# Pix2Prof: fast extraction of sequential information from galaxy imagery via
a deep natural language ‘captioning’ model

Michael J. Smith,1,2,3 Nikhil Arora,3 Connor Stone,3 Stéphane Courteau,3 and
James E. Geach1,2
1Centre for Astrophysics Research, Department of Physics, Astronomy &
Mathematics, University of Hertfordshire, Hatfield, AL10 9AB, UK
2Centre of Data Innovation Research, Department of Physics, Astronomy &
Mathematics, University of Hertfordshire, Hatfield, AL10 9AB, UK
3Department of Physics, Engineering Physics & Astronomy, Queen’s University,
Kingston, ON, K7L 3N6, Canada E-mail<EMAIL_ADDRESS>
###### Abstract

We present ‘Pix2Prof’, a deep learning model that can eliminate any manual
steps taken when extracting galaxy profiles. We argue that a galaxy profile of
any sort is conceptually similar to a natural language image caption. This
idea allows us to leverage image captioning methods from the field of natural
language processing, and so we design Pix2Prof as a float sequence
‘captioning’ model suitable for galaxy profile inference. We demonstrate the
technique by approximating a galaxy surface brightness (SB) profile fitting
method that contains several manual steps. Pix2Prof processes $\sim$1 image
per second on an Intel Xeon E5-2650 v3 CPU, improving on the speed of the
manual interactive method by more than two orders of magnitude. Crucially,
Pix2Prof requires no manual interaction, and since galaxy profile estimation
is an embarrassingly parallel problem, we can further increase the throughput
by running many Pix2Prof instances simultaneously. In perspective, Pix2Prof
would take under an hour to infer profiles for $10^{5}$ galaxies on a single
NVIDIA DGX-2 system. A single human expert would take approximately two years
to complete the same task. Automated methodology such as this will accelerate
the analysis of the next generation of large area sky surveys expected to
yield hundreds of millions of targets. In such instances, all manual
approaches – even those involving a large number of experts – will be
impractical.

###### keywords:

methods: data analysis – methods: statistical – galaxies: photometry.

††pubyear: 2021††pagerange: Pix2Prof: fast extraction of sequential
information from galaxy imagery via a deep natural language ‘captioning’
model–References

##  1 Introduction

Large astrophysical surveys such as the Sloan Digital Sky Survey (SDSS; York
et al., 2000), the Panoramic Survey Telescope and Rapid Response System (Pan-
STARRS; Chambers et al., 2016), the Hyper Suprime Cam (HSC; Aihara et al.,
2017) Subaru Strategic Program Survey, or the upcoming Vera C. Rubin
Observatory Legacy Survey of Space and Time (LSST; Ivezić et al., 2019), carry
an inherent scaling problem. The SDSS has observed over 35 per cent of the
sky, cataloguing over 1 billion astronomical objects (Ahumada et al., 2020)
with a data rate of raw multiband imagery approaching 200 GB per night. In
January 2019, Pan-STARRS second Data Release totalled 1.6 PB of imaging data.
A precursor to LSST, HSC’s 990 megapixel camera has already produced over 1 PB
of imaging data (Aihara et al., 2019). These surveys will be dwarfed by the
upcoming LSST project. LSST’s 3.2 gigapixel camera will be the largest ever
made, and will survey the entire visible sky twice per week, generating
$\sim$500 PB of imaging data over its decade-long mission. The ‘firehose’ of
data from surveys such as LSST will require correspondingly efficient and
fully automated procedures to curate and analyse the data, enabling new
astrophysical findings and making unanticipated discoveries.

In this study, we are concerned with the automated direct analysis of galaxy
imagery towards estimating galaxy properties such as size, luminosity, colour,
and stellar mass. To calculate these properties, one typically applies a
photometric analysis that involves extracting and characterising the spatial
distribution of a galaxy’s light, described by a surface brightness (SB)
profile. The galaxy structural parameters as reflected by the SB profile can
be used to infer a suite of other important characteristics such as light
concentration, age, star formation rate, and assembly history (e.g. Strom et
al., 1976; Bell et al., 2003; Shen et al., 2003; Bernardi et al., 2005;
Fernández Lorenzo et al., 2013; Trujillo et al., 2020).

Numerous catalogues of galaxy structural properties already exist
(Jedrzejewski, 1987; Courteau, 1996; Brinchmann et al., 2004; Blanton et al.,
2005; Hall et al., 2012; Gilhuly & Courteau, 2018). Unfortunately, the methods
used in these compilations are either time consuming, requiring human
supervision, or fast but unreliable since they require a priori assumptions
about the shapes of galaxy components and other features. Similarly to the
study detailed in this paper, Tuccillo et al. (2018) describe a fully
automated neural network based technique (named ‘DeepLeGATo’; Deep Learning
Galaxy Analysis Tool) designed to replicate the GALFIT model-dependent
algorithm. DeepLeGATo is a fine example of an effective application of deep
learning, providing faster and possibly more accurate analysis than its parent
method, GALFIT (Peng et al., 2002). However, DeepLeGATo inherits from its
similarity to GALFIT the same issues stated previously; namely, the hard-coded
assumption of galaxy profile shapes, and other features. Furthermore,
DeepLeGATo can only produce single float outputs, and so cannot infer an SB
profile directly. This means that DeepLeGATo must rely on an intermediary
model to generate a complete SB profile. Therefore, even with semi-automated
methods, the accurate extraction of all the useful information from existing
surveys would take years. With the data volume expected to grow significantly
in the coming years, this becomes an intractable problem. Of great concern is
the possibility that important discoveries and insight could be missed or
delayed significantly due to the technical challenges imposed by the
unprecedented data volume. Clearly, there is a pressing need for entirely new
and efficient automated methods that significantly reduce, and ideally
circumvent, human interactions. Machine learning is ideally suited for this
task, and we apply it in this paper towards the specific problem of extracting
SB profiles from multi-band imaging data. Our approach takes advantage of a
set of SB profiles already determined via classical, interactive methods
(Courteau, 1996; Gilhuly & Courteau, 2018). We describe the classical method
used to produce the training data set in the next section. The remainder of
the paper is organised as follows: Section 2 introduces our approach; our
results and validation are presented in Section 3; Section 4 addresses our
global findings, and concludes with suggestions for broader application of the
algorithm.

##  2 Method

###  2.1 The classical surface brightness profile extraction algorithm

In the surface photometry of galaxies (e.g. Courteau, 1996, and references
therein), the spatially resolved light profile of a galaxy is extracted by
fitting progressively larger isophotes about a common centre. The fitting
technique assumes that projected isophotes are well represented by ellipses. A
galaxy’s centre is found by identifying the brightest pixel in a manually
selected region. Given a manually defined galaxy centre, the classical
algorithm determines the parameters needed to define each ellipse via a least
squares optimisation. The algorithm then generates isophotal solutions at each
radius well into the faint outskirts of the galaxy. In these regions of lower
signal-to-noise, where fitting algorithms are challenged, the algorithm
radially extends the last fitted isophote in the previous operation with a set
of concentric isophotes out to an arbitrarily large radius, usually taken to
be the edge of the image.

The isophotes may vary in ellipticity and position angle as a function of
galactocentric radius. This can become problematic when fitting to non-
axisymmetric structures in galaxies, such as bars and spiral arms that can
cause large twists in the fitted isophotal map. This issue can be corrected by
manually applying a smoothing function to some portions of the image. The
latter consists of manually smoothing the contour fits (i.e. uncrossing
twisted isophotes), and replacing poorly fitted data with a polynomial
smoothing function. Note that, prior to applying isophotal fitting to galaxy
images, some pre-processing is also required: the galaxy centre must be
identified as described above; the ‘sky’ background must be estimated and
removed from the image; nuisance foreground objects (such as unassociated
galaxies or foreground stars) must be identified and masked. These steps add
to the manual supervision of the task.

Besides the assumption that galaxies are circular when viewed face-on, and
thus generally of elliptical appearance when projected on to the plane of the
sky, the algorithm purposefully avoids using a priori knowledge of galactic
disc profile shapes and other features such as bars, rings, and spiral arms.
This avoids biasing the isophotal solution to any pre-determined, and possibly
incorrect, shape – a problem especially acute in the faint outer edges of a
galaxy.

Table 1: A summary of the Courteau (1996) surface brightness profile fitting algorithm’s processes. An approximate wall time per galaxy is given for the manual sections. The automated sections’ time contributions are negligible. Process | Automated? | Wall time (s/gal)
---|---|---
Identify galaxy centre | No | 5
Estimate & remove sky background | Yes | –
Remove foreground objects | No | 120
Fit contours | Yes | –
Extend contours to galaxy extent | No | 30
Smooth isophotes | Yes | –
Interpolate poorly fitted data | No | 120

While the semi-automated steps outlined above yield high quality SB profiles,
the process of obtaining a single profile is slow and systematic variations
may exist between different profile extraction methods. The interactive nature
of certain steps may indeed give rise to marked profile differences,
especially in low SB regimes where the isophotal solutions are less robust.
The low SB regimes will always remain the bane of galaxy image analysis,
whether automated or interactive, but the elimination of subjective steps goes
a long way towards reducing systematic differences between profiles.
Therefore, it becomes desirable to eliminate all interactive steps while
retaining all the benefits of classic algorithms such as Courteau (1996)
described above. In this work, we present a fully automated solution that
incorporates the extant knowledge base of SB profile fitting methodology, but
avoids human interaction.

###  2.2 Borrowing from automated image captioning

In recent years, the field of automated image captioning has benefited greatly
from advances in deep learning. We were strongly influenced by these
developments when designing the architecture of our ‘Pix2Prof’ profile
estimator. In this section, we briefly review pure recurrent neural network
(RNN) based encoder–decoder architectures, or models that only use a single
encoder and decoder to generate captions. A comprehensive review on deep
learning methods applied to image captioning can be found in Hossain et al.
(2019).

We primarily draw inspiration from gated RNN based encoder–decoder
architectures, as seen in Sutskever et al. (2014) for sequence-to-sequence
translation, and in Vinyals et al. (2015), Jia et al. (2015), and Wang et al.
(2016) for image-to-sequence translation. Sutskever et al. (2014) uses a Long
Short Term Memory (LSTM; Hochreiter & Schmidhuber, 1997) network to encode a
given sentence to a latent descriptive vector, and a second LSTM network to
decode the descriptive vector into a different feature space. One can use this
technique to translate text between two different languages, for example.

Figure 1: SDSS images of sample galaxies in the $g$ band (top row), and
corresponding surface brightness ‘ground truth’ profiles (bottom row). $\mu$
is the surface brightness. The galaxy names above each image refer to their
J2000 celestial coordinate.

