# Distribution-based Sketching of Single-Cell Samples

Vishal Athreya Baskaran<EMAIL_ADDRESS>Department of Computer Science,
UNC-Chapel Hill USA ,  Jolene Ranek<EMAIL_ADDRESS>Computational
Medicine Program, Curriculum in Bioinformatics and Computational Biology, UNC-
Chapel Hill USA ,  Siyuan Shan<EMAIL_ADDRESS>Department of Computer
Science, UNC-Chapel Hill USA ,  Natalie Stanley<EMAIL_ADDRESS>Computational Medicine Program, Department of Computer Science, UNC-Chapel
Hill USA and  Junier B. Oliva<EMAIL_ADDRESS>Department of Computer
Science, UNC-Chapel Hill USA

(2022)

###### Abstract.

Modern high-throughput single-cell immune profiling technologies, such as flow
and mass cytometry and single-cell RNA sequencing can readily measure the
expression of a large number of protein or gene features across the millions
of cells in a multi-patient cohort. While bioinformatics approaches can be
used to link immune cell heterogeneity to external variables of interest, such
as, clinical outcome or experimental label, they often struggle to accommodate
such a large number of profiled cells. To ease this computational burden, a
limited number of cells are typically _sketched_ or subsampled from each
patient. However, existing sketching approaches fail to adequately subsample
rare cells from rare cell-populations, or fail to preserve the true
frequencies of particular immune cell-types. Here, we propose a novel
sketching approach based on Kernel Herding that selects a limited subsample of
all cells while preserving the underlying frequencies of immune cell-types. We
tested our approach on three flow and mass cytometry datasets and on one
single-cell RNA sequencing dataset and demonstrate that the sketched cells (1)
more accurately represent the overall cellular landscape and (2) facilitate
increased performance in downstream analysis tasks, such as classifying
patients according to their clinical outcome. An implementation of sketching
with Kernel Herding is publicly available at
https://github.com/vishalathreya/Set-Summarization.

Cytometry, Single-Cell Bioinformatics, Clinical Prediction, Compression

††journalyear: 2022††copyright: acmlicensed††conference: 13th ACM
International Conference on Bioinformatics, Computational Biology and Health
Informatics; August 7–10, 2022; Northbrook, IL, USA††booktitle: 13th ACM
International Conference on Bioinformatics, Computational Biology and Health
Informatics (BCB ’22), August 7–10, 2022, Northbrook, IL, USA††price:
15.00††doi: 10.1145/3535508.3545539††isbn: 978-1-4503-9386-7/22/08

##  1\. Introduction

Advances in high throughput single-cell technologies have transformed our
understanding of cellular heterogeneity across a range of biological and
clinical applications. For example, single-cell measurements have been used to
construct whole organism cell atlases (Regev et al., 2017; Cao et al., 2019)
that can serve as prototypical healthy references that assist in the discovery
of novel disease cell states (Lotfollahi et al., 2021). Translationally,
single-cell modalities have provided insight into complex immune responses
linked to clinical outcomes, such as surgical recovery (Ganio et al., 2020),
pregnancy (Aghaeepour et al., 2017), tumor heterogeneity (Levine et al.,
2015), and infectious disease (Arunachalam et al., 2020).

Figure 1. Overview of Kernel Herding for Sketching Single-Cell Data. Multiple
_sample-sets_ are created after profiling the expression of several genomic or
protein features in single cells across multiple individuals. We used Kernel
Herding to select a limited representative subset of cells from sample-sets
for each individual. Our approach was then evaluated in terms of the quality
of cellular landscape preservation and classification performance of sample-
sets according to the clinical outcomes of the associated patients.

Single-cell flow and mass cytometry (e.g. CyTOF) technologies have been
particularly useful for gaining an in-depth understanding of the diversity of
immune cell-types and their relation to disease, as they simultaneously
profile 20-45 proteins collectively across all of the cells profiled in a
multi-patient cohort (Bendall et al., 2012; Spitzer and Nolan, 2016; Liechti
et al., 2021). This data can be leveraged in statistical models for
identifying differentially abundant cell-populations (Weber et al., 2019; Lun
et al., 2017; Dann et al., 2021), kernel density estimation methods to
characterize treatment resistant cells (Burkhardt et al., 2021), and
classification algorithms to predict a patient’s clinical outcome (Yi and
Stanley, 2021). Similarly, genomic technologies such as single-cell RNA
sequencing (scRNA-seq) technologies can profile $20,000$ or more gene
expression measurements in thousands of cells (Luecken and Theis, 2019). Given
the large number of cells measured in such datasets, downstream analysis tasks
present a computational challenge and quickly become intractable. Traditional
approaches for reducing the size of input data, such as uniform downsampling
or clustering, may fail to identify rare but clinically meaningful cell-
populations. This limits both the utility of these algorithms and the ability
to obtain meaningful results.

Two alternative sketching-based approaches have been developed to
intelligently select a subset of cells such that the overall cellular
landscape of the full data is preserved. Geometric Sketching first
approximates the underlying geometry of the data through a covering of equal
volume hypercubes, then evenly samples points from each at random (Hie et al.,
2019). Alternatively, Hopper constructs a sketch by iteratively minimizing the
Hausdorff distance to ensure points in the full data are well-represented in
the sketch (DeMeo and Berger, 2020). Both methods provide an even sampling of
cells that focuses on capturing rare cell-types. However, as a result, they
fail to preserve the overall distributions of cell-type frequencies which can
be crucial for clinical outcome prediction and biological interpretation
(Ganio et al., 2020; Mathew et al., 2020). In a philosophical divergence with
this previous sketching work, here we prioritize the sketch’s ability to act
as a _stand-in_ for the original sample-set. That is, here we consider
sketches that obtain _similar downstream outcomes_ to the original sample-sets
when analyzed. In some respects, this goal is a generalization of prior
formulations, since rare-cell types must be preserved if the sketch can act as
a stand-in. However, our formulation also allows for the preservation of
general downstream analysis, which may not be entirely dependent on rare-cell
types. Thus, this work presents a novel, more direct approach to sketching
single-cell samples.

We propose to preserve downstream analysis of a single-cell sample by
explicitly preserving the distribution in the sketched sample. In this work,
we show that this can be achieved via Kernel Herding (Chen et al., 2010), a
kernel-based approach to sub-sampling. Furthermore, we show how to efficiently
compute Kernel Herding sketches through random Fourier features (Rahimi et
al., 2007), which allows our sketch approach to scale to large sample-sets. We
first evaluate our approach on three diverse flow and mass cytometry datasets
by computing several statistics to quantify cellular landscape preservation
and performance in downstream tasks, such as patient classification according
to clinical outcome. Furthermore, we demonstrate how this approach can be used
across single-cell data modalities with a single-cell RNA sequencing
application. Our Kernel Herding sketching approach (outlined in Fig. 1) now
makes it possible to select a limited subset of cells from each profiled
sample-set that adequately represents immune cell-types and their relative
frequencies.

##  2\. Methods

Here we are considering the problem of finding a representative subset to an
original sample-set of cells
$\mathcal{X}=\\{x_{i}\in\mathbb{R}^{d}\\}_{i=1}^{n}$. If subsets are to act as
a stand-in for original sample-sets, then they should provide similar outcomes
when processed as the original sets. Thus, we consider a subset,
$\hat{\mathcal{X}}=\\{\hat{x}_{i}\\}_{i=1}^{m}\subset\mathcal{X}$, to be
representative if it maintains the distribution of the original set
$\mathcal{X}$ and yields similar results to $\mathcal{X}$ when processed.

Problem Formulation At an abstract level, we shall construct a subset
$\hat{\mathcal{X}}$ whose statistics match the original sample-set
$\mathcal{X}$. Of course, for any finite number of statistics, there may be an
ambiguity over the original distribution. For instance, a large variety of
samples may share the same mean, thus simply finding a subset of a similar
mean as the original set will not suffice to ensure a representative
subsample. Instead, we shall use the kernel mean embedding (Muandet et al.,
2017) as the target ‘statistic’ to match in the subset. We briefly explain
kernels and kernel mean embedding below.

Kernel Mean Embedding Kernel methods have been successfully applied in a
myriad of machine learning problems such as regression (Vovk, 2013),
classification (Cortes and Vapnik, 1995), and dimensionality reduction (Mika
et al., 1998). Underlying kernel methods is a positive definitive kernel
function $K:\mathbb{R}\times\mathbb{R}\mapsto\mathbb{R}$ 111Kernels may also
generalize over non-real domains, which induces a reproducing kernel Hilbert
space (RKHS) (e.g. see (Berlinet and Thomas-Agnan, 2011) for further details).
In addition to working over typical vector-values data, recently kernels have
also been deployed to operate over distributions (e.g. (Muandet et al.,
2017)). The _kernel mean embedding_ $\mu_{p}:\mathbb{R}\mapsto\mathbb{R}$
represents a distribution, $p$, as:

(1) $\mu_{p}(\cdot)=\mathbb{E}_{x\sim p}[k(x,\cdot)]$

Note that $\mu_{p}$ is itself a function, defined by the expected kernel
evaulation to points drawn from a distribution $p$. For a special class of ‘
_characteristic_ ’ kernels, $k$, such as the common radial-basis function
(RBF) kernel $k(x,x^{\prime})=\exp(-\frac{1}{2\gamma}||x-x^{\prime}||^{2})$,
$||\mu_{p}-\mu_{q}||=0$ if and only if $p=q$. Thus, for characteristic kernels
the kernel mean embedding will be unique to its distribution and matching the
kernel embedding exactly guarantees that one matches a distribution. In
general, the distance222Corresponding to the RKHS norm $||\mu_{p}-\mu_{q}||$
induces a divergence, the maximum mean discrepancy (MMD) (Gretton et al.,
2008), between distributions, which has been useful in comparing distributions
(e.g. (Gretton et al., 2012)).

