Utilizing recordings
spanning two distinct years and seasons, we replicate the established form of
the inter-spheric time distribution described in the literature. Notably, we
demonstrate that by rescaling the time intervals between spherics with the
mean time for a specific recording and given detection threshold, the
distributions collapse to a master curve. This universal pattern is accurately
characterized by a single-parameter mean-scaled Gamma distribution.
Additionally, we note the similarities in the distribution of inter-spheric
times with patterns found in earthquake recurrence times.

##  1 Introduction

Lightning flashes produce a sudden emission of electromagnetic (EM) waves
across a broad domain of the EM spectrum within the radio frequency range [1].
Studies on the radio frequency (RF) spectrum of these sudden emissions
indicate that the peak of the spectral energy is typically around 10 kHz and
decreases with the reciprocal of the frequency [2].The peek of the emitted
energy overlaps with the range of very low frequency (VLF) ($30$ kHz - $3$
kHz) radio waves.In the lower ranges of the RF spectrum the cavity between the
Earth’s surface and the ionosphere acts as a low-attenuation (around $2$-$3$
dB/$1000$ km) [3] waveguide enabling these instantaneous wide bandwidth EM
emissions to travel great distances ($1000$-$2000$ km) [4]. The attenuation
further decreases during the night as a result of the diurnal cycle, as
radiation from the Sun does not reach certain regions of the Earth. This
restructuring of electron density in the upper layers of the ionosphere
creates even more favourable conditions for propagation [5]. In the
literature, these electromagnetic emissions induced by lightning are referred
to as radio atmospheric signals or, more concisely, as spherics [6]. The
global lightning activity is measured in lightning flashes per second. Recent
estimation methods based on satellite data estimate the global yearly average
to be $44\pm 5$ flasher per second, with hemispheric, local and seasonal
variations [7]. The abundant number of spherics generated by global lightning
activity occupy the RF spectrum, contributing to atmospheric electromagnetic
noise. Specially designed radios make it possible to record this noise. In
this article, we investigate the VLF spectrum of spheric emissions for two
main reasons: firstly, because the VLF spectrum of radio waves aligns with the
audible frequency range of sound, making these signals easily audible with
simple amplification; and secondly, because there are readily available data
acquisition tools designed specifically for this frequency range, such as
computer sound cards and high-resolution dictaphones.The fact that the peak of
the lightning spectral energy is also found in this range of VLF domain and
that the attenuation is low at night further enhances the reception of the
spherics [2, 3]. In this paper we present a simple setup of a special radio
that allows us to intercept and record spherics originating from remote
thunderstorms.We present an example spectrogram of such recordings. The
distribution of inter-spheric times for various detection thresholds, as
calculated using the procedure outlined in [8], replicates their findings.

Contrary to the complex description provided in [8], we offer a simpler
account. Drawing inspiration from our earlier studies on earthquakes [9], a
noteworthy observation emerges: by rescaling the time between spherics,
denoted as $t$, with the mean time $\langle t\rangle$ for a particular dataset
of inter-spheric times (across different years and under various $L$ detection
thresholds) to obtain $\tau=\frac{t}{\langle t\rangle}$, the experimental
probability densities, represented by $\rho(\tau)$, collapse to a common
master curve. This universal curve can be described as a single parameter
Gamma-distribution.

##  2 Experiment

###  2.1 The VLF radio device

To perform our experiments we used the Explorer E202 broadband radio
specifically designed for VLF spherics reception. Open-source schematics of
the Explorer E202 [10] with off the shelf electric components were used to
build this device, which is presented in Figure 1. The core of this device is
a TL081C operational amplifier [11] that is connected to the antenna input
through an impedance matching circuit. The device includes two outputs, a
line-level type for recording and a headphone level type for real time field
listening [10]. The device operates on a single 9V battery, making it portable
while it ensures a stable current source and low internal electric noise.

The experimental setup involved a citizens band (CB) radio antenna vertically
positioned at approximately 4 meters in height (refer to Figure 1 B.). This
antenna was connected to the radio through a coaxial cable (see Figure 1 A.).
Additionally, the device was linked to a dictaphone with high-resolution
recording capabilities using a simple two-way audio jack (see Figure 1 A.).
The dictaphone is preferred over directly using a laptop, as it enables longer
recording sessions and is free from the internal electronic noise associated
with laptops. It is important to note that, according to the specifications of
the Explorer E202, the CB antenna used in our setup is sub-optimal. A longer
vertical antenna could have increased the signal-to-noise ratio, although this
did not pose significant issues during the recordings, due to good reception
conditions.

Figure 1: A.) Experimental setup featuring the Explorer E202 connected to both
the dictaphone and the antenna cable. B.) The CB Antenna mounted on a metal
rod with four ground plane radials (suboptimal for VLF signals, but performed
well in our experiments due to its height and location).

###  2.2 Data acquisition

The recordings were performed using the dictaphone, in our case serving as an
analog-digital converter unit (ADC-converter), configured in a line-in mode
that is compatible with the line-out mode of the Explorer E202. A consumer
line-level is a standard in analog audio transmission with a peak-to-peak
voltage of $V_{pp}=0.894\textrm{ V}$ [12]. The ADC-unit digitalizes these
signals with a sample rate $\nu_{s}=48$ kHz and a bit depth of $16$ bit per
sample. According to the Nyquist-Shannon sampling theorem [13], this allows
frequencies to be representable up to $\frac{\nu_{s}}{2}$ frequencies, which
overlaps with the studied VLF frequency range.

Figure 2: The waveform of a single spheric in time.

In Figure 2, we provide an illustration of the waveform of a single spheric.
The signal strength over time is computed as follows: let $\textrm{s}(t)$
denote the value detected by the Analog-to-Digital Converter (ADC) using a
16-bit notation, and $FS$ (full scale) represent the maximum representable
value in the 16-bit notation, given by $FS=2^{16}-1=65535$. The absolute
signal strength relative to $FS$ is then expressed as
$S_{FS}(t)=\frac{\textrm{s}(t)}{FS/2}$. The $S_{FS}(t)$ quantity is
proportional to the electric field strengths detected by the antenna. The
relative signal strength $S_{FS}$ is sufficient for our study as we are
looking for the time signatures of the spherics, to calculate the inter-
spheric time. In Figure 3 we presents the spectrogram of a short signal
sample.

Figure 3: The spectrogram of a $\approx$ 40 seconds excerpt from a recording
performed in 2022. The vertical lines represent the signature of spherics. The
power spectral density values, denoted as
$\textrm{dBFS}[10\textrm{log}(S_{FS}^{2})/\textrm{Hz}]$ are calculated based
on the signal strength relative to the full scale $S_{FS}$ using a Fast
Fourier Transform (FFT), with a time windowing of 2042 samples. At the lower
part of the spectrogram the harmonics of the 50 Hz power line hum is visible.
The vertical lines between $11$ kHz and $13$ kHz, at regular intervals are the
signals of the Alpha (or RSDN-20) navigation [14]. The black rectangle
represents the frequency range of $4$ to $15$ kHz, the frequency domain in
which we investigate spheric phenomena..

As indicated in the highlighted area of Figure 3, the spectral signature of
spheric phenomena in our recordings is observed between $1$ kHz and $15$ kHz.
The spectrograms are generated using Fast Fourier Transform (FFT) applied to a
sample window of $2024$ samples centered around a time-step $t$ in the
digitalized data, with a $50\%$ overlap. The next step involves extracting a
measure of spectral strength within the frequency domain $D=[4\textrm{
kHz},15\textrm{ kHz}]$. We accomplish this by summing the power spectral
density of frequencies within the domain $D$ at each time-step $t$, expressed
as $S_{D}(t)=\sum_{D}S_{FS}^{2}(\nu)_{t}$. This measure, $S_{D}(t)$, is
proportional to the power spectral density of the frequency domain $D$ at the
midpoint $t$ of the FFT sample window. Using this method, we effectively
eliminate electromagnetic noise from the lower end of the spectrum,
originating from the power grid’s $50$ Hz signal and its harmonics, commonly
referred to as mains hum [15]. Additionally, we remove noise from the upper
end of the spectrum, which does not contain relevant information for this
study. From the spectral strength $S_{D}(t)$ over the frequency domain $D$ the
following measure is constructed $b(t)=\textrm{dB}[10\textrm{log}(S_{D}(t)]$
We present it on Figure 4.

Figure 4: On this plot we present the measure
$b(t)=\textrm{dB}[10\textrm{log}(S_{D}(t)]$ employed for spheric detection. In
this presentation the prominent spikes correspond to lightning spherics. The
red line at $L=-85$ dB presents an example threshold to detect spherics,
whereas peaks over this line are considered spherics.

Using the new measure $b(t)$ shown in Figure 4 lightning spherics can be
readily identified along with their precise timing. When the measure $b(t)$
surpasses a predefined threshold level $L$, the exact time $t$ when the
measure reaches a local maximum is located using a peak finder algorithm[16].
Applying this detection method, we generated datasets containing the precise
times of each detected lightning spheric by our device. The recording location
is situated the in the village of Sâncraiu, Romania (46.8304° N, 22.9893° E)
chosen for its rural setting with lower artificial EM noise levels, making it
an optimal environment for recordings. Our experimental measurements
encompassed two recording sessions:

  1. 1.

In 2022-07-04, starting at 21:34:46, a 50 min recording.

  2. 2.

In 2023-04-22, starting at 19:40:53, a 80 min recording.

##  3 Results

In this section we reflect on the probability distribution of inter-spheric
times, as observed in the experimental recording sessions processed using the
methods detailed earlier. A representative example of spheric detection
threshold is depicted in Figure 4 where the red line delineates the threshold
beyond which a peak is deemed a spheric.By reducing the threshold, we
incorporate a greater number of peaks, encompassing weaker spheric signatures.
By applying different threshold levels to the measure $b(t)$, we created
multiple datasets containing the inter-spheric times for the two experimental
sessions. In Table 1 we present the detection threshold $L$ with the
corresponding the number of detected spherics $N$, and the mean time between
spherics $\langle t\rangle$. For each threshold $L$ and year, there
corresponds an inter-spheric time database, as presented in Table 1.

Table 1: The table displays the observed number of spherics $N$, the mean time between two spherics $\langle t\rangle$, and the corresponding threshold level of detection $L$. | Measurement
---
session
| Threshold
---
$L$ [dB]
| Number of Spherics
---
detected $N$
| Mean time between
---
spherics
$\langle t\rangle$[s]
I. 2022 | -83 | 810 | 3.7
| -85 | 1108 | 2.71
| -89 | 2149 | 1.39
| -92 | 3511 | 0.85
| -96 | 6733 | 0.44
II. 2023 | -83 | 162 | 18.3
| -85 | 233 | 12.72
| -90 | 661 | 4.5
| -92 | 999 | 2.99
| -95 | 2644 | 1.13

In Figure 5 we present the probability density of the inter-spheric times
observed in our experiment with different $L$ thresholds of spheric detection
from the datasets with the properties found in Table 1. Based on the observed
shapes of these experimental probability densities, we infer that, for inter-
spheric times exceeding 1.5 seconds, the distribution tends to converge
towards an exponential distribution (indicated by a straight line on a semi-
log plot). This behavior aligns with similar patterns depicted in plots found
in the existing literature [8]. The exponential characteristic of the tail is
attributed to the detection of lightning occurrences from multiple independent
storms around the world at our recording location.

Figure 5: The experimental probability density of inter-spheric times for the
2022 (A.) and 2023 (B.) measurements, considering various thresholds as
described in Table 1. For $t>1.5$ s, the distributions asymptotically converge
towards an exponential pattern over time.

After obtaining the time distribution of inter-spheric intervals, our next
objective was to characterize our findings using a statistical physics
approach, a methodology already applied in geology to describe the recurrence
time of earthquakes [9]. Unlike the prevailing literature, which relies on a
highly parametrized model, our approach seeks simplicity and universality in
capturing the underlying dynamics of the observed inter-spheric time [8]. For
the time datasets created through thresholding (with properties detailed in
Table 1), we constructed the experimental probability density of the mean-
scaled inter-spheric times using the mean time of the dataset, denoted as
$\tau=\frac{t}{\langle t\rangle}$. The rescaled probability densities, denoted
as $\rho(\tau)$, are depicted in Figure 6. We observe a novel universality,
where, for different measurement sessions and threshold levels, the rescaled
distributions collapse onto the same master curve. This curve can be described
by the mean-rescaled Gamma distribution, as represented in equation (1). The
presence of the master curve and the single-parameter Gamma distribution,
describing the recurrence time in earthquakes and also observed in the
distribution of time between acoustic emissions during metal deformation
(compression of micro-capillaries), has been documented in the literature [9].
This suggests universality in the processes producing these diverse natural
phenomena.