Vinyals et al. (2015), Jia et al. (2015), and Wang et al. (2016) all use a
convolutional neural network (CNN; Fukushima, 1980; LeCun et al., 1989) to
first encode an image to a latent descriptive vector, and then use an LSTM
network to decode this vector into a text description (caption) of a given
image. Xu et al. (2015) use a CNN encoder, and a LSTM that attends over the
CNN output. Attention allows this approach to link each word in the caption
with an associated part of the image. This approach works well for images that
are crowded with multiple objects, but a simpler approach is preferred for our
case where each image is dominated by a single, central galaxy.

A galaxy profile can be thought of as analogous to a text caption describing
that galaxy. Both a text caption and galaxy profile can be encoded as a list
of floats or integers, and both have a length and content dependent on the
context of the conditioning image. Both also need to terminate once a complete
sentence or profile is generated, a subjective task well suited to a machine
learning solution. RNNs learn where to terminate a given caption or profile
empirically from the training set. Additionally, galaxy profiles and text
captions can both be approximated with an appropriate RNN. RNNs also produce
spatially consistent captions as a consequence of the architecture, a property
not guaranteed in a straight one-shot CNN. With these benefits in mind, it is
natural to consider an encoder–decoder network for the specific task of
estimating challenging galaxy profiles. Importantly, since the proposed model
directly learns the transformation between an unprocessed galaxy image, and
the galaxy’s corresponding SB profile, it eliminates all of the manual steps
described in Section 2.1 and Table 1.

While we develop Pix2Prof within the context of galaxy SB profile extraction,
the model is equally applicable to any array $\rightarrow$ float sequence
translation task.

###  2.3 Training set

We initially populate our data set with 10 arcmin $\times$ 10 arcmin $g$, $r$,
and $i$ band image crops extracted via the SDSS DR10 (York et al., 2000;
Eisenstein et al., 2011; Ahn et al., 2014) online mosaic interface. Each image
is centred on a galaxy. From these images we extract an SB profile via the
method described in Section 2.1. Fig. 1 presents a random sample of training
set galaxies, and their corresponding, manually extracted SB profiles. The
1953 galaxy image–SB profile pairs in each of the $g$, $r$, and $i$ bands
yield a total of 5859 pairs. This full data set is then divided into training,
validation, and testing sets. There are 5367 galaxy image–profile pairs in the
training set, 192 galaxy image–profile pairs in the validation set, and 300
galaxy image–profile pairs in the test set. The sets are randomly assigned,
with the condition that a given galaxy’s three photometric bands are kept
within the same set. The subset sizes are chosen to maximise the training set
efficacy while retaining most of the training set variance in the test set.

The only destructive pre-processing performed on the galaxy imagery is a
99.9th percentile clipping. This clipping mitigates the issue of single bright
(i.e. ‘hot’) pixels reducing image contrast when the galaxy images are
normalised, which would reduce training efficacy. To this end, we apply a
fixed min-max normalisation, defined as

$\bar{x}=\frac{x-A}{B-A},$ (1)

where $A=2.0~{}\text{nanomaggies}$ is the floor of the minimum value in the
training set, and $B=30.0~{}\text{nanomaggies}$ is the ceiling of the 99.9th
percentile value in the training set.

To reduce information sparsity in the training set images, we further crop
each galaxy image to a shape of [256, 256] pixels before passing the image to
Pix2Prof. We train using the full 32-bit depth of the original data as
measured. Good quality data are paramount when training a neural network, and
we therefore cut the profiles when the signal-to-noise ratio reaches a quality
threshold. We terminate the profile when the signal-to-noise ratio of a 1D
convolution with a length of 40 pixels ($=8~{}\text{arcsec}$) reaches a
threshold of 4. We define signal-to-noise so that it is equivalent to the
ratio of the power of a signal to the power of background noise:
$(\mu/\sigma)^{2}$, where $\mu$ is the mean and $\sigma$ is the standard
deviation of the convolutional window.

###  2.4 Network architecture

We write our model in PyTorch (Paszke et al., 2019), using a ResNet-18
(Srivastava et al., 2015; He et al., 2016) encoder, and a Gated Recurrent Unit
(GRU; Cho et al., 2014) decoder architecture. This architecture takes an
arbitrarily sized single channel image input, and outputs a sequence of floats
of arbitrary length. These floats are spaced along a galaxy’s semimajor axis
at a spacing of 0.2 arcsec per step (the same as the Courteau (1996)
technique). The same network is trained on images in the $g$, $r$, and $i$
bands, and therefore can produce a SB profile in any one of these bands. Fig.
2 shows a representation of the architecture used.

Figure 2: The ResNet $\rightarrow$ GRU encoder–decoder architecture used in
this work. The hidden state $h_{i}$ is the internal state of the GRU, and is
dependent on both the galaxy latent encoding $z$, and the previous profile
predictions $c_{i}$.

We use the standard ResNet-18 architecture as described in He et al. (2016).
The GRU is stacked to three layers. We apply a rectified linear unit (Glorot
et al., 2011, ReLU) activation and a dense neural layer after the three layer
stack to reduce the number of output values to one. As a regularising measure,
we apply dropout at a 20 per cent rate (Srivastava et al., 2014). The ResNet
first encodes the incoming galaxy image to a latent space vector $z$ of length
$512$. This vector is then used as the initial hidden state $h_{0}$ of a GRU.
In this way, Pix2Prof encodes and passes relevant information from the image
to the GRU. The GRU then unrolls to estimate properties of the galaxy from
$z$. In this paper’s case, we demonstrate this process by using $z$ to
estimate a galaxy’s SB profile.

To start estimation, the GRU is fed a start of sequence token. This token is
set as an array of zeros. In place of an end of sequence token, the GRU is
programmed to halt after 100 predictions are output that have a standard
deviation of 0.01 or less. This ensures that the GRU halts estimation once it
encounters the background sky.

We use the Adam optimiser (Kingma & Ba, 2014) to train Pix2Prof via gradient
descent (Robbins & Monro, 1951). Using manual search, we set the learning rate
as $2\times 10^{-4}$. Due to the logarithmic nature of magnitude, we want to
penalise large deviations from our ground truth SB profiles at a higher rate
compared to small deviations, and so we use the mean squared error loss:

$\text{MSE}=\frac{1}{b}\sum^{b}_{i=1}(y_{i}-p_{i})^{2},$ (2)

where $b$ is the batch size, $y$ is the ground truth, and $p$ is a prediction.

###  2.5 Training the model

We augment the galaxy images by applying a ‘wobble’. This wobble is a random
small shift in the centre of the image. Each band is treated independently. We
do this to encourage the network to work with the slightly off-centre galaxies
that will be encountered in real data. This is required to make Pix2Prof
robust to poorly centred galaxy images. We also exploit the rotational
axisymmetry of galaxies and further augment the data by randomly rotating an
input image through $90,180,\text{and}~{}270$ degrees.

We train the model for $100\,000$ global steps on a single NVIDIA TESLA V100
GPU. Training takes approximately 20 minutes per epoch of 500 galaxy images, a
rate of 0.4 galaxies per second.

##  3 Results & Validation

We validate the model during training once per epoch using the validation set.
We test the trained model on 100 randomly sampled observed galaxies in the
$g$, $r$, and $i$ bands (for 300 total image–profile pairs) drawn from the
data set and which are set aside entirely during training. We use the model
with the lowest validation loss; epoch 160. We run an entirely automated
inference on an Intel Xeon CPU E5-2650 v3 CPU at a rate of $0.9$ galaxies per
second.

Figure 3: Randomly sampled test set predicted SB galaxy profiles (orange)
superimposed onto SB profiles measured via the Courteau (1996) method (blue).
$\mu$ is the surface brightness. Distances from centre are in log scale to
emphasise divergences in the high signal-to-noise ratio region closer to the
galaxies’ centres. Below each SB profile plot is the residual defined as
$\text{res}=y-p$, where $y$ is the profile as measured according to Section
2.1, and $p$ is the prediction. The galaxies’ J2000 celestial coordinates and
spectral bands are indicated at the top right of each graph.

Fig. 3 shows a random selection of 25 Pix2Prof inferred test set SB profiles
superimposed on to the Courteau (1996) SB profiles. Since we have trained
Pix2Prof to directly infer a SB profile from a galaxy image we do not produce
intermediate steps (such as the galaxy centre, ellipticity profile, or
position angle profile). However, one could estimate these values if Pix2Prof
is explicitly trained to reproduce them. Fig. 4 shows the error distribution
of the test set as well as the test set error per distance in physical units
from the galaxy centre. We define error as the absolute of Fig. 3’s residual,
the absolute deviation:

$\eta=\left|y-p\right|,$ (3)

where $p$ is a prediction, and $y$ is measured via Courteau (1996). The units
of SB call for additional care in defining our errors. Since SB values are
defined on a logarithmic scale, equation (3) is really a form of fractional
error:

$\eta=\left|2.5\log_{10}\frac{I_{p}}{C}-2.5\log_{10}\frac{I_{y}}{C}\right|=\left|-2.5\log_{10}\frac{I_{y}}{I_{p}}\right|,$
(4)

where $C$ is a constant reference brightness. $\\{I_{p},I_{y}\\}$ are
brightnesses in linear units.

We take the median of this error per galaxy profile to produce the violin
plots in Fig. 4(a), and we take the median of this error across profiles to
produce the line plot in Fig. 4(a). Fig. 4(a)’s line plot shows that the error
increases with radius away from the galaxy centre towards regions containing
less signal, as expected. We find that the median test set absolute deviation
is 0.41 mag arcsec-2 with an interquartile range of 0.21 mag arcsec-2. We also
find that the median test set absolute deviation for $y$ values brighter than
the SDSS limiting SB (26.5 mag arcsec-2) is 0.34 mag arcsec-2, with an
interquartile range of 0.22 mag arcsec-2. Errors of this scale mean that
profiles generated via Pix2Prof will be immediately useful for rough searches;
it would be possible to categorise galaxies roughly by brightness, isophotal
radius, scale length, and other structural parameters. Further refinement of
the model may reduce error, enabling more sophisticated processing and
analysis of generated SB profiles. Possible refinements are described in
Section 4.

(a) Summary statistics for Pix2Prof.

(b) Summary statistics for AutoProf (see §3.1).

(c) Median error per kpc from galaxy centre comparison between Pix2Prof and
AutoProf.