In this work we sketch a sample-set with a subset
$\hat{\mathcal{X}}\subset\mathcal{X}$ that preserves the empirical
distribution found in the original sample-set $\mathcal{X}$ with the mean
embedding $\mu_{\mathcal{X}}$ where:

(2) $\mu_{\mathcal{X}}(\cdot)=\frac{1}{n}\sum_{i=1}^{n}k(x_{i},\cdot).$

That is, we look for a subset $\hat{\mathcal{X}}$ whose empirical distribution
has a small MMD from that of the original set $\mathcal{X}$,
$||\mu_{\hat{\mathcal{X}}}-\mu_{\mathcal{X}}||$. Note that a uniform subsample
$\mathcal{U}\subset\mathcal{X}$ provides a reasonable approximation when the
cardinality $|\mathcal{U}|$ is large enough. That is
$\mu_{\mathcal{U}}(\cdot)=\frac{1}{|\mathcal{U}|}\sum_{x\in\mathcal{U}}k(x,\cdot)$,
shall approximate $\mu_{\mathcal{X}}$ when using enough uniformly subsampled
points. However, as shown by (Chen et al., 2010), one may get a more
representative subsample through _Kernel Herding_ as described below.

Kernel Herding  Kernel herding is a greedy algorithm to approximate a mean
embedding $\mu_{p}$ with a subset of a set of samples $\mathcal{X}\sim p$,
using a subset of points
$\hat{\mathcal{X}}=\\{\hat{x}_{i}\\}_{i=1}^{m}\subset\mathcal{X}$. Here, we
consider the empirical distribution
$p=\frac{1}{n}\sum_{i=1}^{n}\delta(x_{i})$, where $\delta$ is the Dirac-delta
function. Hence, we look to approximate $\mu_{\mathcal{X}}$ with $\mathcal{X}$
being our original sample. Kernel herding is especially applicable for
synthesizing a large number of cells, due to the following properties. First,
the rate of convergence for the error
$\|\mu_{\mathcal{X}}-\frac{1}{m}\sum_{j=1}^{m}k(\hat{x}_{i},\cdot)\|^{2}$ is
$\mathcal{O}(\frac{1}{m})$, which is much faster than the
$\mathcal{O}(\frac{1}{\sqrt{m}})$ rate for a uniformly random subsample. That
is, Kernel Herding can provide a synthesized subset of $\sqrt{m}$ points that
approximates the target distribution as well as $m$ uniformly subsampled
points. This property implies that the Kernel Herding subset shall maintain
the distributional properties found in the original sample-set. Second, it can
be shown that the Kernel Herding subset $\hat{\mathcal{X}}$ also approximates
the expectation of any function $h$333In the corresponding reproducing kernel
Hilbert space (RKHS)., $\mathbb{E}_{x\sim p}[h(x)]$, with a mean over
$\hat{\mathcal{X}}$, $\frac{1}{m}\sum_{j=1}^{m}h({\hat{x}}_{j})$. This
property enables one to obtain similar downstream models when using the
summarized subset as the expectation of functions evaluated.

The Kernel Herding subset $\hat{\mathcal{X}}$ is computed as follows:

Algorithm 1 Compute Subsampled Set using Kernel Herding

0: A set of cells $\mathcal{X}$, from which the number of cells subsampled is
$m$, Radial-Basis Function kernel $k$.

1: Initialize $j\leftarrow 1$, $\mathcal{\hat{X}}\leftarrow\emptyset$

2: while $j\leq m$ do

3: $\hat{\mathbf{x}}\leftarrow\underset{x\in\mathcal{X}}{\mathrm{argmax}}\
\dfrac{1}{N}\sum_{x^{\prime}\in\mathcal{X}}k(x,x^{\prime})-\dfrac{1}{j}\sum_{x^{\prime}\in\hat{\mathcal{X}}}k(x,x^{\prime})$

4: $\mathcal{\hat{X}}\leftarrow\mathcal{\hat{X}}\cup\\{\hat{\mathbf{x}}\\}$

5: $j\leftarrow j+1$

6: end while

7: return $\hat{\mathcal{X}}$

Random Fourier Features  Performing Kernel Herding with a kernel function $k$,
however, may not scale since an iteration requires an $O(n^{2})$ computation
(equivalent to computing the Gram matrix of pairwise kernel evaluations on
$\mathcal{X}$). To rectify this, we propose to use random Fourier frequency
features (Rahimi et al., 2007). For a shift-invariant kernel (such as the RBF
kernel), random Fourier features provide a feature map
$\varphi(x)\in\mathbb{R}^{D}$ such that the dot product in feature space
approximates the kernel evaluation, $\varphi(x)^{T}\varphi(x^{\prime})\approx
k(x,x^{\prime})$ (e.g. see (Rahimi et al., 2007; Oliva et al., 2014; Rahimi et
al., 2007) for further details). Using the dot product of $\varphi(x)$, our
mean embedding becomes
$\mu_{\mathcal{X}}=\frac{1}{n}\sum_{i=1}\varphi(x_{i})$, where
$\mu_{\mathcal{X}}(x^{\prime})=\frac{1}{n}\sum_{i=1}\varphi(x_{i})^{T}\varphi(x^{\prime})$.
In practice, the random features are constructed by drawing frequencies,
$W\in\mathbb{R}^{d\times\frac{D}{2}}$, at random (iid column-wise) once, and
holding them fixed to construct the features

(3)
$\varphi_{W}(x)=[\sin(\mathbf{W}^{T}x),\cos(\mathbf{W}^{T}x)]\in\mathbb{R}^{D},$

where $[\cdot,\cdot]$ denotes concatenation. For instance, to approximate the
RBF kernel, $k(x,x^{\prime})=\exp(-\frac{1}{2\gamma}||x-x^{\prime}||^{2})$, we
would draw the $d$-dimensional frequencies
$W_{i}\overset{iid}{\sim}\mathcal{N}(0,\gamma^{-1}I)$ (Rahimi et al., 2007).
It is simple to show that when using the approximation
$\varphi_{W}(x)^{T}\varphi_{W}(x^{\prime})\approx k(x,x^{\prime})$, one may
compute Algorithm 1 with $O(n)$ iterates by avoiding pairwise kernel
computations (Chen et al., 2010); we detail this below.

Algorithm 2 Compute Subsampled Set using RBF Kernel Herding with Random
Features

0: A set of cells $\mathcal{X}$ with dimensionality $d$, from which the number
of cells subsampled is $m$, dimensionality of the random feature space $D$ and
kernel hyperparameter $\gamma$.

1: # Draw random Fourier frequencies

2: Compute $\mathbf{W}\in\mathbb{R}^{d\times\frac{D}{2}}$ by sampling its
elements independently $\mathbf{W}_{i,j}\sim\mathcal{N}(0,\frac{1}{\gamma})$

3: # Subsampling using Kernel Herding

4: Initialize $j\leftarrow 1$, $\mathcal{\hat{X}}\leftarrow\emptyset$,
$\theta_{0}\leftarrow\frac{1}{n}\sum_{i=1}^{n}\varphi_{\scriptscriptstyle\mathbf{W}}(x_{i})$

5: while $j\leq m$ do

6: $\hat{\mathbf{x}}\leftarrow\underset{x\in\mathcal{X}}{\mathrm{argmax}}\
\theta_{j-1}^{T}\varphi_{\scriptscriptstyle\mathbf{W}}(x_{i})$

7: $\mathcal{\hat{X}}\leftarrow\mathcal{\hat{X}}\cup\\{\hat{\mathbf{x}}\\}$

8:
$\theta_{j}\leftarrow\theta_{j-1}+\theta_{0}-\varphi_{\scriptscriptstyle\mathbf{W}}(\hat{\mathbf{x}})$

9: $j\leftarrow j+1$

10: end while

11: return $\hat{\mathcal{X}}$

###  2.1. Single-Cell Datasets Used in Experiments

In all experiments, we used publicly available, multi-sample flow, mass
cytometry (CyTOF), and single-cell RNA sequencing datasets. For the flow and
mass cytometry datasets, each sample-set consists of a collection of protein
markers (up to $\sim$45) measured across individual cells. For the single-cell
RNA sequencing dataset, each sample-set consists of a collection of gene
expression measurements ($\sim$20k) measured across individual cells. Here, we
briefly introduce the multi-sample publicly available datasets used in our
experiments. All preprocessed data are available in the Zenodo repository:
https://zenodo.org/record/6546964.