$\rho(\tau)=\frac{\alpha^{\alpha}}{\Gamma(\alpha)}\tau^{\alpha-1}e^{-\alpha\tau}$
(1)

Figure 6: The plot illustrates the probability densities $\rho(\tau)$ of mean-
rescaled inter-spheric times $\tau$, at various threshold levels and across
different years. The observed master curve can be described with the mean-
rescaled Gamma distribution in the form of equation (1) with the parameter
$\alpha=0.97$. The inset figure provides a detailed look at the lower segment
of the plot corresponding to 2022 and 2023 datasets, where the distribution
tends towards a power law, exhibiting exponents of $-1.85$ for 2022 and
$-1.25$ for 2023.

For $\tau<0.1$ we also observe a power-law like decay to the Gamma
distribution. A comparable power-law like decay in recurrence time toward the
Gamma distribution has also been observed in the previously mentioned study of
earthquakes [9]. In such cases, the empirical Omori law [17] is often
considered to be the source of these deviating regions in inter-event time
distribution. In seismology it describes the number of aftershocks
(clustering) after a main event as a power law of the time with a negative
exponent. The clustered set of events in the experimental inter-event times
manifests as a short region with low $\tau$ values, which can be approximated
with a power law [18]. We present such cases selected from the experimental
probability densities for spherics in the Figure 6 inset plot, where the
regions dominated by a power-law are highlighted. The inherent clustering
observed in the occurrence of spherics originates from the nature of lightning
events. Typically, lightning is not a singular occurrence; instead, it
comprises an initial main lightning followed by multiple flashes. In this
sequence, the same ion channel from the initial event conducts additional
charge, leading to subsequent flashes [8].

##  4 Conclusions

In this study, we introduced a simple methodology for capturing the radio
waves produced by lightning events within the very low-frequency range. The
atmospheric electromagnetic noise that characterizes this frequency range can
be studied for its statistical characteristics. We studied the temporal
behaviour of the spherics, electromagnetic signatures of remote lightnings.
These spherics signals are present as a sharp and sudden increase in signal
strength over the noise limit of the recordings. Processing data from two
recording sessions conducted in different years and seasons, we generated
datasets of inter-spheric times with varying detection thresholds. We observed
that rescaling individual inter-spheric times with the mean inter-spheric time
of the dataset resulted in collapsed probability distributions onto a common
master curve. This curve can be accurately described by a mean rescaled Gamma
distribution, as indicated by equation (1) with a single parameter
$\alpha\approx 0.97$. To our knowledge, this universality has not been
documented in the literature. Our one-parameter fit provides a more
straightforward description of the observed inter-spheric time distribution
compared to the results reported in existing literature. We also note a
resemblance in the distribution patterns of inter-spheric times to earthquake
recurrence times, and the intervals between acoustic emissions during the
deformation of metals [9]. While all of these phenomena can be modeled with a
Gamma function, we also observed a distinct section with low values in the
density distributions, exhibiting high temporal correlations. In these
regions, the distribution tends to converge towards a power-law. This implies
a universality in the underlying processes governing these diverse natural
phenomena, placing spherics within the broader family of such phenomena.

### Acknowledgements

This work supported by the Romanian UEFISCDI PN-III-P4-ID-PCE-2020-0647
research grant. I would also like to express my gratitude to O. Botos for his
guidance on the implementation of the Explorer E202 radio. Special thanks to
Z. I. Lazar for his invaluable advice throughout the writing of this article.

## References

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# Toric wedge induction and toric lifting property for piecewise linear
spheres with a few vertices

Suyoung Choi  Department of mathematics, Ajou University, 206, World cup-ro,
Yeongtong-gu, Suwon 16499, Republic of Korea<EMAIL_ADDRESS>,  Hyeontae
Jang  Department of mathematics, Ajou University, 206, World cup-ro,
Yeongtong-gu, Suwon 16499, Republic of Korea<EMAIL_ADDRESS>and  Mathieu
Vallée  Université Sorbonne Paris Nord, LIPN, CNRS UMR 7030, F-93430,
Villetaneuse, France<EMAIL_ADDRESS>
###### Abstract.

Let $K$ be an $(n-1)$-dimensional piecewise linear sphere on $[m]$, where
$m\leq n+4$. There are a canonical action of $m$-dimensional torus $T^{m}$ on
the moment-angle complex $\mathcal{Z}_{K}$, and a canonical action of
$\mathbb{Z}_{2}^{m}$ on the real moment-angle complex
$\mathbb{R}\mathcal{Z}_{K}$, where $\mathbb{Z}_{2}$ is the additive group with
two elements. We prove that any subgroup of $\mathbb{Z}_{2}^{m}$ acting freely
on $\mathbb{R}\mathcal{Z}_{K}$ is induced by a subtorus of $T^{m}$ acting
freely on $\mathcal{Z}_{K}$. The proof primarily utilizes a suitably modified
method of toric wedge induction and the combinatorial structure of a specific
binary matroid of rank $4$.

###### Key words and phrases:

PL sphere, lifting problem, toric manifold, real toric manifold, Buchstaber
number, real Buchstaber number, binary matroid

###### 2020 Mathematics Subject Classification:

57S12, 14M25, 52B05

This work was supported by the National Research Foundation of Korea Grant
funded by the Korean Government (NRF-2021R1A6A1A10044950).

###### Contents

  1. 1 Introduction
  2. 2 The case $r\leq 3$
  3. 3 The case $r=4$ : Preliminaries
    1. 3.1 Characteristic and dual characteristic maps
    2. 3.2 Wedge operations
  4. 4 Toric wedge induction
    1. 4.1 Toric wedge induction and its modification
    2. 4.2 Modified Toric wedge induction in terms of dual characteristic maps
  5. 5 Proof of the main theorem
  6. 6 The basis step

##  1\. Introduction

Let $K$ be a simplicial complex on the set $[m]=\\{1,\ldots,m\\}$. We define
the _polyhedral product_ $(\underline{X},\underline{Y})^{K}$ of $K$ with
respect to a pair $(X,Y)$ of topological spaces as follows:

$(\underline{X},\underline{Y})^{K}\coloneqq\bigcup_{\sigma\in
K}\left\\{(x_{1},\ldots,x_{m})\in X^{m}\mid x_{i}\in Y\text{ when
}i\notin\sigma\right\\}.$

Here, $D^{d}$ represents the $d$-dimensional disk, defined as
$D^{d}=\\{\mathbf{x}\in\mathbb{R}^{d}\mid\|\mathbf{x}\|\leq 1\\}$, and
$S^{d-1}$ denotes its boundary sphere of dimension $d-1$. The _moment-angle
complex_ $\mathcal{Z}_{K}$ of $K$ is then defined as
$(\underline{D^{2}},\underline{S^{1}})^{K}$, and the _real moment-angle
complex_ $\mathbb{R}\mathcal{Z}_{K}$ of $K$ is
$(\underline{D^{1}},\underline{S^{0}})^{K}$. We observe that the
$T^{1}$-action on the pair $(D^{2},S^{1})$ leads to the canonical action of
the $m$-dimensional torus $T^{m}=(S^{1})^{m}$ on $\mathcal{Z}_{K}$.
Additionally, there is an $S^{0}$-action on the pair $(D^{1},S^{0})$. For
clarity and consistency in our terminology throughout this paper, we treat
$S^{0}$ as the additive group $\mathbb{Z}_{2}=\mathbb{Z}/2\mathbb{Z}$ with two
elements $\\{0,1\\}$. This, then, yields the canonical
$\mathbb{Z}_{2}^{m}$-action on $\mathbb{R}\mathcal{Z}_{K}$.

It is noteworthy that when an $r$-dimensional subtorus $H$ of $T^{m}$ acts
freely on $\mathcal{Z}_{K}$, the resulting quotient space $\mathcal{Z}_{K}/H$
admits a well-behaved torus action $T^{m}/H\cong T^{m-r}$ with an orbit space
that exhibits a reverse face structure isomorphic to $K$. Such spaces are
commonly referred to as _toric spaces_ or _(partial) quotients_ , and are
fundamental in the study of _toric topology_. Consequently, understanding
which subtori $H$ of $T^{m}$ can act freely on $\mathcal{Z}_{K}$ is of
significant importance. The Buchstaber number $s(K)$ is the maximal integer
$r$ for which there exists a subtorus of rank $r$ acting freely on
$\mathcal{Z}_{K}$. Similarly, taking a subgroup $H$ of $\mathbb{Z}_{2}^{m}$
freely acting on $\mathbb{R}\mathcal{Z}_{K}$ yields the quotient space
$\mathbb{R}\mathcal{Z}_{K}/H$ which is referred to as a _real toric space_ or
a _real (partial) quotient_. The real Buchstaber number $s_{\mathbb{R}}(K)$ is
similarly defined by the existence of a subgroup acting freely on
$\mathbb{R}\mathcal{Z}_{K}$. The determination of (real) Buchstaber numbers is
challenging. We refer to the following publications for details: [4], [17],
[15], [2], and [20].

It is known that the real moment-angle complex $\mathbb{R}\mathcal{Z}_{K}$ is
the fixed point set by the involution on $\mathcal{Z}_{K}$ induced by the
complex conjugation on $D^{2}\subset\mathbb{C}$. This implies that a
$T^{m}$-action on $\mathcal{Z}_{K}$ induces a $\mathbb{Z}_{2}^{m}$-action on
$\mathbb{R}\mathcal{Z}_{K}$, and then $d$-dimensional subtorus of $T^{m}$
acting freely on $\mathcal{Z}_{K}$ induces a rank $d$ subgroup of
$\mathbb{Z}_{2}^{m}$ acting freely on $\mathbb{R}\mathcal{Z}_{K}$. Thus, we
obtain the inequality $s(K)\leq s_{\mathbb{R}}(K)$, and Ayzenberg [1] noted
that the equality does not generally hold; specifically, there exists a
simplicial complex whose real Buchstaber number is strictly bigger than its
Buchstaber number.

From now on, we zero in on the case when $K$ is a PL sphere, since in this
case, all toric spaces over $K$ are PL manifolds [5]. If $K$ is
$(n-1)$-dimensional, we have the inequalities $s(K)\leq s_{\mathbb{R}}(K)\leq
m-n$. Given the condition $s(K)=m-n$, which is a special case often
encountered in various fields of mathematics, the manifold $\mathcal{Z}_{K}/H$
for a maximal subtorus $H\subset T^{m}$ freely acting on $\mathcal{Z}_{K}$ is
termed a _topological toric manifold_ [19] when $K$ is star-shaped. If $K$ is
polytopal, the manifold is referred to as a _quasitoric manifold_ [13]111A
quasitoric manifold was originally called a toric manifold in [13], and was
renamed in [4] to avoid confusion with a smooth compact toric variety..
Similarly, given the condition $s_{\mathbb{R}}(K)=m-n$, the manifold
$\mathbb{R}\mathcal{Z}_{K}/H$ for a maximal subgroup $H$ freely acting on
$\mathbb{R}\mathcal{Z}_{K}$ is called a _real topological toric manifold_ when
$K$ is star-shaped, and it is called a _small cover_ when $K$ is polytopal.
These are real analogs of topological toric and quasitoric manifolds,
respectively.

In the class of PL spheres, no examples have been known where
$s(K)<s_{\mathbb{R}}(K)$. In light of this observation, one may ask whether
$s(K)=s_{\mathbb{R}}(K)$ for a PL sphere $K$, and the following stronger
question can be considered.

######  Problem 1.1.

Let $K$ be a PL sphere on $[m]$. Given a subgroup of $\mathbb{Z}_{2}^{m}$
acting freely on $\mathbb{R}\mathcal{Z}_{K}$, is this action induced by a
subtorus of $T^{m}$ freely acting on $\mathcal{Z}_{K}$?

In particular, when $s_{\mathbb{R}}(K)=m-n$, Problem 1.1 is equivalent to the
(toric) _lifting problem_ (Problem 3.6). In other words, this asks whether
every small cover (or real topological toric manifold) is induced from some
quasitoric manifold (or topological toric manifold, respectively). The lifting
problem was initially proposed by Zhi Lü at the toric topology conference held
in Osaka in 2011, as documented in [8], and remains an open problem in toric
topology, attracting considerable research attention. However, significant
advances in resolving this problem have been elusive. This paper aims to make
a contribution by providing meaningful results to the lifting problem, and
more broadly to Problem 1.1, in the case $m\leq n+4$.

######  Theorem 1.2.

Let $K$ be an $(n-1)$-dimensional PL sphere with $m\leq n+4$ vertices. Then,
any subgroup of $\mathbb{Z}_{2}^{m}$ freely acting on
$\mathbb{R}\mathcal{Z}_{K}$ is induced by a subtorus of $T^{m}$ freely acting
on $\mathcal{Z}_{K}$.