Figure 4: Approximation errors as defined in equation (3). Fig. 4(a) shows
summary statistics for Pix2Prof. For comparison, Fig. 4(b) depicts the same
statistics for AutoProf. In all of the above figures we define absolute
deviation as relative to the Courteau (1996) test set profiles. The leftmost
violin plot in Fig. 4(a) and 4(b) shows the distribution of median test set
errors. The rightmost violin plot in Fig. 4(a) and 4(b) shows the same
distribution for only SB values below the SDSS limiting SB of $26.5$ mag
arcsec-2. The maximum, minimum, mean, and (mean + standard deviation) are
labelled. Below the violin plots are their distribution quartiles. The line
plots show the median error per kpc from the galaxy centre, with the
interquartile range shaded. To reduce the effect of small sample size
variability, the line plots are terminated once 90 per cent of the SB profiles
reach their galaxy’s extent. Fig. 4(c) compares on the same axes the median
errors per kpc from the galaxy centre for Pix2Prof and AutoProf.

In Fig. 5, the three bands’ median errors are separated as a function of
galactocentric radius. Close to the galaxy, there is little difference in the
three bands’ median predictions. However, as we proceed outwards, the
$r$-band’s error is higher than the $g$-band’s, and the $i$-band’s error is
higher still. This is due to a difference in the instrumental noise between
the three bands, as evidenced in the difference in the spectral bands’ median
galaxy image signal-to-noise ratios: $\text{SNR}_{g}=41.6$;
$\text{SNR}_{r}=35.8$; $\text{SNR}_{i}=28.4$.

Fig. 6a shows each test set galaxy’s ellipticity against the galaxy profile’s
median absolute deviation. The ellipticity is defined as the final isophote
ellipticity for a profile calculated using the Courteau (1996) method. Fig. 6b
shows each test set galaxy’s semimajor axis radius for the first isophote
whose value is greater than or equal to $23.5$ mag arcsec-2. In both of these
cases, we run a linear regression and find no significant correlation,
suggesting that Pix2Prof’s predictions are equally robust when inferring
across galaxies with a range of sizes and ellipticities.

Figure 5: Median test set error per kpc from the galaxy centre, with the
interquartile range shaded, split into the three bands present in the test
set. Figure 6: Median test set error over each galaxy predicted profile,
plotted against each galaxy’s ellipticity (left, 6a), and size at $R_{23.5}$
(right, 6b). $p$ values obtained via a linear regression are stated in the
legends.

Figs 3 and 4 show that Pix2Prof can successfully approximate a complicated
astrophysical image processing pipeline with low deviation (0.34 mag arcsec-2
averaged over the test set). Processing $0.9$ galaxies per second on an Intel
Xeon E5-2650 v3 CPU, Pix2Prof improves on the speed of the classical image
analysis method of Courteau (1996) by more than two orders of magnitude. For
comparison, an astronomer trained to use the Courteau (1996) method can
typically process ${\sim}150$ galaxies in a full eight hour working day (or
${\sim}0.005$ galaxies per second). However, even astronomers must rest and so
the true working rate for a human would be ${\sim}150$ galaxies per 24 hours,
or ${\sim}0.002$ galaxies per second.

As Table 2 shows, Pix2Prof eliminates any manual interaction from SB profile
inference, alleviating the issue of subjectivity in the different methods
developed for such tasks; Pix2Prof will infer the same profile every time for
a given galaxy image, whereas a human may not. The full automation of Pix2Prof
enables a complete parallelisation, and thus significant gain in parallel
throughput of galaxy profile estimation.

Table 2: Pix2Prof eliminates all interactive steps in the Courteau (1996) algorithm, alleviating subjectivity and accelerating inference significantly. | Automated in:
---|---
Process |  Courteau (1996)? |  redPix2Prof?
Identify galaxy centre | No |  redYes
Estimate & remove sky background | Yes |  redYes
Remove foreground objects | No |  redYes
Fit contours | Yes |  redYes
Extend contours to galaxy outskirts | No |  redYes
Smooth isophotes | Yes |  redYes
Interpolate poorly fitted data | No |  redYes

###  3.1 Comparison with AutoProf

AutoProf (Stone et al., 2021) is a sibling method to Pix2Prof that uses a more
standard astronomical pipeline to tackle the problem of non-parametric
automated SB profile inference. A combination of standard image analysis
packages from photutils (Bradley et al., 2019) and novel techniques are used
to construct a robust isophotal pipeline. Initial image analysis such as
determining the PSF, centre finding, star masking, and sky background
subtraction are performed using photutils. Next, AutoProf simultaneously fits
an ellipticity and position angle profile by minimising low order FFT
coefficients along each isophote plus a regularisation term (Shalev-Shwartz &
Ben-David, 2014). The regularisation term penalises neighbouring ellipticity
and position angle values that deviate significantly, ultimately favouring
smooth profiles. Once the profiles have converged, AutoProf extracts the
median SB along each isophote and determines the error with a 68.3 per cent
quartile range. A curve of growth is determined by appropriately integrating
the SB profile and propagating errors.

Fig. 4 compares the performance of Pix2Prof and AutoProf. AutoProf is found to
produce SB profiles that are a slightly closer match to those produced via
Courteau (1996) than Pix2Prof. The difference in absolute deviation between
the two methods’ profiles is typically around 0.1–0.2 mag arcsec-2. However,
AutoProf’s accuracy comes with a time penalty; AutoProf takes on average 490 s
to produce a profile on an Intel Xeon E5-2650 v3 CPU, a rate of 0.002 galaxies
per second. This rate is roughly equivalent to the throughput of a human
running the Courteau (1996) method. Pix2Prof processes 0.9 galaxies per second
on the same hardware. Pix2Prof also offers more flexibility; it can be
retrained to recreate any (semi) manual galaxy profile fitting pipeline and is
therefore not limited to automation of the Courteau (1996) method.

AutoProf is presented in Stone et al. (2021), and its code is available at
https://github.com/ConnorStoneAstro/AutoProf.

##  4 Discussion and conclusions

While Pix2Prof can rapidly and accurately produce profiles of arbitrary
length, there are some limitations to this technique. Principally, any profile
produced will be biased to the training set. For instance, if Pix2Prof is
trained on primarily nearby galaxies, it may not yield accurate profiles for
more distant systems whose images will be poorly resolved. Similarly, if the
model is trained on galaxy image–profile pairs as produced by numerical
simulations the model will encode any flaws, incompleteness, or bias inherent
to each simulation and will not encode instrumental effects (e.g. read-out
noise) unless properly included. The same issue will occur if we train on
galaxy image–profile pairs sampled from one survey and deploy the trained
model on a dissimilar survey, for example SDSS, and LSST (York et al., 2000;
Ivezić et al., 2019). It may be possible to mitigate this problem with an
image domain translator (i.e. Zhu et al., 2017; Isola et al., 2017; Choi et
al., 2018) that could transform observations so that they match a given
survey. Of course, the Courteau (1996) measured profiles may also not entirely
reflect the ‘true’ SB profile, due to modelling assumptions, human bias, and
inherent noisiness in measurement. As neural networks typically require very
large data sets, our relatively small data set is likely not reflecting the
true potential of the model. Therefore, a larger set of training data could
improve the results presented here. Generating a larger data set from
simulated galaxies for training Pix2Prof will be a future project.

As described in Jia et al. (2015), due to the vanishing gradient problem a
LSTM or GRU may ‘forget’ an image encoding as it unrolls. For Pix2Prof, this
will manifest in a loss of accuracy at larger galactocentric radius. We see
this effect in Fig. 4(a)’s line plot and Fig. 5, but we cannot disentangle the
individual contributions from image noise and the model architecture. However,
assuming that the noise is significantly caused by GRU ‘forgetfulness’, future
Pix2Prof models could imitate Jia et al. (2015) and counteract the noise by
reinjecting the image encoding into the GRU’s hidden state periodically as it
unrolls. Another solution could involve adopting an architecture that suffers
less from the vanishing gradient problem, such as the Transformer (Vaswani et
al., 2017). The non-sequential nature of a Transformer would also allow us to
parallelise output at inference time, reducing processing time even further.

In Section 1 we stressed the need for efficient and fully automated methods
for timely analysis of ultra-large scale astrophysical imaging survey data. We
believe that Pix2Prof addresses this challenge. Pix2Prof can predict any
galaxy profile, given the right simulated or observed data set. Training
Pix2Prof on simulated galaxy images offers additional benefits; the model
could be trained on information that is only inferred indirectly in
observations. For instance, Pix2Prof could train on sets of galaxy image–mass
profile pairs directly in order to predict dark matter halo profiles, as mass
profiles cannot be recovered classically by direct imaging observations.
Furthermore, Pix2Prof has the potential to automate any galaxy profile fitting
routine and be ported to other forms of galaxy image analysis that may not
rely on isophotal analysis, but still produce a float sequence given a
multidimensional array. These analyses could include galaxy component
decompositions, the characterisation of galaxy interactions and distortions,
pixelised stellar population synthesis, inference of galaxy mass
distributions, and more (e.g. Eneev et al., 1973; Vazdekis, 2001; Peng et al.,
2002). In a future paper, we will demonstrate how Pix2Prof can be used to
recover simultaneously the galaxy SB profile as well as the ellipticity
profile and curve of growth of a galaxy.

An exciting future investigation involves building a system that can predict
properties of unseen classes of objects. This could be achieved by building up
a ‘prior’ that encodes known objects into a latent space and interpolates
between their latent spatial representations at inference time. A generative
model like the Generative Adversarial Network (GAN; Goodfellow et al., 2014)
or Variational Autoencoder (VAE; Kingma & Welling, 2013) could achieve this
(i.e. Spindler et al., 2020). Such a model could quickly identify
astrophysically ‘interesting’ objects in a large field survey. The ability to
search for rare objects in large unstructured data sets will become
increasingly more important as new large scale astronomical surveys come
online (Chambers et al., 2016; Aihara et al., 2017; Ivezić et al., 2019).

In summary, we have introduced a fully automated deep learning model for the
extraction of sequential data from galaxy imagery. We have tested this model
by applying it to the specific problem of estimating galaxy SB profiles, a
process that previously required manual, time-consuming human intervention. We
have tested our model on unseen galaxy images and found that our model has an
average absolute deviation of 0.34 mag arcsec-2 with an interquartile range of
0.22 mag arcsec-2, while inferring SB profiles over two orders of magnitude
faster than the classic (interactive) algorithm it automates.

## Data and Code Availability

The code and trained model used in this paper is available at
https://github.com/Smith42/pix2prof. The profile data set used to train the
network will be released separately (Arora et al., in preparation).