Figure 2. t-SNE visualizations of the sketches produced under each method. The
distribution of points sketched with Kernel Herding (KH) (dark purple points)
follows that of the original data (gray, background points representing a
large number of randomly sampled cells). Sketches with Kernel Herding
adequately represent rare populations and retain their relative frequencies,
while Geometric Sketching and Hopper struggle with these aspects.

  * •

Preeclampsia The preeclampsia CyTOF dataset (Han et al., 2019) includes
samples collected longitudinally from 12 healthy women and 11 women with
preeclampsia throughout their pregnancies. All patient samples were downloaded
from Flow Repository under Repository ID FR-FCM-ZYRQ
(http://flowrepository.org/id/FR-FCM-ZYRQ). Classification tasks distinguished
between healthy from preeclamptic women.

  * •

HVTN The HIV Vaccine Trials Network (HVTN) is a Flow Cytometry dataset and
consists of 96 total sample-sets of T-cells that were each subjected to
stimulation with either Gag or Env (Aghaeepour et al., 2013) (downloaded from
Flow Repository, ID FR-FCM-ZZZV http://flowrepository.org/id/FR-FCM-ZZZV).
Classification tasks distinguished Gag from Env stimulated samples

  * •

NK-Cell The NK-Cell CyTOF dataset profiled NK-Cells across 21 individuals who
were either positive or negative for Cytomegalovirus (CMV) (Arvaniti and
Claassen, 2017) (downloaded from https://github.com/eiriniar/CellCnn).
Classification tasks distinguished CMV positive (CMV+) from CMV negative
(CMV-) samples.

  * •

MS The multiple sclerosis (MS) single-cell RNA sequencing dataset (Schafflick
et al., 2020) consists of peripheral blood samples collected from 4 MS
patients and 4 healthy controls. Patient samples were accessed from the Gene
Expression Omnibus using the accession code GSE138266. We performed standard
single-cell RNA sequencing data preprocessing, including filtering cells
according to read depth and distribution of molecular counts, removing cells
with greater than 20 percent mitochondrial transcripts, and retaining genes
that were expressed amongst a minimum of 5 cells. Following quality control
filtering, we normalized the data to account for differences in sequencing
depth by estimating size factors using Scran pooling normalization v1.20.1
(Lun et al., 2016) and scaling them across batches using Batchelor v1.8.0. We
then performed batch effect correction using ComBat (Johnson et al., 2007).
Lastly, we restricted the feature space by selecting for highly variable genes
on log+1 transformed data using a normalized dispersion measure in Scanpy
v1.8.1 (flavor = Seurat, minimum mean = 0.012, minimum dispersion = 0.25,
maximum mean = 5). In our subsequent experiments, we performed principal
component analysis on the preprocessed dataset and used the top 50 components
for downstream tasks.

##  3\. Results

We compared the performance of Kernel Herding to sketches obtained using
Geometric Sketching, Hopper, and IID subampling for tasks related to the
overall preservation and usefulness of the resulting immunological landscape.
That is, we sought to evaluate whether or not the sketches adequately
represented all major immune cell-types and their relative frequencies and
could be used to produce meaningful immunological features for downstream
tasks. Code for reproducing the results of all subsequent experiments is
publicly available at https://github.com/vishalathreya/Set-Summarization.

###  3.1. Description of Related Algorithms

Here, we briefly define the sketching approaches that were compared to in our
experiments.

  * •

Geometric Sketching Geometric Sketching introduced in Ref. (Hie et al., 2019)
infers a _plaid covering_ of cells in the high-dimensional space. Cells are
sketched through volume-dependent sampling by selecting the same number of
cells from sections of the high-dimensional plaid covering.

  * •

Hopper Hopper introduced in Ref. (DeMeo and Berger, 2020) forms a sketch by
using fastest first traversal, which is a greedy approximation to the
$k$-center problem. Intuitively, this sketching approach sequentially adds
cells to the sketch that are sufficiently different from those that were
already included in the sketch.

  * •

Independent and Identically Distributed Subsampling (IID) IID sketches were
generated by simply selecting a random subsample of cells from each sample-
set. Here, each cell had the same probability of being selected in the sketch.

###  3.2. t-SNE Visualizations of Sketched Regions of the Cellular Landscape

We begin with a qualitative assessment of sketching approaches on cytometry
data (Fig. 2). In order to get a visual understanding of the regions of the
cellular landscape included in the sketches produced by each method, we
plotted a $2d$ t-SNE projection of respective sketches (colored points) along
with a set of overall cells from a large sub-sample of the original set (gray-
colored points). I.e. Fig. 2, plots cells from three samples, one for each
respective dataset (similar results may be obtained from other samples).
Furthermore, we projected cluster centroids (15, from k-means) as large yellow
crosses. Taken together, the respective plots give an overview of the cells
found in each original sample-set (gray), and which cells were then included
in sketches (colored). Note that an IID sketch would stem from a uniform sub-
sample of the gray points, which was omitted for visual clarity. One can
observe that the Geometric Sketching (Geo, pink, Fig. 2) sketches concentrate
over a few sub-regions of the cellular-space, leaving large regions of cells
underrepresented in the sketch. This is also true of Hopper (Hopper, green,
Fig. 2), though to slightly lesser extent. Finally, we can observe that our
proposed Kernel Herding (KH, purple, Fig. 2) sketches yield a more
representative coverage of the cell-space. Following our philosophical goal of
obtaining sketches that may act as general _stand-ins_ for original sample-
sets, it is intuitive that the discrepancy in representation of sketches to
the original shall result in discrepancies of outputs of downstream analysis.
Moreover, following the insights gained from Kernel Herding (Sec. 2), it is
intuitive that IID sketches shall be less representative than those from
Kernel herding. In subsequent experiments below, we show that these intuitions
hold empirically.

Figure 3. t-SNE visualizations of cells sketched from the NK-Cell dataset
colored by their the third nearest neighbor distances in the original protein-
marker feature space.

We provide additional context to the aforementioned t-SNE projections (Fig. 2)
with a volumetric analysis. In particular, as t-SNE projections may not be
volumetrically preserving (areas in the $2d$ space need not scale to areas in
the original space), it may be the case that sketched points seem overly
warped and are not representative of their coverage in the original high-
dimensional protein marker feature space. To obtain a clearer view of the
representation of sketched cells in the original space, we visualize the third
nearest neighbor distance in the original space when scattering the cells in
the $2d$ t-SNE space (Fig. 3 plots this for a sample-set in the NK-Cell
dataset). That is, dark-colored points stem from _sparsely_ populated regions
in the original space, since the nearest neighbor of such points in the
original space was _far_. Similarly, light-colored points stem from _densely_
populated regions in the original space. As suspected above, Geo-Sketch and
Hopper sketches are concentrated in _sparse_ regions of the space. While this
acceptable for certain applications, the resulting sketches largely ignore
denser regions of the cell-space, which prevents sketches from acting as a
_stand-in_ for downstream analysis. In contrast, we observe that Kernel
Herding obtains good coverage of the original space, while still including
cells from sparser regions.

###  3.3. Random Function Fidelity

As previously discussed, our aim is to produce sketches that may act as a
stand-in for general analysis of samples. To test how well sketches estimate
the output of a _wide-range_ of analyses, we begin by evaluating randomly
generated functions on sketches and comparing their outputs to that of the
original sample (Fig. 4).

Figure 4. Random Function Evaluation We evaluated a random function using the
full and sketched versions for the HVTN, Preeclampsia and NK-Cell datasets,
for sketch sizes between 200 and 2500 cells. Sketches with Kernel Herding
generally produce random function evaluations (RFEs) that are the most similar
to that obtained when using all cells.

We generate random cell-wise functions, $f:\mathbb{R}^{d}\mapsto\mathbb{R}$,
that are evaluated on sets, $\mathcal{X}$, as :
$\mathbf{f}(\mathcal{X})=\frac{1}{|\mathcal{X}|}\sum_{x\in\mathcal{X}}f(x)$.
We compare the function evaluation on the original sample, $\mathcal{X}$, to a
sketched sub-sample, $\hat{\mathcal{X}}$, using the $\ell_{1}$ distance:
$|\mathbf{f}(\mathcal{X})-\mathbf{f}(\hat{\mathcal{X}})|$. We parameterize
functions through random features (3):
$f_{W,\beta}(x)=\varphi_{w}(x)^{T}\beta$, where $\varphi_{w}(x)$ are the
random features w.r.t. a drawn set of frequencies, and $\beta$ are
coefficients. Note that $f_{W,\beta}(x)$ is a _highly non-linear_ function in
$x$. Thus, testing the discrepancies
$|\mathbf{f}_{W,\beta}(\mathcal{X})-\mathbf{f}_{W,\beta}(\hat{\mathcal{X}})|$
between multiple $W,\beta$ shall give a robust measure of the representative
power of sketches. For each dataset, we draw 5 random functions (by drawing
$\beta,W$ at random) and report the average $\ell_{1}$ discrepancy for
sketches produced with each respective method for various sketch cardinalities
(see Fig. 4, note the log-scale). Perhaps not surprisingly in light of our
qualitative analysis, we see that Geo-Sketch and Hopper yield sketches that
are poor stand-ins for this task. Interestingly, we also see a large advantage
in Kernel Herding sketches to IID sketches. This highlights that although IID
sketches retain distributional properties, Kernel Herding sketches provide a
more efficient and accurate synthesis.