Let $H$ be a subgroup of $\mathbb{Z}_{2}^{m}$ of rank $r$, freely acting on
$\mathbb{R}\mathcal{Z}_{K}$. Then, $0\leq r\leq\min(s_{R}(K),m-n)\leq 4$. To
address this question, we categorize our approach into two distinct cases: the
case where $r\leq 3$ and the case where $r=s_{\mathbb{R}}(K)=m-n=4$. In
Section 2, we give a positive answer to Problem 1.1 under the condition that
for $r\leq 3$ without the necessity of $m=n+4$. It demonstrates that Theorem
1.2 holds for the case where $m\leq n+3$, or the case $m=n+4$ and $r\leq 3$.

Subsequent sections will focus on the case $r=4$ under the specific condition
$m-n=s_{\mathbb{R}}(K)=4$. Here, we establish the theorem for this case
employing a method we call _toric wedge induction_. This method was firstly
introduced by Choi and Park in [8], and it can be effectively used to
demonstrate properties of toric varieties for certain Picard numbers. In our
proof, we will introduce a more powerful version of toric wedge induction.
Additionally, the combinatorial structure of a binary matroid of rank $4$ will
be crucially used for the basis step of the induction.

##  2\. The case $r\leq 3$

Let $K$ be an $(n-1)$-dimensional simplicial complex on
$[m]=\\{1,2,\ldots,m\\}$, $H\subset T^{m}$ a subtorus of dimension $r\leq
m-n$. After choosing a basis, it can be written as

(2.1) $H=\\{(e^{2\pi i(s_{11}\phi_{1}+\dots+s_{1r}\phi_{r})},\ldots,e^{2\pi
i(s_{m1}\phi_{1}+\dots+s_{mr}\phi_{r})})\in
T^{m}\mid\phi_{j}\in\mathbb{R},j=1,...,r\\},$

where $s_{ij}\in\mathbb{Z}$. We define an $m\times r$ integer matrix
$S=(s_{ij})$. Additionally, the $(m-n)\times r$ matrix
$S_{\hat{i}_{1},\ldots,\hat{i}_{n}}$ is defined as the submatrix of $S$
obtained by excluding the rows corresponding to entries $i_{j}$ for
$j=1,\ldots,n$. The following proposition was proved for polytopal simplicial
complexes in [4], but it can be also proved by a similar argument for general
ones.

######  Proposition 2.1.

Let $K$ be a simplicial complex. Then the subtorus (2.1) acts freely on
$\mathcal{Z}_{K}$ if and only if for any facet $\\{i_{1},\ldots,i_{n}\\}$ of
$K$, the matrix $S_{\hat{i}_{1},\ldots,\hat{i}_{n}}$ defined above gives a
monomorphism $\mathbb{Z}^{r}\longrightarrow\mathbb{Z}^{m-n}$ to a direct
summand.

The latter condition is equivalent to $S_{\hat{i}_{1},\ldots,\hat{i}_{n}}$
having an $r\times r$ submatrix whose determinant is $\pm 1$. A similar
argument holds for $\mathbb{Z}_{2}^{m}$-action on $\mathbb{R}\mathcal{Z}_{K}$.
Then the mod $2$ reduction of the matrix $S$ representing a freely acting
subtorus of $T^{m}$ on $\mathcal{Z}_{K}$ represents a freely acting subgroup
of $\mathbb{Z}_{2}^{m}$ on $\mathbb{R}\mathcal{Z}_{K}$.

######  Theorem 2.2.

Let $K$ be an $(n-1)$-dimensional simplicial complex on $[m]$, and $r\leq 3$ a
non-negative integer. Then any rank $r$ subgroup of $\mathbb{Z}_{2}^{m}$
acting freely on $\mathbb{R}\mathcal{Z}_{K}$ is induced by an $r$-dimensional
subtorus of $T^{m}$ freely acting on $\mathcal{Z}_{K}$.

###### Proof.

Assume an $m\times r$ matrix $S$ over $\mathbb{Z}_{2}$ represents the freely
acting subgroup of $\mathbb{Z}_{2}^{m}$ on $\mathbb{R}\mathcal{Z}_{K}$. Define
the $m\times r$ matrix $\tilde{S}$ over $\mathbb{Z}$ all of whose entries
$\tilde{s}_{ij}$ are in $\\{0,1\\}$ and such that $S\equiv\tilde{S}$ mod $2$.
For any facet $\\{i_{1},\ldots,i_{n}\\}$ of $K$,
$S_{\hat{i}_{1},\ldots,\hat{i}_{n}}$ has an $r\times r$ submatrix $R$ whose
determinant is $1\in\mathbb{Z}_{2}$. Then the corresponding submatrix
$\tilde{R}$ of $\tilde{S}_{\hat{i}_{1},\ldots,\hat{i}_{n}}$ has an odd
determinant. Since the absolute value of every square $\\{0,1\\}$-matrix of
size $r\leq 3$ is less than $r$, the determinant of $\tilde{R}$ is indeed $\pm
1\in\mathbb{Z}$. Hence $\tilde{S}$ defines an $r$-dimensional subtorus of
$T^{m}$ acting freely on $\mathcal{Z}_{K}$. ∎

##  3\. The case $r=4$ : Preliminaries

###  3.1. Characteristic and dual characteristic maps

Let $A$ be an $n\times m$ matrix over $\mathbb{Z}$ for positive integers
$n\leq m$, and $I$ an $n$-subset of $[m]$. Let $A_{I}$ denote the submatrix of
$A$ formed by selecting columns indexed with $i\in I$, and $A^{I}$ the
submatrix of $A$ formed by selecting rows indexed with $i\in I$. Furthermore,
$\overline{A}$ represents a matrix whose columns form a basis of the kernel of
$A$. Note that $\overline{A}$ depends on the choice of a basis of the kernel
of $A$. We introduce one important proposition, known as the _linear Gale
duality_ :

######  Proposition 3.1.

Let $A$ be an $n\times m$ matrix over $\mathbb{Z}$ for positive integers
$n\leq m$. For any $n$-subset $I$ of $[m]$, $\det(A_{I})=\pm 1$ if and only if
$\det(\overline{A}^{I^{c}})=\pm 1$

Let $K$ be an $(n-1)$-dimensional PL sphere on $[m]$, and $H$ an
$r$-dimensional subtorus of $T^{m}$ freely acting on $\mathcal{Z}_{K}$ as in
(2.1). If $r=m-n$, then $H$ is completely described as follows. Let us
consider a map $\lambda\colon[m]\to\mathbb{Z}^{n}$, called a _characteristic
map_ over $K$, such that $\\{\lambda(i_{1}),\ldots,\lambda(i_{k})\\}$ is a
unimodular set for any simplex $\\{i_{1},\ldots,i_{k}\\}$ in $K$. For
convenience, we often represent this map by an $n\times m$ matrix
$\lambda=\begin{bmatrix}\lambda(1)&\cdots&\lambda(m)\end{bmatrix}$ with
elements in $\mathbb{Z}$. This matrix can be interpreted as a linear map
$\mathbb{Z}^{m}\to\mathbb{Z}^{n}$, and concurrently, as an element in
$\operatorname{Hom}(T^{m},T^{n})$. In addition, we call
$\overline{\lambda}\colon[m]\to\mathbb{Z}^{m-n}$ a _dual characteristic map_
over $K$. Similarly, we define _mod $2$ characteristic maps_
$\lambda^{\mathbb{R}}\colon[m]\to\mathbb{Z}_{2}^{n}$ over $K$ and _mod $2$
dual characteristic maps_
$\overline{\lambda}^{\mathbb{R}}\colon[m]\to\mathbb{Z}_{2}^{m-n}$ over $K$. In
particular, an injective mod $2$ dual characteristic map is simply called an
_IDCM_.

For an $n\times m$ matrix $\lambda$, $\overline{\lambda}$ defines a subtorus
$H$ of $T^{m}$ similar to that described in $\eqref{free_action}$. By
Propositions 2.1 and 3.1, $\lambda$ is a characteristic map over $K$ if and
only if the corresponding subtorus $H$ of $\overline{\lambda}$ acts on
$\mathcal{Z}_{K}$ freely.

When considering the toric space $\mathcal{Z}_{K}/H$, the kernel of $\lambda$
itself is essential whereas the choice of a basis of the kernel is not
important. In this context, we consider the concepts of _Davis-Januszkiewicz
equivalence_ , or simply _D-J equivalence_ , for characteristic maps and dual
characteristic maps. Two characteristic maps are said to be D-J equivalent, if
one is obtained by row operations from the other. Two dual characteristic maps
are said to be D-J equivalent if one is obtained by column operations from the
other. This also removes the ambiguity arising from the definition of
$\overline{\lambda}$.

Observe that the mod $2$ reduction of a characteristic map $\lambda$ over $K$
is a mod $2$ characteristic map over $K$. Conversely, given a mod $2$
characteristic map $\lambda^{\mathbb{R}}\colon[m]\to\mathbb{Z}_{2}^{n}$ over
$K$ and a characteristic map $\lambda\colon[m]\to\mathbb{Z}^{n}$ over $K$, if
$\lambda^{\mathbb{R}}$ coincides with the composition of $\lambda$ and the
modulo $2$ reduction map $\mathbb{Z}^{n}\to\mathbb{Z}_{2}^{n}$, then $\lambda$
is called a _lift_ of $\lambda^{\mathbb{R}}$:

$\mathbb{Z}^{n}$$[m]$$\mathbb{Z}_{2}^{n}$.$\lambda^{\mathbb{R}}$$~{}^{\exists}\lambda\;$$\mod
2$

Moreover, it is called the _$\\{0,1\\}$ -lift_ of $\lambda^{\mathbb{R}}$ when
$\lambda$ sends $[m]$ to $\\{0,1\\}$-vectors. Similarly, it is called a
_$\\{0,\pm 1\\}$ -lift_ when it sends $[m]$ to $\\{0,\pm 1\\}$-vectors. Note
that the number of $\\{0,\pm 1\\}$-lifts of a given mod $2$ characteristic map
is finite.

######  Example 3.2.

Let $K$ be the join
$\partial\Delta^{n_{1}}*\partial\Delta^{n_{2}}*\cdots*\partial\Delta^{n_{p}}$
of the boundaries of $p$ simplices. We denote its set of vertices as

$\\{i_{j}\mid 1\leq i\leq p,1\leq j\leq n_{i}+1\\},$

where $i_{1}$, $i_{2}$, $\ldots$, $i_{n_{i}+1}$ comes from the vertices of
$\partial\Delta^{n_{i}}$. Let
$F^{i}_{j}=\\{i_{1},i_{2},\ldots,i_{n_{j}+1}\\}\setminus\\{i_{j}\\}$. The set
of facets of $K$ is

$\\{\cup_{i=1}^{p}F^{i}_{j_{i}}\mid 1\leq j_{i}\leq{n_{i}+1}\\}.$

By [7], up to D-J equivalence and vertex relabeling, a mod $2$ characteristic
map over $K$ is of the form

$\lambda^{\mathbb{R}}=\begin{bmatrix}\lambda_{1}^{\mathbb{R}}&&&&\\\
\ast&\lambda_{2}^{\mathbb{R}}&&&\\\
\hbox{\multirowsetup$\vdots$}&\ast&\ddots&&\\\
&\vdots&&\lambda_{p-1}^{\mathbb{R}}&\\\
\ast&\ast&&\ast&\lambda_{p}^{\mathbb{R}}\\\ \end{bmatrix},$

where $\lambda^{\mathbb{R}}_{i}$ is a mod $2$ characteristic map over
$\Delta^{n_{i}}$ and the empty spaces display zeros. Up to D-J equivalence and
vertex relabeling,

$\lambda^{\mathbb{R}}_{i}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
i_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle i_{n_{i}}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle i_{n_{i}+1}\\\$\hfil\kern
5.0pt&\hfil$I_{n_{i}}$\hfil &5.0pt\hfil$\scriptstyle\begin{array}[]{c}1\\\
\vdots\\\ 1\end{array}$\hfil\kern 5.0pt\crcr}}}}\right]$}}.$

Let $\widetilde{\lambda^{\mathbb{R}}}$ be the $\\{0,1\\}$-matrix over
$\mathbb{Z}$ such that
$\widetilde{\lambda^{\mathbb{R}}}\equiv\lambda^{\mathbb{R}}$ mod $2$. We
denote by $\widetilde{\lambda^{\mathbb{R}}_{i}}(F^{i}_{j})$ the submatrix
obtained by removing the column indexed by $i_{j}$. Note that
$\det(\widetilde{\lambda^{\mathbb{R}}_{i}}(F^{i}_{j}))=\pm 1$. Then for a
facet $\sigma=\cup_{i=1}^{p}F^{i}_{j_{i}}$ of $K$, the determinant of the
submatrix consisting of the columns of $\widetilde{\lambda^{\mathbb{R}}}$
corresponding to $\sigma$ is
$\det(\widetilde{\lambda^{\mathbb{R}}_{i}}(F^{1}_{j_{1}}))\times\dots\times\det(\widetilde{\lambda^{\mathbb{R}}_{i}}(F^{p}_{j_{p}})|)=\pm
1.$ Hence $\widetilde{\lambda^{\mathbb{R}}}$ is the $\\{0,1\\}$-lift of
$\lambda^{\mathbb{R}}$, and it shows that every mod $2$ characteristic map
over the join of the boundaries of simplices has the $\\{0,1\\}$-lift.