## Carbon Emissions

The training of deep learning models requires considerable energy,
contributing to carbon emission and therefore climate change (Strubell et al.,
2019; Lacoste et al., 2019). The energy used while training Pix2Prof on a
single NVIDIA V100 GPU is estimated to be $\sim$20 kWh
($5.54~{}\text{kg}~{}\text{CO}_{2}~{}\text{eq}$) according to the Machine
Learning Impact calculator described in Lacoste et al. (2019). This is
equivalent to driving 28 miles in a typical European passenger car111According
to the European Environmental Agency:
https://www.eea.europa.eu/highlights/average-co2-emissions-from-new-cars-
vans-2019. To counteract further emission from redundant retraining, we follow
the recommendations of Strubell et al. (2019) and make available the fully
trained model, as well as the code to run it. Also, we will make available
trained models for any improvements that we make to Pix2Prof in the future.

## Acknowledgements

This research made use of the University of Hertfordshire’s High Performance
Computing facility (http://uhhpc.herts.ac.uk/). We are grateful to the Natural
Sciences and Engineering Research Council of Canada, the Ontario Government,
Queen’s University, and the Royal Society for critical support through various
scholarships and grants. M.J.S. thanks Queen’s University and the research
group of Stéphane Courteau for their support and hospitality in 2019 while
this paper was conceived and developed. We also thank reviewers at the NeurIPS
2020 conference, as well as the MNRAS editor and referee for helpful comments
and suggestions. The galaxy icon in Fig. 2 is by Agata Kuczmińska and is
available under the CC-BY-4.0 licence at https://goodstuffnononsense.com/hand-
drawn-icons/space-icons/.

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# Equivariantly formal $2$-torus actions of complexity one

Vladimir Gorchakov  Faculty of computer science, Higher School of Economics
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

In this paper we study a specific class of actions of a $2$-torus
$\mathbb{Z}_{2}^{k}$ on manifolds, namely, the actions of complexity one in
general position. We describe the orbit space of equivariantly formal
$2$-torus actions of complexity one in general position and restricted
complexity one actions in the case of small covers. It is observed that the
orbit spaces of such actions are topological manifolds. If the action is
equivariantly formal, we prove that the orbit space is a
$\mathbb{Z}_{2}$-homology sphere. We study a particular subclass of these
$2$-torus actions: restrictions of small covers to a subgroup of index 2 in
general position. The subgroup of this form exists if and only if the small
cover is orientable, and in this case we prove that the orbit space of a
restricted 2-torus action is homeomorphic to a sphere.

###### 2020 Mathematics Subject Classification:

Primary 57S12, 57S17, 57S25, 57N65, 57R91; Secondary 57R18, 55M35, 55R20

The article was prepared within the framework of the HSE University Basic
Research Program

##  1\. Introduction

Let a compact torus $T^{k}=(S^{1})^{k}$ effectively act on a closed manifold
$X^{2n}$ with nonempty finite fixed points set. The number $n-k$ is called the
complexity of the action. The study of orbit spaces of complexity zero is a
well-know subject of toric topology (See [7]). In [2], A. Ayzenberg showed
that the orbit space of a complexity one action in general position is a
topological manifold. In [4], A. Ayzenberg and M. Masuda described the orbit
space of equivariantly formal torus actions of complexity one in general
position. In [3], A. Ayzenberg and V. Cherepanov described torus actions of
complexity one in non-general position.

Similarly, we can study the orbit space of $\mathbb{Z}_{2}^{k}$ action on a
manifold $X^{n}$. The group $\mathbb{Z}_{2}^{k}$ is a real analog of $T^{k}$,
it is called a $2$-torus. Similarly to the torus case the number $n-k$ is
called the complexity of the action. In [20], L. Yu studied the orbit space of
an equivariantly formal $2$-torus action of complexity zero. In [8], V.
Buchstaber and S. Terzic proved that
$Gr_{4,2}(\mathbb{R})/\mathbb{Z}_{2}^{3}\cong S^{4}$ and
$Fl_{3}(\mathbb{R})/\mathbb{Z}_{2}^{2}\cong S^{3}$ for real Grassmann manifold
$G_{4,2}(\mathbb{R})$ of $2$-planes in $\mathbb{R}^{4}$ and the real manifold
$Fl_{3}(\mathbb{R})$ of full flags in $R^{3}$. In [15], D. Gugnin proved that
$T^{n}/\mathbb{Z}_{2}^{n-1}\cong S^{n}$ for a certain action with isolated
fixed points. These are examples of complexity one actions.

The aim of this paper is to describe the orbit space of equivariantly formal
$2$-torus action of complexity one in general position. We also describe
restricted complexity one actions in the case of small covers.

Let us give preliminary definitions and formulate the main results.

Let a $2$-torus $\mathbb{Z}_{2}^{n-1}$ act effectively on a connected closed
smooth manifold $X=X^{n}$ with nonempty set of fixed points. For a fixed point
$x\in X^{\mathbb{Z}_{2}^{n-1}}$ of the action we have the tangent
representation of $\mathbb{Z}_{2}^{n-1}$ at $x$. Consider
$\alpha_{x,1},\ldots,\alpha_{x,n}\in\operatorname{Hom}(\mathbb{Z}_{2}^{n-1},\mathbb{Z}_{2})\cong\mathbb{Z}_{2}^{n-1}$,
the weights of the tangent representation at $x$. The action is said to be in
_general position_ if for any fixed point $x$, any $n-1$ of the weights
$\alpha_{x,1},\dotsc,\alpha_{x,n}$ are linearly independent over
$\mathbb{Z}_{2}$. Now we provide a coordinate description of an action in
general position in the case of $\mathbb{R}^{n}$.

Let $G$ be a subgroup of $\mathbb{Z}^{n}_{2}$ consisting of elements of the
following form:

(1.1) $G=\\{(g_{1},\dots,g_{n})\in\mathbb{Z}_{2}^{n}:\Pi_{i=1}^{n}g_{i}=1\\},$

where $g_{i}\in\\{-1,1\\}$. Since $\mathbb{Z}_{2}^{n}$ acts coordinate-wise on
$\mathbb{R}^{n}$, we have an induced action of $G$ on $\mathbb{R}^{n}$, which
we call the standard complexity one action.

######  Remark 1.1.

If an action of $\mathbb{Z}_{2}^{n-1}$ on $\mathbb{R}^{n}$ is in general
position, then it is weakly equivalent to the standard complexity one action,
see Proposition 3.3.

C. Lange and M. Mikhailova in [12], [17] studied orbit spaces of general
representations of finite groups. It follows from their results that
$\mathbb{R}^{n}/G\cong\mathbb{R}^{n}$ for complexity one representation in
general position. Using this result and some additional condition, which holds
for equivariantly formal actions, we show that for a
$\mathbb{Z}_{2}^{n-1}$-action on $X$ in general position the orbit space
$Q=X/\mathbb{Z}_{2}^{n-1}$ is a topological manifold. If a
$\mathbb{Z}_{2}^{n-1}$-action is in non-general position, then $Q$ is a
topological manifold with boundary similar to the torus case studied in [10]
by V. Cherepanov.

The action of $\mathbb{Z}_{2}^{n-1}$ on $X$ is called equivariantly formal if

(1.2)
$\dim_{\mathbb{Z}_{2}}H^{*}(X^{\mathbb{Z}_{2}^{n-1}};\mathbb{Z}_{2})=\dim_{\mathbb{Z}_{2}}H^{*}(X;\mathbb{Z}_{2}),$

where $X^{\mathbb{Z}_{2}^{n-1}}$ is the fixed point set of
$\mathbb{Z}_{2}^{n-1}$-action.

######  Remark 1.2.

If an $\mathbb{Z}_{2}^{n-1}$-action is in general position, then the fixed
point set is finite and the action is equivariantly formal if and only if
$|X^{\mathbb{Z}_{2}^{n-1}}|=\dim_{\mathbb{Z}_{2}}H^{*}(X;\mathbb{Z}_{2})$.

######  Theorem 1.

Let $\mathbb{Z}_{2}^{n-1}$ act on $X=X^{n}$ equivariantly formal and in
general position. Then the orbit space $Q=X/\mathbb{Z}_{2}^{n-1}$ is a
topological manifold and $H^{*}(Q;\mathbb{Z}_{2})\cong
H^{*}(S^{n};\mathbb{Z}_{2})$.

Let $X$ be a small cover and let $G\cong\mathbb{Z}_{2}^{n-1}$ be a subgroup of
$\mathbb{Z}_{2}^{n}$ of index 2. Then we get the restricted $G$-action of
complexity one on $X$. The subgroup $G$ is called a $2$-subtorus in general
position, if this $G$-action is in general position. In the case of small
covers, we notice that the condition of existence $2$-subtorus in general
position is equivalent to the the condition of orientability of a small cover,
which was proved by H. Nakayama and Y. Nishimura in [18].

######  Theorem 2.

Let $X$ be a small cover. There exists a $2$-subtorus in general position if
and only if $X$ is orientable. If such $2$-subtorus exist, then it is unique.

If the $2$-subtorus in general position exists, then the orbit space is
homeomorphic to the sphere.

######  Theorem 3.

Let $X=X^{n}$ be an orientable small cover and let $G$ be the $2$-subtorus in
general position. Then the orbit space $X/G$ is homeomorphic to the
$n$-dimensional sphere $S^{n}$.

We have the following coordinate description of the $2$-subtorus in general
position.

######  Remark 1.3.

Let $X=X^{n}$ be an orientable small cover over a simple polytope $P$, let
$p=F_{i_{1}}\cap\dots\cap F_{i_{n}}$ be a vertex of $P$ and
$\lambda_{i_{1}},\dots,\lambda_{i_{n}}$ be the corresponding characteristic
vectors. Taking the characteristic vectors as standard generators of
$\mathbb{Z}_{2}^{n}$, consider the following subgroup:

$G=\\{(g_{1},\dots,g_{n})\in\mathbb{Z}_{2}^{n}:\Pi_{i=1}^{n}g_{i}=1\\}.$

Then $G$ is the $2$-subtorus in general position.

Now we provide some examples of complexity one actions. In all examples below
actions are equivariantly formal and in general position. In examples $1.5$,
$1.6$ we get restricted actions on small covers.

######  Example 1.4.

Let $\mathbb{Z}_{2}$ act on $S^{2}$ by rotation on angle $180^{\circ}$ around
an axis. Then $S^{2}/\mathbb{Z}_{2}\cong S^{2}$.