###  3.4. Singular Value Fidelity

Above we studied sketches’ ability to act a stand-in for general non-linear
evaluations on original sample-sets. Here, we now consider a specific analysis
based on singular values. In particular, we study how well the singular values
of the original $n\times d$ (cell $\times$ protein marker matrices) sample
compares to the singular values of corresponding $m\times d$ sketches with the
$\ell_{1}$ metric:
$||\frac{1}{\sqrt{n}}\vec{\sigma}(\mathcal{X})-\frac{1}{\sqrt{m}}\vec{\sigma}(\hat{\mathcal{X}})||_{1}$,
where $\vec{\sigma}$ is the vector of corresponding singular values to sample
or sketch. Note that this $\ell_{1}$ metric directly relates to differences of
eigenvalues of corresponding covariance matrices, and hence is indicative of
how well sketches may act as a stand-in for linear subspace analyses such as
PCA.

Figure 5. Singular Value Distribution Evaluation Differences in singular value
distributions of cell $\times$ protein marker matrices as quantified with an
L1-norm were used to summarize the quality of sketches over a range of sketch
sizes, for preserving the overall cellular landscape in the HVTN, Preeclampsia
and NK-Cell datasets (top, middle, bottom, respectively).

As before, we see that Geo-sketch and Hopper sketches are also unable to act
as stand-ins for a singular-value analysis; a problem that does not improve as
sketch sizes increase. Given that many single cell analyses rely on capturing
the lower dimensional structure in samples (e.g. (Velten et al., 2022; Moon et
al., 2019)), this underlying bias in previous sketching approaches is limiting
to their ability to act as reliable stand-ins. Moreover, we similarly observe
that our proposed Kernel Herding sketches act as more faithful stand-ins to
the naive IID sketches.

###  3.5. Cluster Frequency Fidelity

Figure 6. Cell-Population Frequency Evaluation $k$-means was used to partition
sketched sample-sets (between 200 to 2500 cells) into cell-populations (10,
30, 50 clusters). In comparison to IID, Geometric Sketching, and Hopper,
Kernel Herding sketches most closely preserve the frequencies of cell-
populations observed using all cells in the sample-set.

Next, we study sketches’ abilities to retain the overall cellular frequencies
that were found in the original sample. Cell-type frequencies can be crucial
for clinical outcome prediction and biological interpretation (Bruggner et
al., 2014; Stanley et al., 2020); thus, a sketch’s ability to retain cell-type
frequencies is pivotal for its ability to act as a stand-in to the original
set in many impactful tasks. Here we automatically detect cell-populations
through an unsupervised, kmeans cluster analysis. All cells in the original
sample-set are used to compute the cluster centroids, which then provide the
cluster association for cells from sketches of that set. Once the centroids
are computed for the original sample, each sketch is summarized according to
the frequencies of the clusters in that sketch. That is, for each sketch, we
compute the portion of its cells that were assigned to each cluster,
$\rho(\hat{\mathcal{X}})\in[0,1]^{K}$,
$\rho_{k}(\hat{\mathcal{X}})=\frac{1}{|\hat{\mathcal{X}}|}\sum_{x\in\hat{\mathcal{X}}}\mathbb{I}\\{k=\mathrm{argmin}_{c}||x-\nu_{c}||\\}$,
where $\nu_{1},\ldots,\nu_{K}$ are the cluster centroids. Similarly to above,
we may compare the outcomes of analyzing the original sample-set,
$\mathcal{X}$, to that of a corresponding sketch, $\hat{\mathcal{X}}$, with an
$\ell_{1}$ metric: $||\rho(\mathcal{X})-\rho(\hat{\mathcal{X}})||_{1}$ and
average this distance over all sets in the dataset.

We plot the $\ell_{1}$ discrepancies in frequencies for various sketching
cardinalities and number of clusters in Fig. 6. Following the same major
pattern to previous experiments, we see that alternative sketching approaches
(Geo-Sketch and Hopper) were unable to properly act as a stand-in to computing
the cell-population frequencies in the original set. This trend is also not
assuaged by increasing those sketch’s sub-sampling size. Kernel Herding avoids
these issues, whilst also providing a better representative sketch than IID
sub-sampling. We see similar results across various number of clusters, which
studies the sketches’ fidelity with different granularities of cell-
populations.

Figure 7. _Clinical Outcome Classification Accuracy._ Sketches of 500 samples
per sample set were obtained each with Geometric Sketching, Hopper, IID
subsampling, and Kernel Herding and cells were partitioned amongst 30 clusters
forming cell-populations. The cell-population frequencies were used as
features to predict the clinical outcomes of the associated individuals for
each sample-set in the HVTN, Preeclampsia, and NK-Cell datasets (left, middle,
and right, respectively). We performed leave-one-out cross validation
experiments for this classification task. Kernel Herding and IID produce
sketches and associated features that are more predictive than those obtained
through Geometric Sketching. Kernel Herding also significantly reduces the
variance in classification accuracy in the Preeclampsia dataset.

###  3.6. Classification Effectiveness

Finally, we explicitly test to see if the cell-population discrepancies found
in Fig. 6 are relevant to producing immunological features that are useful for
classifying samples according to the clinical outcomes of their associated
individuals. That is, here we explicitly test that clinically relevant cell
populations (such as _rare-cells_) are maintained in sketches. We hypothesized
that the sketches with more accurately represented frequencies (e.g. as in
Sec. 3.5) may be more predictive of a clinical outcome of interest.

As in Sec. 3.5, for each of the three cytometry datasets, cells were clustered
into one of thirty clusters. Here the clusters are found from an aggregate
collection of the multiple sketched sample-sets in each dataset. That is, the
clusters are determined via a concatenation of all the sketched cells from all
the sample sets that are in respective training sets. The frequency, or
proportion of cells assigned to each population, $\rho(\hat{\mathcal{X}})$,
was used as input for a downstream classification task to predict the clinical
outcome for each sketched set. Note that here we trained a classifier based on
multi-sample dataset
$\left\\{(\rho(\hat{\mathcal{X}_{i}}),y_{i})\right\\}_{i=1}^{N}$, where
$\hat{\mathcal{X}_{i}}$ is the sketch for the $i$-th sample-set, and $y_{i}$
is the corresponding label. Classification experiments were performed in the
HVTN, Preeclampsia and NK-Cell datasets (left, middle, and right of Figure 7,
respectively) by splitting individuals according to leave-one-out cross
validation and training an RBF support vector machine (SVM). Surprisingly, we
found that notwithstanding the explicit focus on maintaining rare-cell types,
Geo-sketch and Hopper were unable to produce sketches that lead to better
accuracies than IID sub-sampling in this setting. In contrast, we found that
our proposed Kernel Herding sketches lead to higher mean accuracies than IID
sketching. This held true for other cluster sizes as well (K=15,K=50). Our
results show that in general, sketching with Kernel Herding produces sketches
that are more clinically predictive than the baseline methods. This is
especially apparent in the preeclampsia dataset, where IID sketching produces
highly variable classification accuracy.

###  3.7. Single-cell RNA Sequencing Fidelity

For a cross-modality comparison, we tested the performance of Kernel Herding
to the sketches obtained using Geometric Sketching, Hopper, and IID
subsampling on single-cell RNA sequencing (scRNA-seq) data. This modality
poses an additional challenge, as scRNA-seq data typically contains 20-30
thousand gene measurements across all cells and has a high degree of sparsity
and technical noise due to capture inefficiency, amplification noise, and
stochasticity (Kharchenko et al., 2014). Leveraging a single cell RNA
sequencing dataset of patients with multiple sclerosis (MS) and healthy
controls, we tested whether sketched cells resulted in similar random function
evaluations, similar singular value distributions, and gave similar cell
population frequencies. As shown in Figure 8, we found that Kernel Herding
produces sketches that are most closely aligned with the results obtained
using the full original sample. More specifically, across a range of sketch
sizes for each sample-set, Kernel Herding achieves the least $\ell_{1}$
distance to the random function estimates in the original dataset (Fig 8A),
has the most similar singular value distributions to the original data (Figure
8B), and best preserves cell population frequencies with the least $\ell_{1}$
norm between true and sketched cell populations (Figure 8C).

Table 1. Runtimes (in seconds) of sketching methods on HVTN, Preeclampsia and NK-Cell datasets (number of cells ($n$ and number of features ($d$) shown underneath in parentheses) as measured on an Intel(R) Xeon(R) Gold 6226R CPU. Note that our Kernel Herding implementation was not optimized and was coded for readability.  Methods | HVTN | Preeclampsia | NK Cell
---|---|---|---
| (n=200k, d=11) | (n=215k, d=33) | (n=13k, d=43)
IID |  0.012 $\pm$ 0.001 |  0.016 $\pm$ 0.003 |  0.004 $\pm$ 0.001
Hopper |  3.13 $\pm$ 0.12 |  4.43 $\pm$ 0.15 |  1.12 $\pm$ 0.09
Geo Sketch |  23.76 $\pm$ 0.06 |  77.92 $\pm$ 0.15 |  12.29 $\pm$ 0.06
Kernel Herding |  26.56 $\pm$ 4.38 |  22.59 $\pm$ 1.38 |  3.64 $\pm$ 0.35
Figure 8. Single-cell RNA Sequencing Evaluation We evaluated Hopper, Geometric
Sketching, IID, and Kernel Herding on producing sketches that preserve the
overall cellular landscape of single-cell RNA sequencing data. Method
performance was quantified using an L1 norm between random function
evaluations of original and sketched samples (A), an L1 norm between singular
value distributions (B), and an L1 norm between true and sketched cell-
population frequencies (C) across a range of sketch sizes (200 - 2500).
Sketches with Kernel Herding produce more similar random function evaluations,
similar singular value distributions, and best preserve the cell population
frequencies that are observed when using all cells.