######  Lemma 3.3.

[8] Let $A$ be an $n\times n$ matrix over $\mathbb{Z}$ whose determinant is
odd. Then there is an $n\times n$ matrix $B$ over $\mathbb{Z}$ such that
$\det(B)=\pm 1$ and $A\equiv B$ mod $2$.

######  Proposition 3.4.

The existence of a lift is a property of the D-J class.

###### Proof.

Let $\lambda^{\mathbb{R}}$ and $\mu^{\mathbb{R}}$ be two D-J equivalent mod
$2$ characteristic maps over $K$. There is an invertible matrix $A$ over
$\mathbb{Z}_{2}$ such that $A\lambda^{\mathbb{R}}=\mu^{\mathbb{R}}$. Suppose
that $\widetilde{\lambda^{\mathbb{R}}}$ is a lift of $\lambda^{\mathbb{R}}$.
There is an integer matrix $\widetilde{A}$ such that $\widetilde{A}\equiv A$
mod $2$ and the determinant of $\widetilde{A}$ is odd. Lemma 3.3 ensures that
there is an invertible integer matrix $B$ such that
$B\widetilde{\lambda^{\mathbb{R}}}\equiv\widetilde{A}\widetilde{\lambda^{\mathbb{R}}}\equiv\mu^{\mathbb{R}}$
mod $2$, that is, $B\widetilde{\lambda^{\mathbb{R}}}$ is a lift of
$\mu^{\mathbb{R}}$ as well. ∎

For the sake of convenience, we define the _dual complex_ $\overline{K}$ of
$K$ as the simplicial complex whose facets are the cofacets of $K$. Also, we
regard $\overline{\lambda}$ as a map from $[m]$ to $\mathbb{Z}^{m-n}$ such
that $\overline{\lambda}(i)$ is the $i$th row of $\overline{\lambda}$, as we
did for characteristic maps. Then by the linear Gale duality,
$\overline{\lambda}$ is a characteristic map over $\overline{K}$.

######  Lemma 3.5.

Let $K$ be a simplicial complex. A mod $2$ characteristic map
$\lambda^{\mathbb{R}}$ over $K$ has a lift if and only if
$\overline{\lambda^{\mathbb{R}}}$ has a lift as a mod $2$ characteristic map
over $\overline{K}$.

###### Proof.

Suppose that $\lambda^{\mathbb{R}}$ has a lift
$\widetilde{\lambda^{\mathbb{R}}}$. Then
$\overline{\widetilde{\lambda^{\mathbb{R}}}}$ is a characteristic map over
$\overline{K}$. From the mod $2$ reduction of the equation
$\widetilde{\lambda^{\mathbb{R}}}\times\overline{\widetilde{\lambda^{\mathbb{R}}}}=\mathds{O}$,
the mod $2$ reduction of the columns of
$\overline{\widetilde{\lambda^{\mathbb{R}}}}$ is a basis of
$\ker\lambda^{\mathbb{R}}$. Hence up to D-J equivalence,
$\overline{\lambda^{\mathbb{R}}}\equiv\overline{\widetilde{\lambda^{\mathbb{R}}}}$
mod $2$, that is, $\overline{\widetilde{\lambda^{\mathbb{R}}}}$ is a lift of
$\overline{\lambda^{\mathbb{R}}}$.

The other direction is essentially the same. ∎

######  Problem 3.6 ((toric) Lifting problem).

Let $K$ be a PL sphere. Does any mod $2$ characteristic map over $K$ have a
lift? Equivalently, does any mod $2$ dual characteristic map over $K$ have a
lift as a mod $2$ characteristic map over $\overline{K}$.

###  3.2. Wedge operations

Let $K$ be a simplicial complex on the vertex set $V$ and $\sigma$ a simplex
in $K$. The _link_ of $\sigma$ in $K$ is the simplicial complex defined by

$\operatorname{Lk}_{K}(\sigma)\coloneqq\\{\tau\in K\mid\sigma\cup\tau\in
K,\sigma\cap\tau=\varnothing\\},$

and the _deletion_ of $\sigma$ in $K$ is the simplicial complex defined by

$K\setminus\sigma\coloneqq\\{\tau\in K\mid\sigma\not\subset\tau\\}.$

For a singleton face $\\{v\\}$ of $K$, its link and deletion are denoted
simply by $\operatorname{Lk}_{K}(v)$ and $K\setminus v$, respectively.

For another simplicial complex $L$ on a disjoint vertex set from $K$, the
_join_ $K\ast L$ of $K$ and $L$ is defined as the simplicial complex

$K\ast L\coloneqq\\{\sigma\cup\tau\mid\sigma\in K,\tau\in L\\}.$

The _suspension_ of $K$ is given by

$\operatorname{\Sigma}(K)\partial I\ast K,$

where $I$ is a $1$-simplex with two new vertices $v_{1}$ and $v_{2}$, and
$\partial I$ is its boundary complex. In $\operatorname{\Sigma}(K)$, the pair
$\\{v_{1},v_{2}\\}$ is referred to as a _suspended pair_ , and each vertex in
it is called a _suspended vertex_.

The _wedge_ of $K$ at a vertex $v$ of $K$ is defined as

$\operatorname{Wed}_{v}(K)\coloneqq(I\ast\operatorname{Lk}_{K}(v))\cup(\partial
I\ast(K\setminus v)),$

where $I$ is a $1$-simplex comprising two new vertices. It is evident that the
link of a new vertex added after applying a wedge to $K$ is isomorphic to $K$.
In that sense, we often use $v_{1}$ and $v_{2}$ to refer to the two copies of
$v$ in $\operatorname{Wed}_{v}(K)$. Consequently, $\operatorname{Wed}_{v}(K)$
has vertex set $(V\setminus\\{v\\})\cup\\{v_{1},v_{2}\\}$. Here, two vertices
$v_{1}$ and $v_{2}$ are referred to as _wedged vertices_ of $v$, and the edge
connecting them as the _wedged edge_ of $v$. Notably,
$\operatorname{\Sigma}(K)$ can be viewed as a wedge at a ghost vertex of $K$.

The wedge operation can be defined equivalently as an easy combinatorial
operation on the minimal non-faces of $K$: we duplicate the vertex $v$ in each
minimal non-face of $K$ it appears in. More precisely, let $\eta\subset V$ be
a subset of the vertex set of $K$.

  1. (1)

If $\eta$ contains $v$, then $\eta$ is a minimal non-face of $K$ if and only
if $\eta\setminus\\{v\\}\cup\\{v_{1},v_{2}\\}$ is a minimal non-face of
$\operatorname{Wed}_{v}(K)$.

  2. (2)

If $\eta$ does not contain $v$, then $\eta$ is a minimal non-face of $K$ if
and only if $\eta$ is a minimal non-face of $\operatorname{Wed}_{v}(K)$.

As for suspensions, one can easily prove that the minimal non-faces of
$K_{1}\ast K_{2}$ is the union of the minimal non-faces of $K_{1}$ and
$K_{2}$. Then the minimal non-faces of $\partial I\ast K$ is obtained by
adding $I$ in the minimal non-faces of $K$. we can add a ghost vertex to $K$
which becomes a minimal non-face of $K$. With this perspective, two
consecutive wedge operations and join operations, including suspension, are
associative and commutative with appropriate vertex identification.

Conversely, suppose that there are two vertices $v_{1}$ and $v_{2}$ such that
for any minimal non-face $\eta$ of $K$, $\\{v_{1},v_{2}\\}\subset\eta$ or
$\\{v_{1},v_{2}\\}\cap\eta=\varnothing$. If $\sigma$ is a facet of $K$
containing neither $\\{v_{1}\\}$ nor $\\{v_{2}\\}$, then
$\\{v_{1}\\}\cup\sigma$ is a non-face, so there is a minimal non-face $\eta$
of $K$ containing $v_{1}$. This contradicts to the assumption. Hence every
facet of $K$ contains $v_{1}$ or $v_{2}$. By the following lemma, if
$\\{v_{1},v_{2}\\}$ is not a minimal non-face of $K$, then it is a wedged edge
of $K$, and otherwise, it is a suspended pair of $K$.

######  Lemma 3.7.

[10] Let $K$ be a PL sphere, and $v_{1}$ and $v_{2}$ be two vertices of $K$.
If every facet of $K$ contains $v_{1}$ or $v_{2}$, then $K$ equals to either
$\operatorname{\Sigma}(L)$ with a suspended pair $\\{v_{1},v_{2}\\}$, or
$\operatorname{Wed}_{v}(L)$ with wedged edge $\\{v_{1},v_{2}\\}$ for some
lower dimensional PL sphere $L$.

######  Corollary 3.8.

If a mod $2$ dual characteristic map $\overline{\lambda^{\mathbb{R}}}$ over a
PL sphere $K$ satisfies
$\overline{\lambda^{\mathbb{R}}}(v_{1})=\overline{\lambda^{\mathbb{R}}}(v_{2})$
for some vertices $v_{1}$ and $v_{2}$ of $K$, then $\\{v_{1},v_{2}\\}$ is a
suspended pair or a wedged edge of $K$.

###### Proof.

By the non-singularity of $\overline{\lambda^{\mathbb{R}}}$, every facet of
$K$ contains $v_{1}$ or $v_{2}$. Then apply the previous lemma. ∎

Consider the vertex set of $K$ to be $[m]=\\{1,\ldots,m\\}$. In light of the
associative and commutative nature of wedge operations, we introduce the
notation $K(J)$, termed a _$J$ -construction_ of $K$ in [3], for a positive
integer $m$-tuple $J=(j_{1},j_{2},\ldots,j_{m})$. This represents the
simplicial complex obtained by applying multiple wedge operations to $K$; for
each $i\in[m]$, wedge operations are applied $j_{i}-1$ times to $K$ at $i$ or
its copied vertices. We will often denote the copied vertices of $i$ by
$i_{1},i_{2},\ldots,i_{j_{i}}$. For the sake of convenience, even when
$j_{i}=1$, we treat $i$ as $i_{1}$, and we say $i_{k}$ is a wedged vertex of
$i$ for each $k\geq 1$.

In addition, due to the commutativity and associativity of the operations
involved, we have the relationship:

(3.1) $(\partial I\ast K)(j_{1},j_{2},\ldots,j_{m+2})=\partial
I(j_{1},j_{2})\ast K(j_{3},\ldots,j_{m+2}),$

where $I$ is the $1$-simplex on $\\{1,2\\}$. This leads to two
characterizations regarding the suspension and wedge operations.

######  Proposition 3.9.

Let $K$ be a simplicial complex, $I$ a $1$-simplex, $v$ a vertex of $K$. Then:

  1. (1)

$K$ is a suspension if and only if so is $\operatorname{Wed}_{v}(K)$ for any
non-suspended vertex $v$ of $K$,

  2. (2)

$K$ is a wedge if and only if so is $\Sigma(K)$.

We define the _Picard number_ of $K$ as $m-n$. One can observe that the wedge
operation preserves the Picard number of $K$ whereas the suspension increases
the Picard number of $K$ by $1$. It is known that the link, wedge, and
suspension operations are closed within the class of PL spheres, see [6] for
details.

A PL sphere not isomorphic to some wedge of another PL sphere is termed a
_seed_. It should be noted that any PL sphere $K$ of Picard number $p$ can be
written as $L(J)$, where $L$ is a seed of the same Picard number $p$. In
addition, one can easily see that $L$ is uniquely determined up to
isomorphism, whereas $J$ can be different.

For our purpose, we are interested in PL spheres of dimension $n-1$ on $[m]$
whose real Buchstaber number coincides with their Picard number $m-n$. Such a
PL sphere is said to be _( $\mathbb{Z}_{2}^{n}$-)colorable_. Ewald [16]
observed that all colorable PL spheres are obtained by colorable seeds, and
Choi and Park [10] proved that the number of colorable seeds with given Picard
number is finite. Although obtaining the list of colorable seeds of given
Picard number is a difficult problem in itself, the list up to Picard number
$4$ has been established in [6].

######  Theorem 3.10.

[6] The number of colorable seeds with Picard number at most $4$ up to
isomorphism is as follows:

$p\backslash n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $>11$ | total
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$1$ | $1$ |  |  |  |  |  |  |  |  |  |  |  | $1$
$2$ |  | $1$ |  |  |  |  |  |  |  |  |  |  | $1$
$3$ |  | $1$ | $1$ | $1$ |  |  |  |  |  |  |  |  | $3$
$4$ |  | $1$ | $4$ | $21$ | $142$ | $733$ | $1190$ | $776$ | $243$ | $39$ | $4$ |  | $3153$

with the empty slots displaying zero.