The action of $\mathbb{Z}_{2}$ on $S^{2}$

######  Example 1.5.

More generally, let $\mathbb{Z}_{2}$ act on a closed orientable surface
$M_{g}$ of genus $g$ by rotation on angle $180^{\circ}$ around the axis. Then
$M_{g}/\mathbb{Z}_{2}\cong S^{2}$. If $g=1$, then it is the particular case of
the next example. Notice that if $g=0$, then $M_{0}=S^{2}$ is not a small
cover.

The action of $\mathbb{Z}_{2}$ on $M_{3}$

The orbit space of the $\mathbb{Z}_{2}$-action on $M_{3}$

######  Example 1.6.

Consider a $\mathbb{Z}_{2}$-action on $S^{1}$ by the map $(x,y)\to(x,-y)$.
Taking the n-fold product, we have a $\mathbb{Z}_{2}^{n}$-action on the
$T^{n}$. Let

$G=\\{(g_{1},\dots,g_{n})\in\mathbb{Z}_{2}^{n}:\Pi_{i=1}^{n}g_{i}=1\\}$

be the index 2 subgroup of orientation-preserving elements. In [15] it was
proved that $T^{n}/G\cong S^{n}$.

Two examples below are not small covers.

######  Example 1.7.

Let $\mathbb{Z}_{2}^{4}$ act on $\mathbb{R}^{4}$ by the standard action. From
this we get the effective action of $\mathbb{Z}_{2}^{3}$ on the real Grassmann
manifold $G_{4,2}(\mathbb{R})$ of $2$-planes in $\mathbb{R}^{4}$. In [8] it
was proved that $G_{4,2}(\mathbb{R})/\mathbb{Z}_{2}^{3}\cong S^{4}$.

######  Example 1.8.

Let $\mathbb{Z}_{2}^{3}$ act on $\mathbb{R}^{3}$ by the standard action. From
this we get the effective action of $\mathbb{Z}_{2}^{2}$ on the real full flag
manifold $Fl_{3}(\mathbb{R})$. In [8] it was proved that
$Fl_{3}(\mathbb{R})/\mathbb{Z}_{2}^{2}\cong S^{3}$.

Now we provide a possible connection of Theorem $3$ with the theory of
n-valued groups. See [6] for the definition of $n$-valued group and other
details. The following construction can be used to produce $n$-valued groups.

Let $G$ be a group, let $A$ be a finite group with $|A|=n$ and $\phi:A\to
Aut(G)$ be homomorphism to the group of automorphisms of $G$. Then we have an
$n$-valued group structure on the orbit space $X=G/\phi(A)$, see [6, Thm. 1].

In example $1.6$ we get a $2^{n-1}$-valued topological group structure on the
$n$-sphere $S^{n}$ for $n\geq 2$. The following problem was posed by V. M.
Buchstaber.

######  Problem 1.

Can Theorem $3$ be applied to other small covers to provide new examples of
$2^{n-1}$-valued group structure on $S^{n}$?

For this we need a small cover $X$ with the property that it is a group and
$\mathbb{Z}_{2}^{n}$ acts by group automorphisms.

######  Non-Example 1.9.

We have that $\mathbb{R}P^{3}$ is diffeomorphic to $SO(3,\mathbb{R})$, hence
there is a Lie group structure on a small cover $\mathbb{R}P^{3}$. However,
$\mathbb{Z}_{2}^{3}$ does not act by Lie group automorphisms. Indeed, since
all automorphisms of $SO(3,\mathbb{R})$ are inner, we have that
$Aut(SO(3,\mathbb{R}))\cong SO(3,\mathbb{R})$. There is no finite subgroup in
$SO(3,\mathbb{R})$ isomorphic to $\mathbb{Z}_{2}^{3}$. Therefore,
$\mathbb{Z}_{2}^{3}$ does not act by group automorphisms.

##  2\. Preliminaries

In this section we recall some general facts about group actions and $2$-torus
actions on manifolds. The main reference about group actions is [5]. The main
reference about equivariantly formal $2$-torus actions is [20].

Let a group $G$ act effectively on closed smooth manifold $X$. In this paper
we consider only smooth action. For a point $x\in X$ let $Stab(x)$ denote the
stabilizer subgroup of $G$ and $Gx$ the orbit of $x$. We define the partition
by orbit types

(2.1) $X=\bigsqcup_{H\subset G}X^{(H)}.$

Here $X^{(H)}=\\{x\in X:Stab(x)=H$}.

We denote the fixed point of subgroup $H$ by $X^{H}=\\{x\in X:Stab(x)\subset
H\\}$.

Let $x\in X^{(H)}$ be a point with the stabilizer subgroup $H$. We can define
the tangent representation of $H$ at $x$:

$H\to GL(T_{x}X/T_{x}Gx).$

Let $V_{x}$ denote $T_{x}X/T_{x}Gx$. There is the following theorem about
$G$-equivariant tubular neighborhood.

######  Theorem 2.1 (The Slice Theorem).

There exist a $G$-equivariant diffeomorphism from the $G\times_{Stab(x)}V_{x}$
onto a $G$-invariant neighborhood of the orbit $Gx$ in $X$, which send the
zero section $G/Stab(x)$ onto the orbit $Gx$.

We now recall the notation of an equivariantly formal action of 2-torus
$\mathbb{Z}_{2}^{k}$, see [20] for the details.

There is a classical result of E. Floyd.

######  Theorem 2.2 ([14]).

For any paracompact $\mathbb{Z}_{2}^{k}$-space $X$ with finite cohomology
dimension, the fixed point set $X^{\mathbb{Z}_{2}^{k}}$ always satisfies

(2.2)
$\dim_{\mathbb{Z}_{2}}H^{*}(X^{\mathbb{Z}_{2}^{k}};\mathbb{Z}_{2})\leq\dim_{\mathbb{Z}_{2}}H^{*}(X;\mathbb{Z}_{2}).$

The next theorem says, when the equality in (2.2) holds.

######  Theorem 2.3 ([16]).

The equality in (2.2) holds if and only if the $E_{2}=E_{\infty}$ for the
Serre spectral sequence of the fibration $X\rightarrow
E\mathbb{Z}_{2}^{k}\times_{\mathbb{Z}_{2}^{k}}X\rightarrow
B\mathbb{Z}_{2}^{k}$.

######  Definition 2.4.

Let a $2$-torus $\mathbb{Z}_{2}^{k}$ act on a closed smooth manifold X. The
action is called _equivariantly formal_ over $\mathbb{Z}_{2}$ if there is
equality in (2.2).

Sometimes the degeneration of the Serre spectral sequence at $E_{2}$ is taken
as the definition of equivariant formality. This definition is equivalent to
Definition 2.4 according to Theorem 2.3. The notation of equivariant formality
is similar to the corresponding notion in the theory of torus actions.

######  Remark 2.5.

If the fixed point set $X^{\mathbb{Z}_{2}^{k}}$ is finite, then $2$-torus
action equivariant formal if and only if

(2.3) $|X^{\mathbb{Z}_{2}^{k}}|=\dim_{\mathbb{Z}_{2}}H^{*}(X;\mathbb{Z}_{2}).$

This remark is useful to characterize equivariant formality of the particular
$2$-torus actions.

######  Definition 2.6.

Consider a non-free effective action of $\mathbb{Z}_{2}^{n}$ on a closed
connected smooth manifold $X^{n}$. Manifold with such action is called a
$2$-torus manifold.

L. Yu proved the following criteria of equivariant formality for $2$-torus
manifolds in terms of its orbit space.

######  Theorem 2.7 ([20]).

Let $X$ be a $2$-torus manifold with orbit space $Q$.

  * (i)

$X$ is equivariantly formal if and only if $X$ is locally standard and $Q$ is
mod $2$ face-acyclic.

  * (ii)

$X$ is equivariantly formal and $H^{*}(X;\mathbb{Z}_{2})$ is generated by its
degree-one part if and only if $X$ is locally standard and $Q$ is a mod $2$
homology polytope.

##  3\. 2-torus actions of complexity one in general position

In this section we show that an orbit space of certain $2$-torus action of
complexity one is a topological manifold. This section extends [2] to
$2$-torus actions.

Let a $2$-torus $\mathbb{Z}_{2}^{n-1}$ act effectively on a connected closed
smooth manifold $X=X^{n}$ with nonempty set of fixed points.

For a fixed point $x\in X^{\mathbb{Z}_{2}^{n-1}}$ of the action we the have
tangent representation of $\mathbb{Z}_{2}^{n-1}$ at $x$. Consider
$\alpha_{x,1},\ldots,\alpha_{x,n}\in\operatorname{Hom}(\mathbb{Z}_{2}^{n-1},\mathbb{Z}_{2})\cong\mathbb{Z}_{2}^{n-1}$,
the weights of the tangent representation at $x$, i.e.

$T_{x}X\cong V(\alpha_{x,1})\oplus\ldots\oplus V(\alpha_{x,1})$

where $V(\alpha_{x,1})$ is a 1-dimensional real representation given by
$t\cdot y=\alpha_{x,1}(t)y$ for $t\in\mathbb{Z}_{2}^{n-1}$ and
$y\in\mathbb{R}$.

######  Definition 3.1.

The action is said to be in _general position_ if for any fixed point $x$, any
$n-1$ of the weights $\alpha_{x,1},\dotsc,\alpha_{x,n}$ are linearly
independent over $\mathbb{Z}_{2}$.

######  Remark 3.2.

From The Slice Theorem it follows that all weights of an action are non-zero
if and only if the fixed point set is discrete. Hence, if $X$ is compact, then
the fixed point set is finite.

Let an action be in general position. Since any $n-1$ of the weights
$\alpha_{x,1},\dotsc,\alpha_{x,n}\in\mathbb{Z}_{2}^{n-1}$ are linearly
independent, we have $\alpha_{x,1}+\dotsc+\alpha_{x,n}=0$. Hence, for any
$t\in\mathbb{Z}_{2}^{n-1}$ we have

(3.1) $\Pi_{i=1}^{n}\alpha(t)_{x,i}=1.$

Moreover, the condition of general position implies that the tangent
representation at any fixed point is faithful. This motivates the following
construction.

Let $G$ be a subgroup of $\mathbb{Z}^{n}_{2}$ consisting of elements of the
following form:

(3.2) $G=\\{(g_{1},\dots,g_{n})\in\mathbb{Z}_{2}^{n}:\Pi_{i=1}^{n}g_{i}=1\\},$

where $g_{i}\in\\{-1,1\\}$. Since $\mathbb{Z}_{2}^{n}$ acts coordinate-wise on
$\mathbb{R}^{n}$, we have an induced action of $G$ on $\mathbb{R}^{n}$, which
we call the standard complexity one action.