##  4\. Discussion and Conclusion

Here, we presented a distribution-based, Kernel Herding approach to select a
limited number of representative cells from each sample-set. Of particular
note, we recast the focus of sketching sample-sets to providing a smaller
sketch that can act as a _stand-in_ to the original set. That is, we
explicitly and quantitatively assess sketch’s ability to faithfully maintain
downstream outcomes when used in place of the original set. In contrast to
existing sketching approaches, such as, Geometric Sketching (Hie et al., 2019)
and Hopper (DeMeo and Berger, 2020), we found that Kernel Herding strikes a
powerful middle-ground between preserving rare cell-populations, while also
representing all major cell-types and retaining their relative frequencies.
Moreover, maintaining the overall distribution of the original sample is
necessary as adequate preservation of cell-population frequencies is important
for linking cellular heterogeneity to clinical phenotype or external variables
of interest (Stanley et al., 2020; Bruggner et al., 2014), and for developing
novel diagnostics or prognostics (Fallahzadeh et al., 2021; Han et al., 2019;
Ganio et al., 2020). Given the modern widespread use of cytometry and scRNA-
sequencing technologies in clinical applications with large patient cohorts,
the presented Kernel Herding based sketching approach makes such data more
manageable for downstream analysis and interpretation. We showed that Kernel
Herding was effective across multiple varied single cell modalities including
flow and mass cytometry, as well as scRNA-seq data.

We demonstrated the usefulness of Kernel Herding at providing sketches that
can serve as a stand-in for the original sample-set by measuring the fidelity
of non-linear function evaluations, singular value distributions, and cell-
population frequencies between the original and sketched sample-sets.
Additionally, we showed the usefulness of sketching through Kernel Herding for
downstream tasks, such as clinical outcome prediction; here, Kernel Herding
had the most stable performance across datasets, number of clusters, and
number of sketched cells. This consistency makes it a reliable method that can
be employed to obtain a representative subset, small or large, of the original
distribution of samples. The runtimes of different methods reported in Table 1
add to the practical usefulness of Kernel Herding.

Future work could consider the implications of sketching with Kernel Herding
in a broader range of tasks required to understand single-cell datasets, such
as, differential abundance analysis (Lun et al., 2017; Dann et al., 2021;
Weber et al., 2019), or for rapid identification of phenotype-associated cells
(Burkhardt et al., 2021). Finally, another area of future work may focus on
generating variable sketch sizes across different subsections of the cellular
landscape, depending on prior knowledge or scientific question.

###### Acknowledgements.

This research was partly funded by NSF grant IIS2133595 and by NIH
1R01AA02687901A1.