##  4\. Toric wedge induction

###  4.1. Toric wedge induction and its modification

Let $K$ be an $(n-1)$-dimensional PL sphere on $[m]$. There are operations on
mod $2$ characteristic maps over $K$ corresponding to wedge and link
operations on $K$. Let $\Lambda^{\mathbb{R}}$ be a mod $2$ characteristic map
over $\operatorname{Wed}_{v}(K)$. Up to D-J equivalence and vertex relabeling,
we may assume that

(4.1) $\Lambda^{\mathbb{R}}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern
0.0pt\kern 2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
v_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle v_{2}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\\\$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\hbox{\multirowsetup$\mathds{O}$}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\textbf{a}\\\$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\textbf{b}\\\$\hfil\kern
5.0pt&\hfil$\mathds{O}$\hfil &5.0pt\hfil$\scriptstyle I_{n-1}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle A$\hfil\kern 5.0pt\crcr}}}}\right]$}},$

where the column indexes $v_{1}$ and $v_{2}$ stand for the associated wedged
vertices, a and b are row vectors of size $m-n$, $I_{n-1}$ is the identity
matrix of size $n-1$, and $A$ is a $\mathbb{Z}_{2}$-matrix of size
$(n-1)\times(m-n)$. The _projection_ of $\Lambda^{\mathbb{R}}$ with respect to
a face $\sigma$ of $\operatorname{Wed}_{v}(K)$ is a map from the vertex set of
$\operatorname{Lk}_{K}(\sigma)$ to $\mathbb{Z}_{2}^{n-|\sigma|}$ defined by

(4.2)
$\operatorname{Proj}_{\sigma}(\Lambda^{\mathbb{R}})(w)=[\Lambda^{\mathbb{R}}(w)]\in\faktor{\mathbb{Z}_{2}^{n}}{\left<\Lambda^{\mathbb{R}}(v)\mid
v\in\sigma\right>}\cong\mathbb{Z}_{2}^{n-|\sigma|}$

for each vertex $w$ of $\operatorname{Lk}_{K}(\sigma)$. If we fix a basis of
$\mathbb{Z}_{2}^{n-|\sigma|}$, we can see that
$\operatorname{Proj}_{\sigma}(\Lambda^{\mathbb{R}})$ is a mod $2$
characteristic map over $\operatorname{Lk}_{K}(\sigma)$. We call
$\operatorname{Proj}_{\sigma}(\Lambda^{\mathbb{R}})$ the _projection_ onto
$\operatorname{Lk}_{K}(\sigma)$.

The links $\operatorname{Lk}_{\operatorname{Wed}_{v}(K)}(v_{1})$ and
$\operatorname{Lk}_{\operatorname{Wed}_{v}(K)}(v_{2})$ are isomorphic to $K$
by identifying $v_{2}$ and $v_{1}$ with $v$, respectively, and the projections
of $\Lambda^{\mathbb{R}}$ with respect to $v_{1}$ and $v_{2}$ are written as

$\lambda^{\mathbb{R}}_{1}=\operatorname{Proj}_{v_{1}}(\Lambda^{\mathbb{R}})=\hbox{}\vbox{\kern
0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
v_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\\\$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\mathds{O}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\textbf{b}\\\$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\mathds{O}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle I_{n-1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
A$\hfil\kern 5.0pt\crcr}}}}\right]$}},\text{ and }$
$\lambda^{\mathbb{R}}_{2}=\operatorname{Proj}_{v_{2}}(\Lambda^{\mathbb{R}})=\hbox{}\vbox{\kern
0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
v_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\\\$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\mathds{O}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\textbf{a}\\\$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\mathds{O}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle I_{n-1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
A$\hfil\kern 5.0pt\crcr}}}}\right]$}}.$

If we consider the first column of each matrix corresponds to the vertex $v$,
then two matrices $\lambda^{\mathbb{R}}_{1}$ and $\lambda^{\mathbb{R}}_{2}$
are mod $2$ characteristic maps over $K$. Hence, $\Lambda^{\mathbb{R}}$
corresponds to a choice of two mod $2$ characteristic maps over $K$ whose
first $n$ columns form an identity matrix such that their submatrices formed
by deleting the first row and the first column are identical up to D-J
equivalence.

Conversely, one may construct at most one $\Lambda^{\mathbb{R}}$ over
$\operatorname{Wed}_{v}(K)$ from an ordered pair of mod $2$ characteristic
maps $\lambda_{1}^{\mathbb{R}}$ and $\lambda_{2}^{\mathbb{R}}$ over $K$ as in
(4.1). We denote
$\Lambda^{\mathbb{R}}=\lambda_{1}^{\mathbb{R}}\wedge_{v}\lambda_{2}^{\mathbb{R}}$
if it exists. If
$\lambda_{1}^{\mathbb{R}}=\lambda_{2}^{\mathbb{R}}=\lambda^{R}$, then
$\lambda^{\mathbb{R}}\wedge_{v}\lambda^{\mathbb{R}}$ always exists for any
$v$, and it is called the _canonical extension_ of $\lambda^{\mathbb{R}}$ at
$v$.

It should be noted that we can represent each characteristic map over $K(J)$
by a combination of characteristic maps over $K$. From this viewpoint, we
shall introduce one powerful inductive tool to demonstrate some properties on
real toric spaces, for example the existence of a lift of
$\Lambda^{\mathbb{R}}$ as in this paper.

For a PL sphere $K$, let $\mathcal{X}$ be a collection of pairs
$(L,\lambda^{\mathbb{R}})$ such that every $L$ is expressed as $K(J)$ for some
$J$, and $\lambda^{\mathbb{R}}$ is a mod $2$ characteristic map over $L$. Then
$\mathcal{X}$ is called a _wedge-stable set based on $K$_ if
$(\operatorname{Wed}_{v}(L),\lambda_{1}^{\mathbb{R}}\wedge\lambda_{2}^{\mathbb{R}})\in\mathcal{X}$
whenever both $(L,\lambda_{1}^{\mathbb{R}})$ and
$(L,\lambda_{2}^{\mathbb{R}})$ are in $\mathcal{X}$.

We present the concept of _toric wedge induction_ which is a method employed
to demonstrate the validity of a given property across $\mathcal{X}$.

######  Proposition 4.1 (Toric wedge induction).

For a PL sphere $K$, let $\mathcal{X}$ be a wedge-stable set based on $K$, and
$\mathcal{P}$ a property. Suppose that the following holds;

  1. (1)

Basis step: All $(K,\lambda^{\mathbb{R}})\in\mathcal{X}$ satisfies
$\mathcal{P}$ .

  2. (2)

Inductive step: If $(L,\lambda_{1}^{\mathbb{R}})$,
$(L,\lambda_{2}^{\mathbb{R}})\in\mathcal{X}$ satisfy $\mathcal{P}$, then so
does
$(\operatorname{Wed}_{v}(L),\lambda_{1}^{\mathbb{R}}\wedge_{v}\lambda_{2}^{\mathbb{R}})$
for any vertex $v$ of $L$.

Then $\mathcal{P}$ holds on $\mathcal{X}$.

The credit of original idea of toric wedge induction should be given to Choi
and Park [8]. They used it for showing the projectivity of certain toric
manifolds in [8] or [9]. Later, the authors of this paper used it for
classifying toric manifolds satisfying equality within an inequality regarding
the number of minimal components in their rational curve space [6].

However, it is sometimes challenging to perform the inductive step. In that
situation, we can relax it by strengthening the basis step. In this paper, we
introduce a new, easier version of this method that helps with the toric
lifting property we want to show.

In order to do it, we briefly review the notions and properties, following
[10], where the reader may find a much more details about the relations
between characteristic maps over $K(J)$ and puzzles explained below.

The _pre-diagram $D^{\prime}(K)$_ of $K$ is an edge-colored non-simple graph
such that

  1. (1)

the node set of $D^{\prime}(K)$ is the set of D-J classes of the mod $2$
characteristic maps over $K$,

  2. (2)

for a vertex $v$ of $K$ and two mod $2$ characteristic maps
$\lambda^{\mathbb{R}}_{1}$ and $\lambda^{\mathbb{R}}_{2}$ over $K$, a pair
$(\\{\lambda^{\mathbb{R}}_{1},\lambda^{\mathbb{R}}_{2}\\},v)$ is a colored
edge of $D^{\prime}(K)$ if and only if there is a mod $2$ characteristic map
over $\operatorname{Wed}_{v}(K)$ whose two projections onto $K$ are
$\lambda^{\mathbb{R}}_{1}$ and $\lambda^{\mathbb{R}}_{2}$.

We denote $G(J)$ the $1$-skeleton of the simple polytope
$P(J)\coloneqq\Delta^{j_{1}-1}\times\Delta^{j_{2}-1}\times\dots\times\Delta^{j_{m}-1}$.
Each edge $\bm{\epsilon}$ of $G(J)$ is uniquely written as

$\displaystyle\bm{\epsilon}=\alpha_{1}\times\alpha_{2}\times\dots\times\alpha_{v-1}\times\epsilon_{v}\times\alpha_{v+1}\times\dots\times\alpha_{m},$

where $\alpha_{i}$ is a vertex of $\Delta^{j_{i}-1}$, $1\leq i\leq m,i\not=v$,
and $\epsilon_{v}$ is an edge of $\Delta^{j_{v}-1}$. Then color
$\bm{\epsilon}$ by $v$.

Then, a mod $2$ characteristic map $\lambda^{\mathbb{R}}$ over $K(J)$ can be
expressed by an edge-colored graph homomorphism $\phi\colon G(J)\to
D^{\prime}(K)$; When $\bm{\alpha}$ is a vertex of $G(J)$, we can write

$\displaystyle\bm{\alpha}=\alpha_{1}\times\alpha_{2}\times\dots\times\alpha_{m},$

where $1\leq\alpha_{i}\leq j_{i}$ is a vertex of $\Delta^{j_{i}-1}$ for $1\leq
i\leq m$. Observe that

$\displaystyle\mathcal{F}(\bm{\alpha})\coloneqq\\{1_{1},1_{2},\ldots,1_{j_{1}},\ldots,m_{1},m_{2},\ldots,m_{j_{m}}\\}\setminus\\{1_{\alpha_{1}},2_{\alpha_{2}},\ldots,m_{\alpha_{m}}\\}$

does not contain any minimal non-face of $K(J)$, so it is a face of $K(J)$.
Since each vertex $i_{j_{k}}$ ($j_{k}\not=\alpha_{i}$) in
$\mathcal{F}(\bm{\alpha})$ is a wedged vertex of $i_{\alpha_{i}}$,
$\operatorname{Lk}_{K(J)}(\mathcal{F}(\bm{\alpha}))$ is isomorphic to $K$, and
the projection $\operatorname{Proj}_{\mathcal{F}(\bm{\alpha})}(\lambda^{R})$
is a mod $2$ characteristic map over $K$ by the natural bijection between
$\\{1,2,\ldots,m\\}$ and
$\\{1_{\alpha_{1}},2_{\alpha_{2}},\ldots,m_{\alpha_{m}}\\}$. Define $\phi$ by
$\phi(\bm{\alpha})=\operatorname{Proj}_{\mathcal{F}(\bm{\alpha})}(\lambda^{\mathbb{R}})$
for each vertex $\bm{\alpha}$ of $G(J)$. Let $\epsilon$ be an edge of $G(J)$.
Up to relabeling the vertices of $K$, $\epsilon$ consists of two vertices
$\alpha$ and
$\alpha^{\prime}\coloneqq\alpha^{\prime}_{1}\times\alpha_{2}\times\dots\times\alpha_{m}$.
Then
$\phi(\epsilon)=\\{\operatorname{Proj}_{\mathcal{F}(\bm{\alpha})}(\lambda^{\mathbb{R}}),\operatorname{Proj}_{\mathcal{F}(\bm{\alpha^{\prime}})}(\lambda^{\mathbb{R}})\\}$.
By the following proposition,
$\overline{\operatorname{Proj}_{\mathcal{F}(\bm{\alpha})}(\lambda^{\mathbb{R}})}$
and
$\overline{\operatorname{Proj}_{\mathcal{F}(\bm{\alpha^{\prime}})}(\lambda^{\mathbb{R}})}$
are same except their first rows. Hence $\phi$ is an edge-colored graph
homomorphism.

######  Remark 4.2.

In the above situation, it is worthy to note that if
$\overline{\lambda^{\mathbb{R}}}(1_{\alpha_{1}})=\overline{\lambda^{\mathbb{R}}}(1_{\alpha^{\prime}_{1}})$,
then $\phi$ is not irreducible.

######  Proposition 4.3.

Let $K$ be a PL sphere, and $\sigma$ a face of $K$ such that the Picard
numbers of $K$ and $\operatorname{Lk}_{K}(\sigma)$ are same. Up to D-J
equivalence, the dual of the projection of a mod $2$ characteristic map
$\lambda^{\mathbb{R}}$ with respect to a face $\sigma$ of $K$ is obtained by
removing rows corresponding to $\sigma$ in $\overline{\lambda^{\mathbb{R}}}$.