We show below that the orbit space of the standard complexity one action is
homeomorphic to $\mathbb{R}^{n}$. Moreover, $G$ is the unique subgroup of
$\mathbb{Z}_{2}^{n}$ of index 2 such that
$\mathbb{R}^{n}/G\cong\mathbb{R}^{n}$.

Let $\chi:\mathbb{Z}_{2}^{n-1}\to GL_{n}(\mathbb{R})$ be an action in general
position with the weights $\alpha_{1},\dots,\alpha_{n}$. Define
$\phi:\mathbb{Z}_{2}^{n-1}\to G$ by the following formula
$\phi(t)=(\alpha_{1}(t),\dots,\alpha_{n}(t))$. Since the action in general
position is faithful representation and $\eqref{eqGenPos}$ holds, we have that
$\phi$ is an isomorphism and the following diagram commutes:

(3.3)
${\mathbb{Z}_{2}^{n-1}}$${GL_{n}(\mathbb{R})}$${G}$$\scriptstyle{\chi}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$

Where $\chi$ is the action in general position and $\psi$ is the standard
complexity one action of $G$ on $\mathbb{R}^{n}$ by coordinates, i.e. $g\cdot
x=(g_{1}x_{1},\dots,g_{n}x_{n})$. Hence, we get the following:

######  Proposition 3.3.

Let $\mathbb{Z}_{2}^{n-1}$-action on $\mathbb{R}^{n}$ be in general position.
Consider $G$ as in (3.2). Then this action is weakly equivalent to the
standard complexity one action of $G$ on $\mathbb{R}^{n}$, i.e. diagram
$\eqref{IsoGenPos}$ commutes.

In the following proposition we prove that all stabilizers of a $2$-torus
action in general position are generated by rotations, i.e. by orthogonal
transformations whose fixed-point subspace has codimension two.

######  Proposition 3.4.

Consider the standard complexity one action of $G$ on $\mathbb{R}^{n}$. Let
$H=Stab(x)$ be the stabilizer subgroup of any $x\in\mathbb{R}^{n}$. Then the
orbit space $\mathbb{R}^{n}/H$ is homeomorphic to $\mathbb{R}^{n}$. Moreover,
$G$ is the only one subgroup of $\mathbb{Z}^{n}_{2}$ of index $2$ such that
$\mathbb{R}^{n}/G\cong\mathbb{R}^{n}$.

###### Proof.

Let us describe the stabilizer $H$ of a point $x\in\mathbb{R}^{n}$. For
$x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$ let

$I=\\{i\in\\{1,\dots,n\\}:i\in I\text{ if }x_{i}=0\\}=\\{i_{1},\dots,i_{k}\\}$

be the set of indices with zero coordinates of $x$. If $|I|<2$, then the
stabilizer subgroup is trivial, hence we can assume that $|I|\geq 2$. Consider
the following subgroup of $\mathbb{Z}_{2}^{n}$:

(3.4)
$\mathbb{Z}_{2}^{I}=(\mathbb{Z}_{2},1)^{I}=\\{(g_{1},\dots,g_{n})\in\mathbb{Z}_{2}^{n}:g_{i}\in\mathbb{Z}_{2}\text{
if }i\in I,\text{ otherwise }g_{i}=1\\}.$

We have that $H=\mathbb{Z}_{2}^{I}\cap G$. Taking $g_{i_{1}},\dots,g_{i_{k}}$
as generators of $\mathbb{Z}_{2}^{I}$, we have that
$H=\\{(g_{1},\dots,g_{k})\in\mathbb{Z}_{2}^{I}:\prod^{k}_{1}g_{i}=1\\}$. This
means that $H$ generated by rotations. It follows from [17, Thm. A] that the
orbit space of this action is homeomorphic to $\mathbb{R}^{n}$. From this
theorem also follows that any subgroup $H$ of $\mathbb{Z}_{2}^{n}$ such that
$\mathbb{R}^{n}/H$ is homeomorphic to $\mathbb{R}^{n}$ must be generated by
rotations, but $G=\\{(g_{1},\dots,g_{n}):\Pi_{i=1}^{n}g_{i}=1\\}$ is the only
subgroup of index $2$ generated by rotations. ∎

######  Corollary 3.5.

Suppose $\mathbb{Z}_{2}^{n-1}$ action on $\mathbb{R}^{n}$ is in general
position. Then $\mathbb{R}^{n}/Stab(x)\cong\mathbb{R}^{n}$ for the stabilizer
subgroup $Stab(x)$ of any $x\in\mathbb{R}^{n}$.

###### Proof.

Since action is in general position, we have that the diagram (3.3) commutes.
Therefore, the action is weakly equivalent to the action in Proposition 3.4. ∎

######  Remark 3.6.

From the previous corollary it follows that the stabilizer subgroup of an
action in general position can not be arbitrary. For example, it can not be
$H=\\{(1,1,1,1),(-1,-1,-1,-1)\\}$ and, indeed, $\mathbb{R}^{4}/H$ is
homeomorphic to the open cone over $\mathbb{R}P^{3}$. This example shows
importance of the condition that subgroup is generated by rotations.
Description of all linear representations of finite groups whose orbit spaces
are homeomorphic to $\mathbb{R}^{n}$ can be found in [17].

For the global statement of the previous corollary we need the following
condition on an action:

(3.5) $\mbox{Every connected component of }X^{H}\mbox{ has a global fixed
point,}$

where $X^{H}=\\{x\in X:H\subset Stab(x)$}.

The following lemma shows that this condition holds for equivariantly formal
action of $2$-torus.

######  Lemma 3.7 ([20, Lem. 3.2]).

Suppose a $\mathbb{Z}_{2}^{k}$-action on a compact manifold $X$ is
equivariantly formal. Then for every subgroup $H$ of $\mathbb{Z}_{2}^{k}$, the
induced action of $\mathbb{Z}^{k}_{2}$( or $\mathbb{Z}_{2}^{k}/H$) on every
connected component $N$ of $X^{H}$ is equivariantly formal, hence $N$ has a
$\mathbb{Z}^{k}_{2}$-fixed point.

Now we can prove the following theorem about the orbit space of complexity one
actions in general position. This is the first part of Theorem 1 from the
introduction.

######  Theorem 3.8.

Let $\mathbb{Z}_{2}^{n-1}$ act on a connected closed smooth manifold
$X=X^{n}$. Suppose that the action is in general position and the condition
(3.5) holds. Then the orbit space $Q=X/\mathbb{Z}_{2}^{n-1}$ is a topological
manifold.

###### Proof.

Let us denote $G=\mathbb{Z}_{2}^{n-1}$. Let $H=Stab(x)$ be the stabilizer
subgroup of any $x$ in $X$, or equivalently $x\in X^{(H)}$. Hence, $x\in N$,
where $N$ is a connected component of $X^{H}$. It follows from the condition
(3.5) that $N$ has a global fixed point $x^{\prime}$. By The Slice Theorem
there exist a $G$ \- equivariant neighborhood $U(x^{\prime})$ of $x^{\prime}$
such that $U(x^{\prime})$ is $G$-diffeomorphic to $T_{x^{\prime}}X$. Since the
tangent representation of $H$ at $x$ depends only on a connected component of
$X^{H}$, we can assume that $x$ is near $x^{\prime}$, i.e. $x\in
U(x^{\prime})$. Therefore, $H$ is a stabilizer subgroup of an action
$\mathbb{Z}_{2}^{n-1}$ on $\mathbb{R}^{n}$ in general position. By The Slice
Theorem every orbit has a $G$-equivariant neighborhood $U$ such that $U$ is
$G$-equivariantly homeomorphic to $G\times_{H}V_{x}$, where
$V_{x}=T_{x}X/T_{x}Gx$. Since the orbit $Gx$ is a discrete set, we have
$V_{x}=T_{x}X$. Therefore, $U/G\cong T_{x}X/H\cong\mathbb{R}^{n}$ by Corollary
3.5. ∎

##  4\. Equivariantly formal actions of complexity one

In this section we show that the orbit space of an equivariantly formal action
in general position is a $\mathbb{Z}_{2}$-homology sphere. First of all, we
introduce the notion of a face.

For an action of $\mathbb{Z}_{2}^{n-1}$ on $X$ consider the equivariant
filtration

$\varnothing=X_{-1}\subset X_{0}\subset X_{1}\subset\cdots\subset X_{n-1}=X,$

where $X_{i}$ is the union of orbits of size at most $2^{i}$ and
$X_{-1}=\varnothing$. Notice that

$X_{i}=\\{x\in X:rk(Stab(x))\geq n-1-i\\}=\bigsqcup_{rkH\geq n-1-i}X^{(H)}$

and each $X^{(H)}$ is the disjoint union of submanifolds of X. There is an
orbit type filtration of the orbit space $Q=X/\mathbb{Z}_{2}^{n-1}$:

$\varnothing=Q_{-1}\subset Q_{0}\subset Q_{1}\subset\cdots\subset Q_{n-1}=Q,$

where $Q_{i}=X_{i}/\mathbb{Z}_{2}^{n-1}$ and $Q_{-1}=\varnothing$.

######  Definition 4.1.

A closure of any connected component of $Q_{i}\setminus Q_{i-1}$ is called _a
face of rank i_. For a face $F$ of rank $i$ we define $F_{-1}=F\cap Q_{i-1}$.

Let $F$ be a face of $Q$, consider the quotient map $p:X\to Q$, denote the
preimage of $F$ by $X_{F}=p^{-1}(F)$. Let $G_{F}\subset\mathbb{Z}_{2}^{n-1}$
be the non-effective kernel of the $\mathbb{Z}_{2}^{n-1}$-action on $X_{F}$.
Then $X_{F}$ is a connected component of $X^{G_{F}}$, therefore $X_{F}$ is a
submanifold of $X$.

######  Definition 4.2.

Let $F$ be a face of $Q$. The preimage $X_{F}=p^{-1}(F)$ of $F$ is called
_face submanifold_ corresponding to $F$. For a face submanifold $X_{F}$ we
define $X_{F_{-1}}=X_{F}\cap X_{i}$.

In the next proposition we show that each $X_{F}$ is an equivariantly formal
$2$-torus manifold.

######  Proposition 4.3.

Let $\mathbb{Z}_{2}^{n-1}$ act on $X$ equivariantly formal and in general
position. Let $X_{F}$ be the face submanifold corresponding to a face $F$ of
rank $i<n-1$. Then $\dim X_{F}=i$, hence $X_{F}$ is a equivariantly formal
$2$-torus manifold with an action $\mathbb{Z}_{2}^{n-1}/G_{F}$.