## References

  * (1)
  * Aghaeepour et al. (2013) Nima Aghaeepour, Greg Finak, Holger Hoos, Tim R Mosmann, Ryan Brinkman, Raphael Gottardo, and Richard H Scheuermann. 2013.  Critical assessment of automated flow cytometry data analysis techniques.  _Nature methods_ 10, 3 (2013), 228–238.
  * Aghaeepour et al. (2017) Nima Aghaeepour, Edward A Ganio, David Mcilwain, Amy S Tsai, Martha Tingle, Sofie Van Gassen, Dyani K Gaudilliere, Quentin Baca, Leslie McNeil, Robin Okada, Mohammad S Ghaemi, David Furman, Ronald J Wong, Virginia D Winn, Maurice L Druzin, Yaser Y El-Sayed, Cecele Quaintance, Ronald Gibbs, Gary L Darmstadt, Gary M Shaw, David K Stevenson, Robert Tibshirani, Garry P Nolan, David B Lewis, Martin S Angst, and Brice Gaudilliere. 2017\.  An immune clock of human pregnancy.  _Science immunology_ 2, 15 (2017).
  * Arunachalam et al. (2020) Prabhu S Arunachalam, Florian Wimmers, Chris Ka Pun Mok, Ranawaka APM Perera, Madeleine Scott, Thomas Hagan, Natalia Sigal, Yupeng Feng, Laurel Bristow, Owen Tak-Yin Tsang, et al. 2020\.  Systems biological assessment of immunity to mild versus severe COVID-19 infection in humans.  _Science_ 369, 6508 (2020), 1210–1220.
  * Arvaniti and Claassen (2017) Eirini Arvaniti and Manfred Claassen. 2017.  Sensitive detection of rare disease-associated cell subsets via representation learning.  _Nature communications_ 8, 1 (2017), 1–10.
  * Bendall et al. (2012) Sean C Bendall, Garry P Nolan, Mario Roederer, and Pratip K Chattopadhyay. 2012.  A deep profiler’s guide to cytometry.  _Trends in immunology_ 33, 7 (2012), 323–332.
  * Berlinet and Thomas-Agnan (2011) Alain Berlinet and Christine Thomas-Agnan. 2011.  _Reproducing kernel Hilbert spaces in probability and statistics_.  Springer Science & Business Media.
  * Bruggner et al. (2014) Robert V Bruggner, Bernd Bodenmiller, David L Dill, Robert J Tibshirani, and Garry P Nolan. 2014\.  Automated identification of stratifying signatures in cellular subpopulations.  _Proceedings of the National Academy of Sciences_ 111, 26 (2014), E2770–E2777.
  * Burkhardt et al. (2021) Daniel B Burkhardt, Jay S Stanley, Alexander Tong, Ana Luisa Perdigoto, Scott A Gigante, Kevan C Herold, Guy Wolf, Antonio J Giraldez, David van Dijk, and Smita Krishnaswamy. 2021.  Quantifying the effect of experimental perturbations at single-cell resolution.  _Nature Biotechnology_ (2021), 1–11.
  * Cao et al. (2019) Junyue Cao, Malte Spielmann, Xiaojie Qiu, Xingfan Huang, Daniel M Ibrahim, Andrew J Hill, Fan Zhang, Stefan Mundlos, Lena Christiansen, Frank J Steemers, Cole Trapnell, and Jay Shendure. 2019\.  The single-cell transcriptional landscape of mammalian organogenesis.  _Nature_ 566, 7745 (feb 2019), 496–502.
  * Chen et al. (2010) Yutian Chen, Max Welling, and Alex Smola. 2010.  Super-samples from kernel herding. In _Proceedings of the Twenty-Sixth Conference on Uncertainty in Artificial Intelligence_. 109–116.
  * Cortes and Vapnik (1995) Corinna Cortes and Vladimir Vapnik. 1995.  Support-vector networks.  _Machine learning_ 20, 3 (1995), 273–297.
  * Dann et al. (2021) Emma Dann, Neil C Henderson, Sarah A Teichmann, Michael D Morgan, and John C Marioni. 2021\.  Differential abundance testing on single-cell data using k-nearest neighbor graphs.  _Nature Biotechnology_ (2021), 1–9.
  * DeMeo and Berger (2020) Benjamin DeMeo and Bonnie Berger. 2020.  Hopper: a mathematically optimal algorithm for sketching biological data.  _Bioinformatics_ 36, Supplement_1 (2020), i236–i241.
  * Fallahzadeh et al. (2021) Ramin Fallahzadeh, Franck Verdonk, Ed Ganio, Anthony Culos, Natalie Stanley, Ivana Marić, Alan L Chang, Martin Becker, Thanaphong Phongpreecha, Maria Xenochristou, et al. 2021\.  Objective Activity Parameters Track Patient-Specific Physical Recovery Trajectories After Surgery and Link With Individual Preoperative Immune States.  _Annals of Surgery_ (2021).
  * Ganio et al. (2020) Edward A Ganio, Natalie Stanley, Viktoria Lindberg-Larsen, Jakob Einhaus, Amy S Tsai, Franck Verdonk, Anthony Culos, Sajjad Ghaemi, Kristen K Rumer, Ina A Stelzer, Dyani Gaudilliere, Eileen Tsai, Ramin Fallahzadeh, Benjamin Choisy, Henrik Kehlet, Nima Aghaeepour, Martin S Angst, and Brice Gaudilliere. 2020\.  Preferential inhibition of adaptive immune system dynamics by glucocorticoids in patients after acute surgical trauma.  _Nature communications_ 11, 1 (2020), 1–12.
  * Gretton et al. (2008) Arthur Gretton, Karsten M Borgwardt, Malte J Rasch, Bernhard Schölkopf, and Alexander Smola. 2008\.  A Kernel Method for the Two-Sample Problem.  _Journal of Machine Learning Research_ 1 (2008), 1–10.
  * Gretton et al. (2012) Arthur Gretton, Karsten M Borgwardt, Malte J Rasch, Bernhard Schölkopf, and Alexander Smola. 2012\.  A kernel two-sample test.  _The Journal of Machine Learning Research_ 13, 1 (2012), 723–773.
  * Han et al. (2019) Xiaoyuan Han, Mohammad S Ghaemi, Kazuo Ando, Laura S Peterson, Edward A Ganio, Amy S Tsai, Dyani K Gaudilliere, Ina A Stelzer, Jakob Einhaus, Basile Bertrand, Natalie Stanley, Anthony Culoar, Athena Tanada, Julien Hedou, Eileen S Tsai, Ramin Fallahzadeh, Ronald J Wong, Amy E Judy, Virginia D Winn, Maurice L Druzin, Yair J Blumendeld, Mark A Hlatky, Cecele C Quaintance, Ronald S Gibbs, Brendan Carvalho, Gary M Shaw, David K Stevenson, Martin S Angst, Aghaeepour, and Brice Gaudilliere. 2019\.  Differential dynamics of the maternal immune system in healthy pregnancy and preeclampsia.  _Frontiers in immunology_ 10 (2019), 1305.
  * Hie et al. (2019) Brian Hie, Hyunghoon Cho, Benjamin DeMeo, Bryan Bryson, and Bonnie Berger. 2019.  Geometric sketching compactly summarizes the single-cell transcriptomic landscape.  _Cell systems_ 8, 6 (2019), 483–493.
  * Johnson et al. (2007) W Evan Johnson, Cheng Li, and Ariel Rabinovic. 2007.  Adjusting batch effects in microarray expression data using empirical Bayes methods.  _Biostatistics_ 8, 1 (jan 2007), 118–127.
  * Kharchenko et al. (2014) Peter V Kharchenko, Lev Silberstein, and David T Scadden. 2014\.  Bayesian approach to single-cell differential expression analysis.  _Nat. Methods_ 11, 7 (jul 2014), 740–742.
  * Levine et al. (2015) Jacob H Levine, Erin F Simonds, Sean C Bendall, Kara L Davis, D Amir El-ad, Michelle D Tadmor, Oren Litvin, Harris G Fienberg, Astraea Jager, Eli R Zunder, Rachel Finck, Amanda L Gedman, Ina Radtkae, James R Downing, Dana Pe’er, and Garry P Nolan. 2015\.  Data-driven phenotypic dissection of AML reveals progenitor-like cells that correlate with prognosis.  _Cell_ 162, 1 (2015), 184–197.
  * Liechti et al. (2021) Thomas Liechti, Lukas M Weber, Thomas M Ashhurst, Natalie Stanley, Martin Prlic, Sofie Van Gassen, and Florian Mair. 2021.  An updated guide for the perplexed: cytometry in the high-dimensional era.  _Nature Immunology_ 22, 10 (2021), 1190–1197.
  * Lotfollahi et al. (2021) Mohammad Lotfollahi, Mohsen Naghipourfar, Malte D Luecken, Matin Khajavi, Maren Büttner, Marco Wagenstetter, Žiga Avsec, Adam Gayoso, Nir Yosef, Marta Interlandi, Sergei Rybakov, Alexander V Misharin, and Fabian J Theis. 2021.  Mapping single-cell data to reference atlases by transfer learning.  _Nat. Biotechnol._ (aug 2021).
  * Luecken and Theis (2019) Malte D Luecken and Fabian J Theis. 2019.  Current best practices in single-cell RNA-seq analysis: a tutorial.  _Molecular systems biology_ 15, 6 (2019), e8746.
  * Lun et al. (2017) Aaron TL Lun, Arianne C Richard, and John C Marioni. 2017\.  Testing for differential abundance in mass cytometry data.  _Nature methods_ 14, 7 (2017), 707–709.
  * Lun et al. (2016) Aaron T L Lun, Karsten Bach, and John C Marioni. 2016\.  Pooling across cells to normalize single-cell RNA sequencing data with many zero counts.  _Genome Biol._ 17 (apr 2016), 75.
  * Mathew et al. (2020) Divij Mathew, Josephine R Giles, Amy E Baxter, Derek A Oldridge, Allison R Greenplate, Jennifer E Wu, Cécile Alanio, Leticia Kuri-Cervantes, M Betina Pampena, Kurt D’Andrea, Sasikanth Manne, Zeyu Chen, Yinghui Jane Huang, John P Reilly, Ariel R Weisman, Caroline A G Ittner, Oliva Kuthuru, Jeanette Dougherty, Kito Nzingha, Nicholas Han, Justin Kim, Ajinkya Pattekar, Eileen C Goodwin, Elizabeth M Anderson, Madison E Weirick, Sigrid Gouma, Claudia P Arevalo, Marcus J Bolton, Fang Chen, Simon F Lacey, Holly Ramage, Sara Cherry, Scott E Hensley, Sokratis A Apostolidis, Alexander C Huang, Laura A Vella, UPenn COVID Processing Unit, Michael R Betts, Nuala J Meyer, and E John Wherry. 2020.  Deep immune profiling of COVID-19 patients reveals distinct immunotypes with therapeutic implications”.  _Science_ 369, 6508 (sep 2020).
  * Mika et al. (1998) Sebastian Mika, Bernhard Schölkopf, Alexander J Smola, Klaus-Robert Müller, Matthias Scholz, and Gunnar Rätsch. 1998.  Kernel PCA and De-noising in feature spaces.. In _NIPS_ , Vol. 11. 536–542.
  * Moon et al. (2019) Kevin R Moon, David van Dijk, Zheng Wang, Scott Gigante, Daniel B Burkhardt, William S Chen, Kristina Yim, Antonia van den Elzen, Matthew J Hirn, Ronald R Coifman, Natalia B Ivanova, Guy Wolf, and Smita Krishnaswamy. 2019.  Visualizing structure and transitions in high-dimensional biological data.  _Nat. Biotechnol._ 37, 12 (dec 2019), 1482–1492.
  * Muandet et al. (2017) Krikamol Muandet, Kenji Fukumizu, Bharath Sriperumbudur, and Bernhard Schölkopf. 2017.  Kernel mean embedding of distributions: A review and beyond.  _Foundations and Trends in Machine Learning_ 10, 1-2 (2017), 1–141.
  * Oliva et al. (2014) Junier Oliva, Willie Neiswanger, Barnabás Póczos, Jeff Schneider, and Eric Xing. 2014\.  Fast distribution to real regression. In _Artificial Intelligence and Statistics_. PMLR, 706–714.
  * Rahimi et al. (2007) Ali Rahimi, Benjamin Recht, et al. 2007\.  Random Features for Large-Scale Kernel Machines.. In _NIPS_ , Vol. 3. Citeseer, 5\.
  * Regev et al. (2017) Aviv Regev, Sarah A Teichmann, Eric S Lander, Ido Amit, Christophe Benoist, Ewan Birney, Bernd Bodenmiller, Peter Campbell, Piero Carninci, Menna Clatworthy, Hans Clevers, Bart Deplancke, Ian Dunham, James Eberwine, Roland Eils, Wolfgang Enard, Andrew Farmer, Lars Fugger, Berthold Göttgens, Nir Hacohen, Muzlifah Haniffa, Martin Hemberg, Seung Kim, Paul Klenerman, Arnold Kriegstein, Ed Lein, Sten Linnarsson, Emma Lundberg, Joakim Lundeberg, Partha Majumder, John C Marioni, Miriam Merad, Musa Mhlanga, Martijn Nawijn, Mihai Netea, Garry Nolan, Dana Pe’er, Anthony Phillipakis, Chris P Ponting, Stephen Quake, Wolf Reik, Orit Rozenblatt-Rosen, Joshua Sanes, Rahul Satija, Ton N Schumacher, Alex Shalek, Ehud Shapiro, Padmanee Sharma, Jay W Shin, Oliver Stegle, Michael Stratton, Michael J T Stubbington, Fabian J Theis, Matthias Uhlen, Alexander van Oudenaarden, Allon Wagner, Fiona Watt, Jonathan Weissman, Barbara Wold, Ramnik Xavier, Nir Yosef, and Human Cell Atlas Meeting Participants. 2017.  The Human Cell Atlas.  _Elife_ 6 (dec 2017).
  * Schafflick et al. (2020) David Schafflick, Chenling A Xu, Maike Hartlehnert, Michael Cole, Andreas Schulte-Mecklenbeck, Tobias Lautwein, Jolien Wolbert, Michael Heming, Sven G Meuth, Tanja Kuhlmann, Catharina C Gross, Heinz Wiendl, Nir Yosef, and Gerd Meyer Zu Horste. 2020\.  Integrated single cell analysis of blood and cerebrospinal fluid leukocytes in multiple sclerosis.  _Nat. Commun._ 11, 1 (jan 2020), 247\.
  * Spitzer and Nolan (2016) Matthew H Spitzer and Garry P Nolan. 2016.  Mass cytometry: single cells, many features.  _Cell_ 165, 4 (2016), 780–791.
  * Stanley et al. (2020) Natalie Stanley, Ina A Stelzer, Amy S Tsai, Ramin Fallahzadeh, Edward Ganio, Martin Becker, Thanaphong Phongpreecha, Huda Nassar, Sajjad Ghaemi, Ivana Maric, Anthony Culos, Alan L Chang, Maria Xenochristou, Xiaoyuan Han, Camilo Espinosa, Kristen Rumer, Laura Peterson, Franck Verdonk, Dyani Gaudilliere, Eileen Tsai, Dorien F Feyaerts, Jakob Einhaus, Kazuo Ando, Ronald J Wong, Gerlinde Obermoser, Gary M Shaw, David K Stevenson, Martin S Angst, Brice Gaudilliere, and Nima Aghaeepour. 2020.  VoPo leverages cellular heterogeneity for predictive modeling of single-cell data.  _Nature communications_ 11, 1 (2020), 1–9.
  * Velten et al. (2022) Britta Velten, Jana M Braunger, Ricard Argelaguet, Damien Arnol, Jakob Wirbel, Danila Bredikhin, Georg Zeller, and Oliver Stegle. 2022\.  Identifying temporal and spatial patterns of variation from multimodal data using MEFISTO.  _Nat. Methods_ (jan 2022).
  * Vovk (2013) Vladimir Vovk. 2013\.  Kernel ridge regression.  In _Empirical inference_. Springer, 105–116.
  * Weber et al. (2019) Lukas M Weber, Malgorzata Nowicka, Charlotte Soneson, and Mark D Robinson. 2019.  diffcyt: Differential discovery in high-dimensional cytometry via high-resolution clustering.  _Communications biology_ 2, 1 (2019), 1–11.
  * Yi and Stanley (2021) Haidong Yi and Natalie Stanley. 2021.  CytoSet: predicting clinical outcomes via set-modeling of cytometry data. In _Proceedings of the 12th ACM Conference on Bioinformatics, Computational Biology, and Health Informatics_ (Gainesville, Florida) _(BCB ’21”, Article 47)_. Association for Computing Machinery, New York, NY, USA, 1–8.