###### Proof.

It is sufficient to prove the result for a vertex $\sigma=\\{v\\}$. To project
$\lambda^{\mathbb{R}}$ with respect to $\\{v\\}$, let us left multiply
$\lambda^{\mathbb{R}}$ by an invertible matrix $g$ so that the vector
$g\lambda^{\mathbb{R}}(v)$ has a single nonzero entry, say at index $k$. From
(4.2), if we delete the $v$th column and $k$th row of $g\lambda^{\mathbb{R}}$,
then we obtain the projection $\operatorname{Proj}_{v}(\lambda^{\mathbb{R}})$.
Let $\overline{\mu^{\mathbb{R}}}$ be the matrix obtained by removing $v$th row
in $\overline{\lambda^{\mathbb{R}}}$. Then, any column of
$\overline{\mu^{\mathbb{R}}}$ is an element of the kernel of
$\operatorname{Proj}_{v}(\lambda^{\mathbb{R}})$ since every component of the
$v$th column of $\lambda^{\mathbb{R}}$ except the $k$th one is $0$. Note that
$\overline{\mu^{\mathbb{R}}}$ remains of full rank since there must exist a
cofacet of $K$ that does not contain $v$. ∎

However, not all edge-colored graph homomorphism $\phi$ is obtained from a mod
$2$ characteristic map over $K(J)$. If it is, $\phi$ is called a _(realizable)
puzzle_ over $K(J)$ (or, over $K$, if there is no confusion), and
$\lambda_{\phi}^{\mathbb{R}}$ denotes the corresponding mod $2$ characteristic
map over $K(J)$. In particular, a puzzle which does not contain any edge
corresponding to a canonical extension is called _irreducible_.

Consider a realizable puzzle $\phi$ over $K(J)$. In $G(J)$, two edges

$\displaystyle\bm{\epsilon}$
$\displaystyle=\alpha_{1}\times\alpha_{2}\times\dots\times\alpha_{v-1}\times\epsilon_{v}\times\alpha_{v+1}\times\dots\times\alpha_{m},$
$\displaystyle\bm{\epsilon}^{\prime}$
$\displaystyle=\alpha^{\prime}_{1}\times\alpha^{\prime}_{2}\times\dots\times\alpha^{\prime}_{v-1}\times\epsilon^{\prime}_{v}\times\alpha^{\prime}_{v+1}\times\dots\times\alpha^{\prime}_{m}$

are called _parallel_ if $\epsilon_{v}=\epsilon^{\prime}_{v}$. By [10,
Corollary 4.4], if there is an edge $\bm{\epsilon}$ of $\phi$ corresponding to
a canonical extension, then so does every parallel edge to $\bm{\epsilon}$.
Therefore, every puzzle is obtainable from an irreducible puzzle by a sequence
of canonical extensions.

######  Proposition 4.4 (Modified toric wedge induction).

For a PL sphere $K$, let $\mathcal{X}$ be a wedge-stable set based on $K$, and
$\mathcal{P}$ a property. Suppose that the following holds;

  1. (1)

Basis step: For any positive integer tuple $J$ and any irreducible realizable
puzzle $\phi$ over $K(J)$, $(K(J),\lambda_{\phi}^{\mathbb{R}})$ satisfies
$\mathcal{P}$.

  2. (2)

Inductive step: If $(L,\lambda^{\mathbb{R}})\in\mathcal{X}$ satisfies
$\mathcal{P}$, then so does the pair consisting of the wedge of $L$ at $v$ and
the canonical extension of $\lambda^{\mathbb{R}}$ at $v$ for any vertex $v$ of
$L$.

Then $\mathcal{P}$ holds on $\mathcal{X}$.

It should be noted that the basis step consists of finitely many cases as the
lemma below.

######  Lemma 4.5.

For a simplicial complex $K$, there are finitely many irreducible puzzles over
$K$.

###### Proof.

Let $J=(j_{1},j_{2},\ldots,j_{m})$. Fix a vertex
$\bm{\alpha}=\alpha_{1}\times\alpha_{2}\times\dots\times\alpha_{m}$ of $P(J)$.
For each $j_{v}$, $\bm{\alpha}$ is also a vertex of the simplex
$\alpha_{1}\times\alpha_{2}\times\dots\times\alpha_{v-1}\times\Delta^{j_{v}-1}\times\alpha_{v+1}\times\dots\times\alpha_{m}$.
Because any two vertices of $\Delta^{j_{v}-1}$ forms an edge, the number
$j_{v}$ can not exceed the number of mod $2$ characteristic maps over $K$. ∎

######  Remark 4.6.

The concept of wedge of characteristic maps and puzzle is not only described
for mod $2$ characteristic maps, but also for characteristic maps. See [10]
for details. This means that we can apply toric wedge induction to a
collection of toric spaces. In addition, the number of PL spheres which admit
a characteristic map is less than or equal to the number of PL spheres which
admit a mod $2$ characteristic map, so it is finite. However, the number of
characteristic maps over a seed is not finite. Hence the basis step may not be
implemented by direct computations in finite steps.

###  4.2. Modified Toric wedge induction in terms of dual characteristic maps

Even if the modified version (Proposition 4.4) of the toric wedge induction
has an easier inductive step than the original version (Proposition 4.1),
there still remains a challenging part to deal with: constructing
(irreducible) puzzles. In this subsection, we characterize irreducible puzzles
in terms of dual characteristic maps over seeds, and restate the modified
toric wedge induction based on a seed using dual characteristic maps instead
of irreducible puzzles.

Let $K$ be a colorable seed of dimension $n-1$ on $[m]$,
$\lambda^{\mathbb{R}}$ a mod $2$ characteristic map over $K$. Assume that
there are two vertices of $K$ such that
$\overline{\lambda^{\mathbb{R}}}(v)=\overline{\lambda^{\mathbb{R}}}(w)$. By
Remark 3.8, $K$ has to be a suspension, and then by Proposition 3.9, $K$ is
the suspension of a seed. Let us call a seed that is a suspension a _suspended
seed_. Hence if $K$ is a non-suspended seed, then every dual characteristic
map over $K$ is injective. An _IDCM_ denotes an injective mod $2$ dual
characteristic map. By the following lemma, every irreducible puzzle over a
non-suspended seed corresponds to an IDCM.

######  Lemma 4.7.

Let $\phi$ be an irreducible puzzle over $K(J)$ for a PL sphere $K$ on $[m]$
and a positive integer $m$-tuple $J$. Then
$\overline{\lambda_{\phi}^{\mathbb{R}}}$ is injective if and only if
$\overline{\phi(\bm{\alpha})}$ is injective for any vertex $\bm{\alpha}$ of
$G(J)$.

###### Proof.

Assume that $\overline{\lambda_{\phi}^{\mathbb{R}}}$ is injective. By
Proposition 4.3, each projection $\overline{\phi(\bm{\alpha})}$ of
$\lambda_{\phi}^{\mathbb{R}}$ has no repeated rows, so it is injective.

Conversely, suppose that
$\overline{\lambda_{\phi}^{\mathbb{R}}}(v)=\overline{\lambda_{\phi}^{\mathbb{R}}}(w)$
for some vertices $v$, $w$ of $K(J)$. By Remark 4.2 and the irreducibility of
$\phi$, $\\{v,w\\}$ cannot be copies of one vertex of $K$ by wedge operations.
Hence there is a vertex $\bm{\alpha}$ of $G(J)$ such that $v=k_{\alpha_{k}}$
and $w=l_{\alpha_{l}}$ for some $\alpha_{k}\leq j_{k}$, $\alpha_{l}\leq
j_{l}$, and distinct vertices $k,l\in[m]$ and . This yields
$\overline{\operatorname{Proj}_{\mathcal{F}(\bm{\alpha})}(\lambda_{\phi}^{\mathbb{R}})}(v)=\overline{\operatorname{Proj}_{\mathcal{F}(\bm{\alpha})}(\lambda_{\phi}^{\mathbb{R}})}(w)$,
so $\overline{\phi(\alpha)}$ is not injective. ∎

In general, a seed $K$ is of the form $\partial I_{1}\ast\dots\ast\partial
I_{q}\ast L$, where $I_{k}$ is the $1$-simplex with vertices $2k-1$ and $2k$
for each $k\leq q$, and $L$ is a non-suspended seed. By (3.1),

(4.3) $K(J)=\partial I_{1}(J_{1})\ast\dots\ast\partial I_{q}(J_{q})\ast
L(J_{q+1}).$

Before studying mod $2$ characteristic maps over $K(J)$, we need the following
analysis of mod $2$ characteristic maps over the join of two simplicial
complexes. Research on (integral) characteristic maps over the join $K*L$ is
well-established in references such as [11] and [14], and it can be converted
well to mod $2$ characteristic maps. Refer to these for further information.
Consider the join of simplicial complexes $K_{1}$ of dimension $n_{1}-1$ with
$m_{1}$ vertices and $K_{2}$ of dimension $n_{2}-1$ with $m_{2}$ vertices. One
can observe that any (mod $2$) characteristic map $\lambda^{\mathbb{R}}$ over
$K_{1}\ast K_{2}$ has the following form;

$\lambda^{\mathbb{R}}=\begin{bmatrix}\lambda^{\mathbb{R}}_{11}&\lambda^{\mathbb{R}}_{12}\\\
\lambda^{\mathbb{R}}_{21}&\lambda^{\mathbb{R}}_{22}\end{bmatrix},$

where $\lambda^{\mathbb{R}}_{11}$, $\lambda^{\mathbb{R}}_{22}$ are mod $2$
characteristic maps over $K_{1}$, $K_{2}$ respectively, see [11, Lemma 3.1]
for instance. Moreover, we can assume that the first $n_{1}$ columns of
$\lambda^{\mathbb{R}}_{11}$, and the first $n_{2}$ columns of
$\lambda^{\mathbb{R}}_{22}$ form identity matrices by D-J equivalence and
vertex relabeling. Then, up to D-J equivalence,

(4.4) $\lambda^{\mathbb{R}}=\begin{bmatrix}I_{n_{1}}&A&\mathds{O}&B\\\
\mathds{O}&C&I_{n_{2}}&D\end{bmatrix}\text{ and
}\overline{\lambda^{\mathbb{R}}}=\begin{bmatrix}A&B\\\
I_{m_{1}-n_{1}}&\mathds{O}\\\ C&D\\\ \mathds{O}&I_{m_{2}-n_{2}}\end{bmatrix}.$

We call this form of $\lambda^{\mathbb{R}}$ the _joining representative_ of
$\lambda^{\mathbb{R}}$. It should be noted that $\begin{bmatrix}A\\\
I_{m_{1}-n_{1}}\end{bmatrix}$ and $\begin{bmatrix}D\\\
I_{m_{2}-n_{1}}\end{bmatrix}$ are mod $2$ dual characteristic maps over
$K_{1}$ and $K_{2}$, respectively.

######  Lemma 4.8.

Let $K$ be a seed and $\phi$ an irreducible puzzle over its $J$-construction
(4.3). Suppose that there are two vertices $v$ and $w$ of $K(J)$ such that
$\overline{\lambda_{\phi}^{\mathbb{R}}}(v)=\overline{\lambda_{\phi}^{\mathbb{R}}}(w)$.
Then there exists a suspended pair $\\{2k-1,2k\\}$ of $K$ such that
$v=(2k-1)_{s}$, $w=(2k)_{t}$ for some $s\leq j_{2k-1}$ and $t\leq j_{2k}$.

###### Proof.

By Remark 3.8, $\\{v,w\\}$ is a suspended pair or a wedged edge of $K(J)$. In
the former case, we are done.

Let $\\{v,w\\}$ be a wedged edge of $K(J)$. As written in the proof of Lemma
4.7, there are two distinct vertices $l$ and $l^{\prime}$ of $K$ such that
$v=l_{s}$, $w=l^{\prime}_{t}$ for some $s\leq j_{l}$ and $t\leq
j_{l^{\prime}}$. By commutativity of wedge and join operations, $\\{v,w\\}$ is
a wedged edge of $L(J_{q+1})$ or $\partial I_{k}(J_{k})$ for some $k$. If $l$
and $l^{\prime}$ are contained in some $\partial I_{k}(J_{k})$, then they form
a suspended pair since they are distinct vertices.