###### Proof.

Let $G_{F}$ be the non-effective kernel of the $\mathbb{Z}_{2}^{n-1}$-action
on $X_{F}$, we have that

(4.1) $|G_{F}|=2^{n-1-i},$

since $\operatorname{rk}F=i$. Consider $p^{-1}(F^{\circ})=\\{x\in
X_{F}:Stab(x)=G_{F}\\}$ is an open subset of $X_{F}$. Let $x^{\prime}\in
X_{F}$ be a global fixed point of action $\mathbb{Z}_{2}^{n-1}$ on $X$. Let
$U(x^{\prime})$ be $\mathbb{Z}_{2}^{n-1}$-equivariant neighborhood of
$x^{\prime}$ such that $U(x^{\prime})$ is $\mathbb{Z}_{2}^{n-1}$-equivariant
diffeomorphic to $T_{x}^{\prime}X$. We have that $p^{-1}(F^{\circ})\cap
U(x^{\prime})\cong V$, where $V=\\{x\in T_{x^{\prime}}X:Stab(x)=G_{F}\\}$ is a
linear subspace such that for any $x=(x_{1},\ldots,x_{n})\in V$ we have
$x_{i}=0$, where $i\in I\subset\\{1,\ldots,n\\}$, i.e. some coordinates of $x$
are zero. Let $|I|=k$ be the number of zero coordinates. $G_{F}$ is generated
by rotations, since it is the stabilizer subgroup of the standard complexity
one action. Therefore, $G_{F}$ contains all
$(g_{1},\ldots,g_{n})\in\mathbb{Z}_{2}^{n}$ such that $g_{j}=-1$ for all $j\in
J$, where $J\subset I$ and $|J|$ is even. Hence, we have

(4.2)
$|G_{F}|=\binom{k}{0}+\binom{k}{2}+\dots+\binom{k}{2\lfloor\frac{k}{2}\rfloor}=2^{k-1}.$

On the other hand, we have $|G_{F}|=2^{n-1-i}$, therefore $k=n-i$. Hence,
$\dim X_{F}=n-k=i$ and $\operatorname{rk}\mathbb{Z}_{2}^{n-1}/G_{F}=i$. This
action is equivariantly formal by Lemma 3.7. ∎

In this paper all cohomology groups are taken with coefficients in
$\mathbb{Z}_{2}$.

######  Corollary 4.4.

For any face $F$ of rank $i<n-1$, we have
$H^{*}(F,F_{-1})=H^{*}(D^{i},\partial D^{i})$.

###### Proof.

From Theorem 2.7 it follows that $F$ is mod 2 face-acyclic. Hence,

$H^{*}(F,F_{-1})=H^{*}(D^{i},\partial D^{i})$

by Lefschetz duality. ∎

Now we introduce Atiyah–Bredon–Franz–Puppe sequence for equivariant cohomology

(4.3) $0\to H^{*}_{\mathbb{Z}_{2}^{k}}(X)\stackrel{{\scriptstyle
i^{*}}}{{\to}}H^{*}_{\mathbb{Z}_{2}^{k}}(X_{0})\stackrel{{\scriptstyle\delta_{0}}}{{\to}}H^{*+1}_{\mathbb{Z}_{2}^{k}}(X_{1},X_{0})\stackrel{{\scriptstyle\delta_{1}}}{{\to}}\cdots\\\
\cdots\stackrel{{\scriptstyle\delta_{k-2}}}{{\to}}H^{*+k-1}_{\mathbb{Z}_{2}^{k}}(X_{k-1},X_{k-2})\stackrel{{\scriptstyle\delta_{k-1}}}{{\to}}H^{*+k}_{\mathbb{Z}_{2}^{k}}(X,X_{k-1})\to
0,$

where $\delta_{i}$ is the connecting homomorphism in the long exact sequence
of equivariant cohomology of the triple $(X_{i+1},X_{i},X_{i-1})$ and $X_{i}$
is the union of orbits of size at most $2^{i}$. If an
$\mathbb{Z}_{2}^{k}$-action on $X$ is equivariantly formal, then this sequence
is exact. For the proof see [1],[9]. Also see [13] for the torus case.

######  Theorem 4.5 ([1, Thm. 10.2]).

Suppose that an action of $\mathbb{Z}_{2}^{k}$ on $X$ is equivariantly formal.
Then sequence (4.3) is exact.

From exactness of this sequence and previous corollary we immediately get the
following result:

######  Proposition 4.6.

Let $\mathbb{Z}_{2}^{n-1}$ act on $X$ equivariantly formal and in general
position. Then for the orbit space $Q$ we have $H^{i}(Q,Q_{n-2})=0$ for
$i<n-1$.

###### Proof.

To be short denote $G=\mathbb{Z}_{2}^{n-1}$. Consider $i$-th term in (4.3)
with $i\leq n-2$:

$H^{*+i}_{G}(X_{i},X_{i-1})\cong\bigoplus_{F\colon\dim
F=i}H^{*+i}_{G}(X_{F},X_{F}\cap X_{i-1})\cong\bigoplus_{F\colon\dim
F=i}H^{i}(F,F_{-1})\otimes H^{*}(BG_{F}).$

The first isomorphism follows from the equivariant version of Mayer-Vietoris
sequence and the second isomorphism follows from the two facts: the action of
group $G/G_{F}$ on $X_{F}\setminus(X_{F}\cap X_{i-1})$ is free and from
Corollary 4.4. Therefore, $H^{*+i}_{G}(X_{i},X_{i-1})=0$ for $*<0$ and $i\leq
n-2$. Consider $*<0$. Then from exactness of sequence $\eqref{eqABseqForX}$ we
get that $H_{G}^{i}(X,X_{n-2})=0$ for $i<n-1$. On the other hand, we have
$H_{G}^{i}(X,X_{n-2})\cong H^{i}(Q,Q_{n-2})$ since the action of $G$ on
$X\setminus X_{n-2}$ is free. Hence, we have $H^{i}(Q,Q_{n-2})=0$ for $i<n-1$.
∎

Now we can prove Theorem 1 from the introduction.

######  Theorem 4.7.

Let $\mathbb{Z}_{2}^{n-1}$ act on $X=X^{n}$ equivariantly formal and in
general position. Then the orbit space $Q=X/\mathbb{Z}_{2}^{n-1}$ is a
$\mathbb{Z}_{2}$-homology $n$-sphere, i.e. $H^{*}(Q;\mathbb{Z}_{2})\cong
H^{*}(S^{n};\mathbb{Z}_{2})$.

###### Proof.

Consider the cohomology spectral sequence associated with the filtration

$\varnothing=Q_{-1}\subset Q_{0}\subset Q_{1}\subset\cdots\subset Q_{n-1}=Q,$

where $Q_{i}=X_{i}/\mathbb{Z}_{2}^{n-1}$ and $Q_{-1}=\varnothing$:

(4.4) $E_{1}^{p,q}=H^{p+q}(Q_{p},Q_{p-1})\Rightarrow H^{p+q}(Q).$

We get that

(4.5)
$E_{1}^{p,q}=H^{p+q}(Q_{p},Q_{p-1})=\bigoplus_{F:rkF=p}H^{p+q}(F,F_{-1}).$

From the Corollary 4.4 and Proposition 4.6 it follows that the only non-zero
terms of the first page of the spectral sequence are the $0$-th row
$(E_{1}^{p,0},d^{1})$ and the $n-1$-th column $(E_{1}^{n-1,q},d^{1})$. We have
that the first page of the spectral sequence (4.4) as shown on Figure 1.

$\ast$$\ast$$\cdots$$\vdots$$\ast$$\ast$$\ast$$\ast$$E_{1}^{p,q}$$0$$n-1$$0$$p$$q$$0$$0$$0$
Figure 1. The first page of the spectral sequence (4.4)

We claim that the differential complex in the $0$-th row $(E_{1}^{p,0},d^{1})$
is isomorphic to the degree $0$ part of the non-augmented version of Atiyah-
Bredon-Franz-Puppe sequence (4.3)

$0\to
H^{0}_{\mathbb{Z}_{2}^{n-1}}(X_{0})\stackrel{{\scriptstyle\delta_{0}}}{{\to}}H^{1}_{\mathbb{Z}_{2}^{n-1}}(X_{1},X_{0})\stackrel{{\scriptstyle\delta_{1}}}{{\to}}\cdots\stackrel{{\scriptstyle\delta_{n-3}}}{{\to}}H^{n-2}_{\mathbb{Z}_{2}^{n-1}}(X_{n-2},X_{n-3})\stackrel{{\scriptstyle\delta_{n-2}}}{{\to}}H^{n-1}_{\mathbb{Z}_{2}^{n-1}}(X,X_{n-2})\to
0.$

Indeed, we have the natural projection map $\pi:X_{G}\to X/G$ such that
$\pi((X_{i})_{G})\subset Q_{i}$ for any $i$. Therefore, $\pi$ induces a map
between long exact sequences of $(X_{i+1},X_{i},X_{i-1})_{G}$ and
$(Q_{i+1},Q_{i},Q_{i-1})$, hence $\pi^{*}$ commutes with $\delta_{i}$ for any
$i$. We claim that the induced maps in cohomology

$\pi^{*}:H^{i}(Q_{i},Q_{i-1})\to H^{i}_{G}(X_{i},X_{i-1})$

are isomorphisms.

We have that

$H^{i}_{G}(X_{i},X_{i-1})=\bigoplus_{F\colon
rkF=i}H^{i}_{G}(X_{F},X_{F_{-1}}),$ $H^{i}(Q_{i},Q_{i-1})=\bigoplus_{F\colon
rkF=i}H^{i}(F,F_{-1}).$

Since $(X_{F},X_{F_{-1}})$ fixed by $G_{F}$ and $EG=E(G_{F})\times
E(G/G_{F})$, we have that
$H_{G}^{i}(X_{F},X_{F_{-1}})=H^{i}_{G/G_{F}}(X_{F},X_{F_{-1}})\otimes_{\mathbb{Z}_{2}}H^{0}(BG_{F})$.
Therefore, it is enough to prove that the map

$\pi^{*}:H^{i}(F,F_{-1})\to H^{i}_{G/G_{F}}(X_{F},X_{F_{-1}})$

is an isomorphism. We have that the action of $G/G_{F}$ is free on
$X_{F}\setminus X_{F_{-1}}$, therefore $\pi^{-1}(x)=E(G/G_{F})$ for any $x\in
F\setminus F_{-1}$. Hence, from a relative version of the Vietoris–Begle
Theorem (see [19]), it follows that the induced map

$\pi^{*}:H^{i}(F,F_{-1})\to H^{i}_{G/G_{F}}(X_{F},X_{F_{-1}})$

is an isomorphism. Since the action is equivariantly formal, the Atiyah-
Bredon-Franz-Puppe sequence (4.3) is exact. Therefore, the differential
complex in the $0$-th row $(E_{1}^{p,0},d^{1})$ is exact.