# Exotic almost complex circle actions on $6$-manifolds

Panagiotis Konstantis and Nicholas Lindsay

###### Abstract

Jang has proven a remarkable classification of $6$-dimensional manifolds
having an almost complex circle action with $4$ fixed points. Jang classifies
the weights and associated multigraph into six cases, leaving the existence of
connected manifolds fitting into three of the cases unknown. We show that one
of the unknown cases may be constructed by a surgery construction of Kustarev,
and the underlying manifold is diffeomorphic to $S^{4}\times S^{2}$. We show
that the action is not equivariantly diffeomorphic to a linear one, thus
giving a new exotic $S^{1}$-action of on a product of spheres that preserves
an almost complex structure. We also prove a uniqueness statement for the
almost complex structures produced by Kustarev’s construction and prove some
topological applications of Jang’s classification.

##  1 Introduction

In [9, Theorem 1.1], Jang proved a remarkable classification of
$6$-dimensional manifolds having an almost complex circle action with $4$
fixed points. This generalized earlier classifications by Ahara when the
manifold satisfies $c_{1}^{3}\neq 0$, and $\frac{c_{1}c_{2}}{24}=1$ [1] and
Tolman in the case when the action is Hamiltonian [15].

In [9, Theorem 1.1], such manifolds are classified into 6 cases, depending on
their weights and associated multigraph. The existence of connected, almost
complex $6$-manifolds with circle actions fitting into case 4-6 of the
classification was left open. These three new unknown cases are characterized
by the fact that they have vanishing Todd genus. Jang points out that a
disconnected example fitting into case 4 of the classification is furnished by
$S^{6}\cup S^{6}$. This case stands out from the others because the associated
multigraph is disconnected, in fact the disjoint union two copies of the GKM
graph associated to $S^{6}$.

Firstly, we observe that a connected example may be obtained via surgery
construction of Kustarev [11].

######  Theorem 1.1.

There is a connected, almost complex $6$-manifold with an almost complex
$S^{1}$ action fitting into case 4 of Jang’s classification. It is obtained by
performing the Kustarev surgery operation on two copies of $S^{6}$, and is
diffeomorphic to $S^{4}\times S^{2}$.

The proof follows from a topological result which expresses the fiber connect
sum with a sphere along a subsphere as a surgery operation, see Section 4 for
the relevant definitions. The most technichal step is to determine the class
of the framing associated to a free orbit, which is done in subsection 4.4.

Moreover, we prove the following corollary, showing that these circle actions
are exotic.

######  Corollary 1.2.

There exists a circle action on $S^{4}\times S^{2}$, preserving an almost
complex structure and not equivariantly diffeomorphic to a linear circle
action.

Our second result, involves the more general surgery construction of Kustarev
for almost complex $2n$-manifolds with an almost complex $T^{k}$-action [12].
Kustarev proves that for $2n-k=2,4,5,6\mod 8$, the the original construction
may be generalized directly. I.e. the result of gluing two such manifolds
along the complement of open neighborhoods of free orbits has a
$T^{k}$-invariant almost complex structure.

Our next result is the following:

######  Theorem 1.3.

Suppose that $2n-k=4,5\mod 8$, then the almost complex structure produced by
Kustarev’s surgery is unique up to homotopy. In particular, for
$S^{1}$-actions the homotopy class of the almost complex structure is unique,
provided $n=3\mod 4$.

In particular, for Kustarev’s original surgery construction for
$S^{1}$-actions on $6$-manifolds, the homotopy class is uniquely determined by
the original almost complex $S^{1}$-manifolds. In the following section we
return our attention to Jang’s classification, and prove the following
topological restriction on examples fitting into another of the unknown cases.

######  Theorem 1.4.

Let $M$ be a simply connected, $\mathbb{Z}$-equivariantly formal, almost
complex $6$-manifold fitting into Case 6 of Jang’s classification. Then, $M$
is diffeomorphic to the quadric $3$-fold.

The proof follows from the ABBV localization theorem and the Wall-Jupp-Zubr
classification theorem for $6$-manifolds.

We finish the article with a brief section discussing open questions and
further directions. Because of the wealth of examples produced via Kustarev’s
surgery construction it is necessary to restrict the class of manifolds in
Jang’s classification to hope to obtain finiteness up to for example
diffeomorphism (homotopy equivalence could be equally appropriate here). We
pose some specific questions, which appear to be interesting and perhaps
approachable.

Acknowledgements. We would like to thank Silvia Sabatini for helpful
discussions.

##  2 Preliminaries

We start by recalling some standard results about almost complex circle
actions, which we will need.

######  Lemma 2.1.

Let $M$ be a closed, almost complex $6$-manifold with a almost complex
$S^{1}$-action having finitely many fixed points. Label the points
$p_{1}\ldots p_{k}$ and let the weights at $p_{j}$ be denoted
$\\{w_{j,1},\ldots,w_{j,n}\\}$. Then,

$\int_{M}c_{1}(TM)^{3}=\sum_{p_{j}}\frac{(\sum_{j}w_{i,j})^{3}}{\prod_{j}w_{i,j}}.$

###### Proof.

This follows by applying Applying ABBV localization [2, 3] theorem to
$(c_{1}^{S^{1}}(TM))^{3}$. ∎

######  Lemma 2.2.

Let $M$ be closed $6$-manifold having a almost complex $S^{1}$-action with
finitely many fixed points. Let $N_{0}$ denote the number of fixed points such
that none of the weights are negative. Then,

$N_{0}=\frac{c_{1}c_{2}(M)}{24}=\mathrm{Td}(M),$

is the Todd genus of $M$.

###### Proof.

This follows from the localization formula for the Hirzebruch
$\chi_{y}$-polynomial [8, Section 5.7], of which the Todd genus is the
constant term. ∎

The following theorem is the main result of [9].

######  Theorem 2.3 ([9, Theorem 1.1]).

Let $M$ be an almost complex $6$-manifold with an almost complex circle action
with $4$ fixed points. Then one of the following cases occurs:

  1. (a)

$\mathbb{CP}^{3}$ type. $\mathrm{Td}(M)=1$ and the weights of the circle
action are $\\{a,b,c\\}$, $\\{-a,b-a,c-a\\}$, $\\{-b,a-b,c-b\\}$,
$\\{-c,a-c,b-c\\}$ for mutually distinct positive integers $a,b,c$.

  2. (b)

$Q^{3}$ type. $\mathrm{Td}(M)=1$ and the weights of the circle action are
$\\{a,a+b,a+2b\\}$, $\\{-a,b,a+2b\\}$, $\\{-a-2b,-b,a\\}$,
$\\{-a-2b,-a-b,-a\\}$, for some positive integers $a,b$.

  3. (c)

Fano $3$-fold type . $\mathrm{Td}(M)=1$ and the weights of the circle action
are $\\{1,2,3\\}$, $\\{-1,1,a\\}$, $\\{-1,1,-a\\}$, $\\{-1,-2,-3\\}$ for some
integer $a$.

  4. (d)

$S^{6}\cup S^{6}$ type. $\mathrm{Td}(M)=0$ and the weights of the circle
action are $\\{a,b,-a-b\\}$, $\\{-a,-b,a+b\\}$, $\\{c,d,-c-d\\}$,
$\\{-c,-d,c+d\\}$, for some positive integers $a,b,c,d$.