Assume that $l$ and $l^{\prime}$ are contained in $L$. Let $\bm{\alpha}$ be a
vertex of $G(J)$ such that its $l$th and $l^{\prime}$th components are $s$ and
$t$, respectively. Then $\phi(\bm{\alpha})$ is a mod $2$ characteristic map
over $K$ such that
$\overline{\phi(\bm{\alpha})}(l)=\overline{\phi(\bm{\alpha})}(l^{\prime})$.
Consider the joining representative (4.4) of $\phi(\bm{\alpha})$ over $K$ by
setting $K_{1}=\partial I_{1}\ast\dots\ast\partial I_{q}$ and $K_{2}=L$. Then
the mod $2$ dual characteristic map $\begin{bmatrix}D\\\
I_{m_{2}-n_{2}}\end{bmatrix}$ over the non-suspended seed $L$ has repeated two
rows at $l$ and $l^{\prime}$, which contradicts that a non-suspended seed only
admits IDCMs. ∎

Suppose that $\phi$ is an irreducible puzzle over $K(J)$, and there are two
vertices $v$ and $w$ of $K(J)$ such that
$\overline{\lambda^{\mathbb{R}}_{\phi}}(v)=\overline{\lambda^{\mathbb{R}}_{\phi}}(w)$.
By Lemma 4.8, there is a suspended pair of $\\{2k-1,2k\\}$ of $K$ such that
$v=(2k-1)_{s}$ and $w=(2k)_{t}$. If $\\{v,w\\}$ is not a suspended pair, that
is $J_{k}\not=(1,1)$, then it is a wedged edge of $K(J)$ by Corollary 3.8.
Without loss of generality, we can assume that $j_{2k-1}\geq 2$. The
irreducibility of $\phi$ ensures
$\overline{\lambda^{\mathbb{R}}_{\phi}}((2k-1)_{1})\not=\overline{\lambda^{\mathbb{R}}_{\phi}}((2k-1)_{2})$.

Consider a positive integer tuple
$J^{\prime}=(j^{\prime}_{1},\dots,j^{\prime}_{m})$ such that
$j^{\prime}_{2k-1}=1$, $j^{\prime}_{2k}=j_{2k-1}+j_{2k}-1$, and
$j^{\prime}_{i}=j_{i}$ for $i\not=2k-1,2k$. Define a simplicial map $f\colon
K(J)\to K(J^{\prime})$ by

(4.5) $f(x)=\begin{cases}(2k)_{j_{2k}+l-1},&\mbox{if }x=(2k-1)_{l}\mbox{ for
}l>1\\\ x,&\mbox{otherwise}.\end{cases}$

Since any J-constructions of $\partial I_{k}$ is a simplex, $f$ is an
isomorphism. Then $\lambda^{\mathbb{R}}_{\phi}\circ f^{-1}$ is a mod $2$
characteristic map over $K(J^{\prime})$. Note that this does not correspond to
an irreducible puzzle over $K(J^{\prime})$. Hence we obtain the following
theorem.

######  Theorem 4.9.

Let $K$ be a seed, and $\phi$ an irreducible puzzle over $K(J)$. Suppose that
there are two vertices $v$ and $w$ of $K(J)$ such that
$\overline{\lambda_{\phi}^{\mathbb{R}}}(v)=\overline{\lambda_{\phi}^{\mathbb{R}}}(w)$.
If $\\{v,w\\}$ is not a suspended pair, then there exists an isomorphism
$f\colon K(J^{\prime})\to K(J)$ for a positive integer tuple $J^{\prime}$ such
that $\lambda_{\phi}\circ f$ does not correspond to an irreducible puzzle over
$K(J^{\prime})$.

We call $\overline{\lambda^{\mathbb{R}}}$ _quasi-injective_ if
$\overline{\lambda^{\mathbb{R}}}(v)=\overline{\lambda^{\mathbb{R}}}(w)$
implies that $\\{v,w\\}$ is a suspended pair of $K$. From the above theorem,
we can restate the modified toric wedge induction based on a seed as the
following.

######  Proposition 4.10.

Let $K$ be a colorable seed, and

$\mathcal{X}_{K}=\\{(L,\lambda^{\mathbb{R}})\mid L=K(J)\text{ for some }J,\
\lambda^{\mathbb{R}}\text{ is a mod~{}$2$ characteristic map over }L\\}.$

For a property $\mathcal{P}$, suppose that the following holds;

  1. (1)

Basis step: For any positive integer tuple $J$ and any quasi-IDCM
$\overline{\lambda^{\mathbb{R}}}$ over $K(J)$, $(K(J),\lambda^{\mathbb{R}})$
satisfies $\mathcal{P}$.

  2. (2)

Inductive step: If $(L,\lambda^{\mathbb{R}})\in\mathcal{X}_{K}$ satisfies
$\mathcal{P}$, then so does the pair consisting of the wedge of $L$ at $v$ and
the canonical extension of $\lambda^{\mathbb{R}}$ at $v$ for any vertex $v$ of
$L$.

Then $\mathcal{P}$ holds on $\mathcal{X}_{K}$.

Even though Lemma 4.5 ensures that the basis step of modified toric wedge
induction is a finite problem, we can see directly that in this form. The
number of suspended pairs can not exceed the Picard number, so the number of
rows of a quasi-IDCM can not exceed $p+2^{p}-1$.

######  Remark 4.11.

In particular, (modified) toric wedge induction is useful when we want to see
a property for all real toric spaces over PL spheres of Picard number $p$. By
the injectivity of mod $2$ dual characteristic maps over non-suspended seeds,
if $K$ is a non-suspended seed, then we have $m\leq 2^{p}-1$, so there are
finitely many non-suspended seeds of Picard number $p$. Since suspended seeds
of Picard number $p$ are suspensions of non-suspended and suspended seeds of
Picard number $p-1$, there are finitely many suspended seeds of Picard number
$p$ as well. Hence, Lemma 4.5 guarantees that the basis steps of (modified)
toric wedge induction based on all seeds of Picard number $p$ can be also
solved in finite time.

##  5\. Proof of the main theorem

For the reader’s convenience, we recall the statement of the main theorem. See
1.2

###### Proof.

Let $r$ be the rank of a subgroup acting freely on
$\mathbb{R}\mathcal{Z}_{K}$. For the case $r\leq 3$, the statement is verified
by Theorem 2.2. Hence, it is enough to consider the case when $m=n+4$, and
$r=m-n=4$.

We apply the modified version of toric wedge induction with $\mathcal{P}$ as
in Proposition 4.10. Let $K$ be a seed. For a positive integer tuple $J$ and a
mod $2$ characteristic map $\lambda^{\mathbb{R}}$ over $K(J)$, we say the pair
$(K(J),\lambda^{\mathbb{R}})$ satisfies $\mathcal{P}$ if and only if the mod
$2$ characteristic map $\lambda^{\mathbb{R}}$ has a lift. The basis step of
any seed of Picard number $4$ will be accomplished in Section 6. The inductive
step follows from Lemma 5.1 below. Therefore, by Proposition 4.10,
$\mathcal{P}$ holds on the set of all real toric spaces of Picard number $4$.
∎

######  Lemma 5.1 (Inductive Step).

For a PL sphere $K$, if a mod $2$ characteristic map $\lambda^{\mathbb{R}}$
over $K$ has a lift, then the canonical extension of $\lambda^{\mathbb{R}}$ at
$v$ has a lift for any vertex $v$ of $K$.

###### Proof.

Let $K$ be an $(n-1)$-dimensional PL sphere on $[m]$. By relabeling the
vertices and D-J equivalence, we may assume that the first $n$ columns of
$\lambda^{\mathbb{R}}$ and the last $m-n$ columns of
$\overline{\lambda}^{\mathbb{R}}$ are of form the identity matrix, and the
wedge operation is performed at $v=1\in[m]$. Let
$\lambda=\widetilde{\lambda^{\mathbb{R}}}$ be a lift of
$\lambda^{\mathbb{R}}$. By Proposition 3.4, up to D-J equivalence, we can also
assume that its first $n$ columns form the $n\times n$ identity matrix.

Set
$\lambda^{\mathbb{R}}=\begin{bmatrix}\hbox{\multirowsetup$I_{n}$}&\overline{\lambda^{\mathbb{R}}}(1)\\\
&\overline{\lambda^{\mathbb{R}}}(2)\\\ &\vdots\\\
&\overline{\lambda^{\mathbb{R}}}(n)\end{bmatrix}$ and its lift
$\lambda=\begin{bmatrix}\hbox{\multirowsetup$I_{n}$}&\overline{\lambda}(1)\\\
&\overline{\lambda}(2)\\\ &\vdots\\\ &\overline{\lambda}(n)\end{bmatrix}$.
Then,

$\Lambda^{\mathbb{R}}=\begin{bmatrix}\hbox{\multirowsetup$I_{n+1}$}&\overline{\lambda^{\mathbb{R}}}(1)\\\
&\overline{\lambda^{\mathbb{R}}}(1)\\\ &\overline{\lambda^{\mathbb{R}}}(2)\\\
&\vdots\\\ &\overline{\lambda^{\mathbb{R}}}(n)\end{bmatrix}\text{ and
}\Lambda=\begin{bmatrix}\hbox{\multirowsetup$I_{n+1}$}&\overline{\lambda}(1)\\\
&\overline{\lambda}(1)\\\ &\overline{\lambda}(2)\\\ &\vdots\\\
&\overline{\lambda}(n)\end{bmatrix}$

are the canonical extension of $\lambda^{\mathbb{R}}$ at $1$ and its lift,
respectively. ∎

##  6\. The basis step

In this section, we prove that every quasi-IDCM over a PL sphere $K$ of Picard
number $4$ has a lift as a mod $2$ characteristic map over $\overline{K}$.
Then the basis step of Proposition 4.10 is accomplished by Lemma 3.5.

Let $K$ be an $n-1$ dimensional PL sphere of Picard number $4$. Suppose that
$K$ admits a quasi-IDCM $\lambda^{\mathbb{R}}$ that is not an IDCM. Then if
$K=L(J)$ for some seed $L$, then by Theorem 3.10, $L$ is one of the following:

  1. (1)

$L_{1}=\partial I\ast\partial P_{5}$,

  2. (2)

$L_{2}=\partial I_{1}\ast\partial I_{2}\ast\partial I_{3}\ast\partial I_{4}$,

  3. (3)

$L_{3}=\partial I\ast\partial C^{4}(7)$,

where $P_{5}$ is a pentagon, and $C^{4}(7)$ is a $4$-dimensional cyclic
polytope with $7$ vertices. By Example 3.2, if $K=L_{2}(J)$, then every mod
$2$ characteristic map over $K$ has a lift, so assume that $L\not=L_{2}$ in
addition. Since there is no seed of Picard number $2$ with $n=1$ and $3$ by
Theorem 3.10, we can see that $\partial P_{5}$ and $\partial C^{4}(7)$ are
non-suspended seeds. Hence $\lambda^{\mathbb{R}}$ has exactly one pair of
vertices with row repetition.

For convenience, let $L^{\prime}=\partial P_{5}$ or $\partial C^{4}(7)$. By
(3.1), we can write $K=L(J)=\partial I\ast L^{\prime}(J_{2})$. With setting
$K_{1}=\partial I$ and $K_{2}=L^{\prime}(J_{2})$, $\lambda^{\mathbb{R}}$ is of
the form by (4.4):

$\lambda^{\mathbb{R}}=\begin{bmatrix}1&1&\mathds{O}\\\
\mathds{O}&\ast&\mu^{\mathbb{R}}\end{bmatrix},$

where $\mu^{\mathbb{R}}$ is a mod $2$ characteristic map over
$L^{\prime}(J_{2})$. By the definition of the join, a subset $\sigma$ of the
vertex set of $K$ is facet of $K$ if and only if $\sigma=\\{v\\}\cup\tau$ for
a vertex $v$ of $\partial I$ and a facet $\tau$ of $L^{\prime}(J_{2})$. By
Theorem 2.2, there is a lift $\widetilde{\mu^{\mathbb{R}}}$ of
$\mu^{\mathbb{R}}$. Then

$\begin{bmatrix}1&1&\mathds{O}\\\
\mathds{O}&\ast&\widetilde{\mu^{\mathbb{R}}}\end{bmatrix}$

is a lift of $\lambda^{\mathbb{R}}$.

In the rest of the section, let us focus only on IDCMs.

######  Theorem 6.1.

Let $K$ be a PL sphere with less than $168$ facets. Then every injective mod
$2$ dual characteristic map over $K$ has a lift as a mod $2$ characteristic
map over $\overline{K}$.