Hence, we have that the second page of the spectral sequence (4.4) as shown on
Figure 2.

$\mathbb{Z}_{2}$$0$$\vdots$$\ast$$\ast$$\ast$$0$$E_{2}^{p,q}$$0$$n-1$$0$$p$$q$$0$$0$$0$
Figure 2. The second page of the spectral sequence (4.4)

We have that $E^{p,q}_{\infty}=0$ for $0<p+q\leq n-1$, therefore $H^{i}(Q)=0$
for $i\leq n-1$. On the other hand, $Q$ is a topological manifold, hence
$H^{i}(Q)\cong H^{i}(S^{n})$. ∎

##  5\. Complexity one actions in case of small covers

In this section we recall the definition of a small cover and prove Theorem 2,
3 from the introduction. For the details about small covers see [11].

######  Definition 5.1.

Let $P=P^{n}$ be a simple polytope of dimension $n$. A small cover over $P$ is
a smooth manifold $X=X^{n}$ with a locally standard
$\mathbb{Z}_{2}^{n}$-action such that the orbit space $X/\mathbb{Z}_{2}^{n}$
is diffeomorphic to a simple polytope $P$ as a manifold with corners.

Let $\pi:X\to P$ be a small cover over $P$. For every face $F$ of $P$ and for
every $x,y\in\pi^{-1}(F^{\circ})$ the stabilizer group of $x$ and $y$ is the
same, i.e. $Stab(x)=Stab(y)$. Denote this stabilizer group by $G_{F}$. In
particular, if $F$ is a facet, then $G_{F}$ is subgroup of rank one, hence
$G_{F}=\langle\lambda(F)\rangle$ for some $\lambda(F)\in\mathbb{Z}_{2}^{n}$.
Hence, we get the characteristic function

$\lambda:\mathcal{F}\to\mathbb{Z}_{2}^{n},$

from the set $\mathcal{F}$ of all facets of $P$. We denote $\lambda(F_{i})$ by
$\lambda_{i}$. For a codimension k face $F$ we have that $F=F_{1}\cap\dots\cap
F_{k}$ for some facets $F_{1},\dots,F_{k}\in\mathcal{F}$, then $G_{F}$ is a
subgroup with rank equal to $k$ and generated by
$\lambda_{1},\dots,\lambda_{k}$. Therefore, we have the following
$(*)$-condition

$(*)$ Let $F=F_{1}\cap\dots\cap F_{k}$ be any codimension k face of $P$. Then
$\lambda_{1},\dots,\lambda_{k}$ are linearly independent in
$\mathbb{Z}_{2}^{n}$.

Conversely, a simple polytope $P$ and a map
$\lambda:\mathcal{F}\to\mathbb{Z}_{2}^{n}$ satisfying the $(*)$-condition
determine a small cover $X(P,\lambda)$ over $P$. For the construction of
$X(P,\lambda)$ and other details see [11].

######  Theorem 5.2 ([11, Prop. 1.8]).

Let $X$ be a small cover over $P$ with characteristic function
$\lambda:\mathcal{F}\to\mathbb{Z}_{2}^{n}$. Then $X$ and $X(P,\lambda)$ are
equivariantly homeomorphic.

Let $X$ be a small cover, let $G\cong\mathbb{Z}_{2}^{n-1}$ be a subgroup of
$\mathbb{Z}_{2}^{n}$ of index 2, then we get an restricted $G$-action of
complexity one on $X$. The subgroup $G$ is called a $2$-subtorus in general
position, if this $G$-action is in general position.

######  Remark 5.3.

Notice that it is possible that a $2$-subtorus in general position does not
exist. For example, for the small cover $\mathbb{R}P^{2}$ over $\Delta^{2}$
there is no $2$-subtorus in general position.

However, we will see that if $X$ is an orientable small cover, then such
$2$-subtorus exist. Every subgroup $G\subset\mathbb{Z}_{2}^{n}$ with rank
$n-1$ is determined by some non-zero linear functional
$\xi\in(\mathbb{Z}_{2}^{n})^{*}$ by the following $G=\operatorname{Ker}(\xi)$.
We have the following criteria when $G$ is in general position.

######  Proposition 5.4.

Let $X=X^{n}$ be a small cover, let $\lambda_{i}=\lambda(F_{i})$ be
characteristic vectors. The $2$-subtorus
$G=\operatorname{Ker}(\xi:\mathbb{Z}_{2}^{n}\to\mathbb{Z}_{2})$ is in general
position if and only if $\xi(\lambda_{i})=1$, in other words
$\lambda_{i}\notin G$, for every facet $F_{i}\in\mathcal{F}$.

###### Proof.

If $p=F_{i_{1}}\cap\dots\cap F_{i_{n}}$ is a vertex of $P$, then the elements
of the dual basis $\lambda^{*}_{i_{1}},\dots,\lambda_{i_{n}}^{*}$ are the
weights of the tangent representation of $\mathbb{Z}_{2}^{n}$ in the
corresponding fixed point. Therefore, $G=\operatorname{Ker}(\xi)$ is in
general position at this fixed point if and only if any $n-1$ of the weights
$\lambda^{*}_{i_{1}},\dots,\lambda_{i_{n}}^{*}$ are linearly independent in
$(\mathbb{Z}_{2}^{n})^{*}/\langle\xi\rangle.$

Since $\lambda^{*}_{i_{1}},\dots,\lambda_{i_{n}}^{*}$ is a basis, we have that
$\xi=a_{1}\lambda^{*}_{i_{1}}+\dots+a_{n}\lambda^{*}_{i_{n}}$ in
$\mathbb{Z}_{2}^{n}$ for $a_{i}\in\mathbb{Z}_{2}$. Therefore, we have
$a_{1}\lambda^{*}_{i_{1}}+\dots+a_{n}\lambda^{*}_{i_{n}}=0$ in
$(\mathbb{Z}_{2}^{n})^{*}/\langle\xi\rangle$ for $a_{i}\in\mathbb{Z}_{2}$. We
have that any $n-1$ of the weights
$\lambda^{*}_{i_{1}},\dots,\lambda_{i_{n}}^{*}$ are linearly independent in
$(\mathbb{Z}_{2}^{n})^{*}/\langle\xi\rangle$ if and only if all $a_{i}$ are
non-zero, i.e. $a_{i}=1$ for every $i$. The last statement is equivalent to
that $\xi(\lambda_{i_{k}})=1$ for all $i_{k}$. ∎

######  Corollary 5.5.

If $G\subset\mathbb{Z}_{2}^{n}$ is in general position, then
$Stab(x)\not\subset G$ for every $x\in X$.

###### Proof.

Indeed, Stab(x) is generated by $\lambda_{i}$ for some $i$. ∎

######  Remark 5.6.

From the proof, we see that there exist only one $2$-subtorus in general
position. If we choose a vertex $p=F_{i_{1}}\cap\dots\cap F_{i_{n}}$, then
corresponding characterestic vectors $\lambda_{i_{1}},\dots,\lambda_{i_{n}}$
is a basis of $\mathbb{Z}_{2}^{n}$. Taking these vectors as generators of
$\mathbb{Z}_{2}^{n}$, we get that

$G=\\{(g_{1},\dots,g_{n})\in\mathbb{Z}_{2}^{n}:\Pi_{i=1}^{n}g_{i}=1\\}$

in the multiplicative notation.

The condition from Proposition 5.4 is related to the following result of H.
Nakayama and Y. Nishimura.

######  Theorem 5.7 ([18, Thm. 1.7]).

A small cover $X=X^{n}$ is orientable if and only if there exist
$\xi\in(\mathbb{Z}_{2}^{n})^{*}$ such that $\xi(\lambda_{i})=1$ for every face
$F_{i}\in\mathcal{F}$.

From this result we get Theorem 2 from the introduction.

######  Corollary 5.8.

Let $X$ be a small cover. There exists a $2$-subtorus in general position if
and only if $X$ is orientable.

Now we can prove Theorem 3 from the introduction.

######  Theorem 5.9.

Let $X=X^{n}$ be an orientable small cover over, let $G$ be the $2$-subtorus
in general position. Then the orbit space $X/G$ is homeomorphic to the
$n$-dimensional sphere $S^{n}$.

###### Proof.

Let $Q=X/G$ be the orbit space of the $G$-action, let $P=X/\mathbb{Z}_{2}^{n}$
be the orbit space of the $\mathbb{Z}_{2}^{n}$-action. By the definition of a
small cover, $P$ is a simple polytope and $\text{dim}P=n$. Notice that
$P=Q/(\mathbb{Z}_{2}^{n}/G)$ and we have the quotient map

$p:Q\to Q/(\mathbb{Z}_{2}^{n}/G).$

If $x\in P$ is a free $\mathbb{Z}_{2}^{n}$-orbit, i.e. $x$ in the interior of
$P$, then $p^{-1}(x)=\mathbb{Z}_{2}$. Otherwise, by Corollary 5.5 there exist
$t\in Stab(x)$ such that $t\notin G$. Therefore, if a
$\mathbb{Z}_{2}^{n}$-stabilizer group of $x$ is non-trivial, i.e. $x$ in the
boundary of $P$, then it is a fixed point of $\mathbb{Z}_{2}^{n}/G$-action.
Hence, for $x\in\partial P$ we have that $p^{-1}(x)$ is a single point. Since
$P$ is contractible, the map $p:Q\to Q/(\mathbb{Z}_{2}^{n}/G)$ admit a section
over the interior of $Q/(\mathbb{Z}_{2}^{n}/G)=P$. Therefore, we have

$Q\cong P\times\mathbb{Z}_{2}/\\!\\!\sim$

where $(x,1)\sim(x,-1)$ if $x\in\partial P$. Since $P$ is homeomorphic to the
$n$-disc $D^{n}$, we have that

$Q\cong D^{n}\sqcup D^{n}/\\!\\!\sim\cong S^{n},$

which proves the statement. ∎

## Acknowledgements

The author would like to thanks his advisor A. A. Ayzenberg for an invitation
to toric topology, his help and guidance during this work, T. E. Panov for
helpful discussions about small covers. The author would like to thanks Eva
Vlasova for her support and for drawing pictures.

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