###### Proof.

Let $A$ be a $4\times 15$ matrix over $\mathbb{Z}$ consisting of only $0,1$
entries with neither repeated columns nor the zero column. Consider the binary
matroid $\mathcal{M}$ representing the mod $2$ linear independence relations
between the columns of $A$. More explicitly, it is the simplicial complex
whose facets are the set of the indices of $4$ columns with an odd determinant
in $\mathbb{Z}$. Through direct computation, we know that it has
$|GL(4,\mathbb{Z}_{2})|/4!=840$ facets, where $GL(4,\mathbb{Z}_{2})$ is the
general linear group of degree $4$ over $\mathbb{Z}_{2}$. Among them, $835$
correspond to sets of $4$ vectors with determinants $\pm 1$, and the other $5$
correspond to the following sets of $4$ vectors with determinants $\pm 3$:

$A_{1}=\left\\{\begin{bmatrix}1\\\ 1\\\ 0\\\
0\end{bmatrix},\begin{bmatrix}1\\\ 0\\\ 1\\\
0\end{bmatrix},\begin{bmatrix}1\\\ 0\\\ 0\\\
1\end{bmatrix},\begin{bmatrix}0\\\ 1\\\ 1\\\ 1\end{bmatrix}\right\\}$,
$A_{2}=\left\\{\begin{bmatrix}1\\\ 1\\\ 0\\\
0\end{bmatrix},\begin{bmatrix}0\\\ 1\\\ 1\\\
0\end{bmatrix},\begin{bmatrix}0\\\ 1\\\ 0\\\
1\end{bmatrix},\begin{bmatrix}1\\\ 0\\\ 1\\\ 1\end{bmatrix}\right\\}$,
$A_{3}=\left\\{\begin{bmatrix}1\\\ 0\\\ 1\\\
0\end{bmatrix},\begin{bmatrix}0\\\ 1\\\ 1\\\
0\end{bmatrix},\begin{bmatrix}0\\\ 0\\\ 1\\\
1\end{bmatrix},\begin{bmatrix}1\\\ 1\\\ 0\\\ 1\end{bmatrix}\right\\}$,

$A_{4}=\left\\{\begin{bmatrix}1\\\ 0\\\ 0\\\
1\end{bmatrix},\begin{bmatrix}0\\\ 1\\\ 0\\\
1\end{bmatrix},\begin{bmatrix}0\\\ 0\\\ 1\\\
1\end{bmatrix},\begin{bmatrix}1\\\ 1\\\ 1\\\ 0\end{bmatrix}\right\\}$, and
$A_{5}=\left\\{\begin{bmatrix}1\\\ 1\\\ 1\\\
0\end{bmatrix},\begin{bmatrix}1\\\ 1\\\ 0\\\
1\end{bmatrix},\begin{bmatrix}1\\\ 0\\\ 1\\\
1\end{bmatrix},\begin{bmatrix}0\\\ 1\\\ 1\\\ 1\end{bmatrix}\right\\}$.

Let $a_{1}=\begin{bmatrix}1\\\ 1\\\ 0\\\ 0\end{bmatrix}$,
$a_{2}=\begin{bmatrix}1\\\ 0\\\ 1\\\ 0\end{bmatrix}$,
$a_{3}=\begin{bmatrix}1\\\ 0\\\ 0\\\ 1\end{bmatrix}$, and
$a_{4}=\begin{bmatrix}0\\\ 1\\\ 1\\\ 1\end{bmatrix}$ be the four vectors in
$A_{1}$. In mod $2$, we can observe that

$\displaystyle A_{2}$
$\displaystyle=\\{a_{1},a_{1}+a_{2},a_{1}+a_{3},a_{1}+a_{4}\\},$
$\displaystyle A_{3}$
$\displaystyle=\\{a_{1}+a_{2},a_{2},a_{2}+a_{3},a_{2}+a_{4}\\},$
$\displaystyle A_{4}$
$\displaystyle=\\{a_{1}+a_{3},a_{2}+a_{3},a_{3},a_{3}+a_{4}\\},\text{ and }$
$\displaystyle A_{5}$
$\displaystyle=\\{a_{1}+a_{4},a_{2}+a_{4},a_{3}+a_{4},a_{4}\\}.$

Notice that this combinatorial structure does not depend on the choice of
$A_{i}$’s and $a_{j}$’s.

Define $ind_{A}(A_{i})=$ the set of indices of vectors in $A_{i}$ in $A$. For
an element $g$ of $GL(4,\mathbb{Z}_{2})$, $gA$ is obtained by a column
permutation of $A$, so the five sets appear again with some other column
indices. Suppose that
$\\{ind_{A}(A_{1}),ind_{A}(A_{1}),\ldots,ind_{A}(A_{5})\\}\cap\\{ind_{gA}(A_{1}),ind_{gA}(A_{2}),\ldots,ind_{gA}(A_{5})\\}\not=\varnothing$,
that is they contain a common element $ind_{A}(A_{i})=ind_{gA}(A_{j})$. This
means that $g(A_{i})=A_{j}$. Then by the property we discussed above, $g$ maps
the collection of the five sets $A_{i}$ on itself. Hence for any $g\in
GL(4,\mathbb{Z}_{2})$, there are only two possibilities:

  * •

$\\{ind_{A}(A_{1}),ind_{A}(A_{2}),\ldots,ind_{A}(A_{5})\\}\cap\\{ind_{gA}(A_{1}),ind_{gA}(A_{2}),\ldots,ind_{gA}(A_{5})\\}=\varnothing$
or

  * •

$\\{ind_{A}(A_{1}),ind_{A}(A_{2}),\ldots,ind_{A}(A_{5})\\}=\\{ind_{gA}(A_{1}),ind_{gA}(A_{2}),\ldots,ind_{gA}(A_{5})\\}$.

Since every subset of column vectors of $A$ consisting of $4$ vectors with
determinant $3$ or $-3$ can be transformed into $A_{1}$ in mod $2$ by
multiplying with a suitable $g\in GL(4,\mathbb{Z}_{2})$, this yields a
partition
$\mathcal{A}=\\{\\{ind_{gA}(A_{1}),ind_{gA}(A_{2}),\ldots,ind_{gA}(A_{5})\\}\mid
g\in GL(4,\mathbb{Z}_{2})\\}$ of the set of facets of $\mathcal{M}$ with
$|\mathcal{A}|=840/5=168$. In this partition, only one among the $168$ sets
contains the set of vectors of determinant $\pm 3$ in $\mathbb{Z}$.

Now, let $\overline{\lambda^{\mathbb{R}}}$ be an IDCM over $K$. Then, there is
an embedding of $\overline{K}$ in $\mathcal{M}$ according to the index of
$\overline{\lambda^{\mathbb{R}}}(i)$ in $A$ for each vertex $i$. If
$\overline{K}$ has less than $168$ facets, then there exists an element of
$\mathcal{A}$ which does not intersect the set of the facets of the embedding
of $\overline{K}$ by the reverse pigeonhole principle. This means that there
exists $g\in GL(4,\mathbb{Z}_{2})$ such that for any facet $\sigma$ of
$\overline{K}$, the determinant of the matrix consisting of the $4$ vectors in
$\overline{\lambda^{\mathbb{R}}}g(\sigma)$ is $1$ or $-1$ when we see the
matrix as an integer $\\{0,1\\}$-matrix. Hence, it provides a lift, as
desired. ∎

Let $K$ be a seed of Picard number $4$. In turn, let us consider the case
where $K$ is an $(n-1)$-dimensional PL sphere that has more than or exactly
$168$ facets and that supports an IDCM. By using the list of colorable seeds
of Picard number $4$ (as in Theorem 3.10), one can see that if $n<10$, then
any colorable PL sphere has less than $168$ facets, and, hence, $n\geq 10$. On
the other hand, the condition of supporting an IDCM implies that $m\leq 15$,
so $n\leq 11$. In addition, we can check whether there is an IDCM over a given
$K$ with $10\leq n\leq 11$ by the Garrison-Scott algorithm [18] for finding
all mod $2$ characteristic maps over $K(J)$ or a modified algorithm, which is
faster with small Picard numbers, for finding only IDCMs introduced in [12].

######  Algorithm 6.2.

  * •

Input: a seed $K$.

  * •

Initialization:

    * $\mathcal{J}\leftarrow\\{(1,\dots,1)\\}$,

    * $\mathcal{J}_{1}\leftarrow\varnothing$,

    * $\mathcal{J}_{2}\leftarrow\varnothing$,

    * $\mathcal{R}\leftarrow\varnothing$.

  * •

Output: the list of $J$ such that $K(J)$ admits an IDCM and has $\geq 168$
facets.

  * •

Procedure:

    1. (1)

Set $\mathcal{J}\leftarrow$ the first element $\mathcal{J}[1]$ of
$\mathcal{J}$, and remove it from $\mathcal{J}$.

    2. (2)

If there is no IDCM over $K(J)$, then add $J$ to $\mathcal{J}_{1}$, and go to
(1).

    3. (3)

Add $J$ to $\mathcal{J}_{2}$.

    4. (4)

If $K(J)$ has $\geq 168$ facets, then add $J$ to $\mathcal{R}$.

    5. (5)

If $\mathcal{J}\not=\varnothing$, then go to (1).

    6. (6)

If $\dim(K(J))=10$, then return $\mathcal{R}$.

    7. (7)

For $i=1,2$, set $\mathcal{J}^{\prime}_{i}\leftarrow$ the set of $J^{\prime}$
such that $J^{\prime}$ is equal to $J$ for some $J$ in $\mathcal{J}_{i}$
except $k$th component for some $k$, and $k$th component of $J^{\prime}$ is 1
larger than the one of $J$.

    8. (8)

Set $\mathcal{J}_{1}\leftarrow\mathcal{J}^{\prime}_{1}$,
$\mathcal{J}_{2}\leftarrow\varnothing$, and
$\mathcal{J}\leftarrow\mathcal{J}^{\prime}_{2}\setminus\mathcal{J}^{\prime}_{1}$,
and go to (1).

After all these refining by Algorithm 6.2, there only remain twenty-one PL
spheres that have more than or exactly $168$ facets and ninety-six IDCMs over
them. Table 1 shows that the number of such PL spheres and IDCMs over them.

$n$ |  | 10 | 11
---|---|---|---
$K$ | 11 | 10
| seeds | 11 | 4
| non-seeds | 0 | 6
$(K,\overline{\lambda^{\mathbb{R}}})$ | 11 | 85
| seeds | 11 | 5
| non-seeds | 0 | 80
Table 1. The numbers of PL sphere $K$ supporting an IDCM and having $\geq 168$
facets (above) and the total number of IDCMs over them (below).

Two IDCMs $\Lambda_{1}^{\mathbb{R}}$ and $\Lambda_{2}^{\mathbb{R}}$ over a
non-seed $K$ are said to be _symmetric_ if they can be expressed by
$\Lambda_{1}^{\mathbb{R}}=\lambda_{1}^{\mathbb{R}}\wedge_{v}\lambda_{2}^{\mathbb{R}}$
and
$\Lambda_{2}^{\mathbb{R}}=\lambda_{2}^{\mathbb{R}}\wedge_{v}\lambda_{1}^{\mathbb{R}}$
for some IDCMs $\lambda_{1}^{\mathbb{R}}$ and $\lambda_{2}^{\mathbb{R}}$.
Although they are distinguished as IDCMs, the existence of their lifts are
equivalent. Therefore, it is enough to consider all IDCMs up to symmetry.

Reducing symmetries, there is only one IDCM over each non-seed $K$ of Table 1.
See Table 2.

$n$ |  | 10 | 11
---|---|---|---
$(K,\overline{\lambda^{\mathbb{R}}})$ | 11 | 11
| seeds | 11 | 5
| non-seeds | 0 | 6
Table 2. The number of IDCMs up to symmetry over the PL spheres $K$ having
$\geq 168$ facets.

The final step is to check whether all twenty-two pairs
$(K,\lambda^{\mathbb{R}})$ have $\\{0,\pm 1\\}$-lifts by the following simple
algorithm.

######  Algorithm 6.3.

  * •

Input: the cofacets $CF$ of $K$ and a mod $2$ dual characteristic map
$\overline{\lambda^{\mathbb{R}}}$ over $K$

  * •

Initialization:

    * $I\leftarrow$ the list of the indices of nonzero entries in $\lambda^{\mathbb{R}}$,

    * $i\leftarrow 0$.

  * •

Output: a $\\{0,\pm 1\\}$-lift of $\overline{\lambda^{\mathbb{R}}}$ if it
exists and $0$ otherwise.

  * •

Procedure:

    1. (1)

If $i=|I|$, then return $0$.

    2. (2)

Set $S\leftarrow$ the list of all $i$-subsets of $I$.

    3. (3)

If $S=\varnothing$, then set $i\leftarrow i+1$, and go to (1).

    4. (4)

Set $s\leftarrow$ the first element $S[1]$ of $S$, and remove it from $S$.

    5. (5)

Replace $1$’s in $\overline{\lambda^{\mathbb{R}}}$ with indices in $s$ by
$-1$’s.

    6. (6)

If there is a cofacet $cf\in CF$ such that the determinant of the matrix
consisting of the rows of $\overline{\lambda^{\mathbb{R}}}$ corresponding to
$cf$ is not $\pm 1$, then go to (3).

    7. (7)

Return $\overline{\lambda^{\mathbb{R}}}$.

In conclusion, we have the following theorem.

######  Theorem 6.4.

For a PL sphere of Picard number $4$, any quasi-injective mod $2$ dual
characteristic map over $K$ has a lift as a mod $2$ characteristic map over
$\overline{K}$.

The database containing the PL spheres of Picard number $4$ admitting an IDCM
are available on the second author’s Github repository:

https://github.com/Hyeontae1112/TWI

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