  5. (e)

$Bl_{p}(S^{6})$ type. $\mathrm{Td}(M)=0$ and the weights of the circle action
are $\\{-3a-b,a,b\\}$, $\\{-2a-b,3a+b,3a+2b\\}$, $\\{-a,-a-b,2a+b\\}$,
$\\{-b,-3a-2b,a+b\\}$, for some positive integers $a,b$ and reversing the
circle action if neccesary.

  6. (f)

$Bl_{C}(S^{6})$ type. $\mathrm{Td}(M)=0$ and the weights of the circle action
are $\\{-a-b,2a+b,b\\}$, $\\{-2a-b,a,b\\}$, $\\{-b,-2a-b,a+b\\}$,
$\\{-a,-b,2a+b\\}$, for some positive integers $a,b$.

##  3 The Kustarev surgery

In this section we will give a presentation of Kustarev’s surgery construction
from [11, 12]. We feel that the current presentation may save some time for
readers less familiar with obstruction theory to verify the details. Moreover,
we set up notation to show in some cases the Kustarev construction yields a
unique invariant almost complex structure up to homotopy.

Suppose $M_{0}$ and $M_{1}$ are two compact, smooth $2n$-manifolds equipped
with a smooth faithful action of a connected, compact Lie group $G$ of
dimension $k$. Assume there are free $G$-orbits $O_{i}\subset M_{i}$.

Then, by the equivariant slice theorem there is a tubular neighborhood $U_{i}$
of $O_{i}$ which is canonically, equivariant diffeomorphic to $G\times
D^{2n-k}$, where $D^{2n-k}$ denotes here the open disc of radius $1$ in
$\mathbb{R}^{2n-k}$. The action is the standard action on the left factor and
the trivial action on the right fact. Thus $U_{i}\setminus O_{i}\times\\{0\\}$
is diffeomorphic to $G\times S^{2n-k-1}\times(0,1)$. We define the _fibred
connected sum_ by

$M_{0}\\#_{G}M_{1}:=(M_{0}\setminus O_{0}\times\\{0\\}\cup M_{1}\setminus
O_{1}\times\\{0\\})/\sim$

where we identify $(t,v,r)\in U_{0}\setminus G\times\\{0\\}\cong G\times
S^{2n-k-1}\times(0,1)$ with $(t,v,1-r)\in U_{1}\setminus G\times\\{0\\}$. It
is clear that there is an induced $G$-action on the fiber connected sum.

######  Theorem 3.1 ([11, 12]).

Let $M_{0}$ and $M_{1}$ be two compact manifolds of dimension $2n$, having a
smooth faithful almost complex action of a compact, connected Lie group $G$ of
dimension $k$, such that each manifold possess a free orbit. If $2n-k$ is
$2,4,5,6$ mod $8$ then $M_{0}\\#_{G}M_{1}$ admits a $G$-invariant almost
complex structure, which extends the almost complex structures on
$M_{i}\setminus U_{i}\subset M_{0}\\#_{G}M_{1}$ for $i=0,1$.

It is known that the tangent bundle of a smooth manifold $M$ induces up to
homotopy a unique map $M\to B\mathrm{SO}(2n)$ (we denote by $BH$ the
classifying space of a topological group $H$). The existence of an almost
complex structure on $M$ is equivalent to the existence of a lift $M\to BU(n)$
of $M\to B\mathrm{SO}(2n)$ through the map $p\colon BU(n)\to BSO(2n)$. Let $F$
be the homotopy fiber of $p$, thus $F$ is homotopy equivalent to
$\mathrm{SO}(2n)/\mathrm{U}(n)$. We recall a foundational theorem of
obstruction theory, namely the existence and uniqueness for the relative
lifting problem. The existence is proved on [7, Page 418], the uniqueness is
left as a (straightforward) exercise [7, Ex. 24, page 420].

######  Theorem 3.2.

Suppose $p:X\rightarrow Y$ is a map admitting a Moore-Postnikov tower of
principal fibrations. Suppose that the fibre $F$ of $p$ has abelian
fundamental group. Suppose there exists a CW pair $(W,A)$, a map
$F:W\rightarrow Y$ and a $p$-lift $\tilde{F}:A\rightarrow X$. Then if for all
$i>0$, $H^{i+1}(W,A,\pi_{i}(F))$ vanishes, then the lift $\tilde{F}$ may be
extended to a $p$-lift over $W$. Furthermore the extension is unique up to
homotopy if $H^{i}(W,A;\pi_{i}(F))$ vanishes for all $i>0$.

It is known that $p:BU(n)\rightarrow BSO(n)$ admits a Moore-Postnikov tower of
principal fibrations, and that the fiber is simply connected. It follows that,
Theorem 3.2 is applicable to this situation.

The manifold $M_{0}\\#_{G}M_{1}$ can be written as the union

$M_{0}\setminus U_{0}\cup V\cup M_{1}\setminus U_{1}$

where $V$ is diffeomorphic to $G\times S^{2n-k-1}\times[0,1]$.

Denote by $J_{i}$ the invariant almost complex structures on $M_{i}$. Then
$J_{i}$ restricted to $G\times S^{n-k-1}\times\\{i\\}$ is determined by
$J_{i}$ restricted to $\\{\mathbf{1}\\}\times S^{n-k-1}\times\\{i\\}$, where
$\mathbf{1}\in G$ denotes the neutral element. Therefore it suffices to extend
the almost complex structure on $\\{\mathbf{1}\\}\times
S^{n-k-1}\times\\{0,1\\}$ to an almost complex structure $J^{\prime}$ on
$\\{\mathbf{1}\\}\times S^{2n-k-1}\times[0,1]$. This defines an invariant
almost complex structure $J$ on $V$ by

$J(t,v,r):=g_{\ast}(J(\mathbf{1},v,r))$

where $g_{\ast}\colon TV_{(\mathbf{1},v,r)}\to TV_{(t,v,r)}$ is the
differential of the action of $g\in G$. Note that $J$ coincides with $J_{i}$
restricted to $G\times S^{2n-k-1}\times\\{i\\}$ and thus it defines an
invariant almost complex structure on $M_{0}\\#_{G}M_{1}$ as claimed in the
theorem. We define now

$W:=\\{\mathbf{1}\\}\times S^{2n-k-1}\times[0,1],\quad
A=\\{\mathbf{1}\\}\times S^{n-k-1}\times\\{0,1\\}.$

Since $(W,A)$ is a good pair (see [7, p. 124]) and $W/A$ the suspension of
$S^{2n-k-1}$ we have for $i>0$

$H^{\ast}(W,A;\pi_{i}(F))\cong
H^{\ast}(W/A;\pi_{i}(F))=H^{\ast}(S^{2n-k};\pi_{i}(F)).$

For $\pi_{q}(\mathrm{SO}(2n)/U(n))$ does not depend on $n$ provided $q<2n-1$
and in this range $\pi_{q}(\mathrm{SO}(2n)/U(n))=0$ if $q\equiv 1,3,4,5$ mod
$8$. This follows from Bott periodicity, see [4, Page 432] for the precise
statement. Since $k>0$ we have that $\pi_{2n-k-1}(\mathrm{SO}(2n)/U(n))$ does
not depend on $n$ and therefore from Theorem 3.2 we have the existence of an
invariant almost complex structure if $2n-k\equiv 2,4,5,6$ mod $8$.

Having recalled Kustarev’s construction, we show that under certain conditions
on the dimension of the manifold and the group acting the almost complex
structure produced is unique.

######  Theorem 3.3.

Suppose that additionally $2n-k=5,6\mod 8$. Then, there is a unique homotopy
class of almost complex structures extending invariant almost complex
structures on $M_{i}\setminus G$ ($i=1,2$).

###### Proof.

In [11] it was proven, that the problem of extending the almost complex
structure across the collar $G\times S^{2n-k-1}\times[0,1]$ equivariantly, is
equivalent to the (non-equivariant) relative lifting problem for maps from
$S^{2n-k-1}$ along the map $p$, relative to the boundary. The proof of this,
follows from the fact the group action is free on this region, so an invariant
almost complex structure is determined by its restriction to the slice
$\mathbf{1}\times S^{2n-k-1}\times[0,1]$.

By Theorem 3.2, the uniqueness obstruction for the existence property for this
lifting property is

$H^{i}(W/A,\pi_{i}(F)).$

Since $W/A$ is homeomorphic to $S^{2n-k}$, the existence and uniqueness holds
when

$\pi_{2n-k}(F)=\pi_{2n-k+1}(F)=0.$

By the known form of the stable homotopy groups of $SO(2n)/U(n)$ [4, Page
432], this happens exactly when $2n-k=5,6\mod 8$.

∎

We note that in the cases which are not covered by the above, the obstruction
to the uniqueness of almost complex structures lies either in $\mathbb{Z}$ or
$\mathbb{Z}_{2}$, depending on $n$ and $k$. Moreover, the uniqueness proved
above indeed applies to the original construction of Kustarev on $6$-manifolds
with circle action, and more generally.

######  Corollary 3.4.

In case of a circle actions, the invariant almost complex structure of Theorem
3.1 on $M_{0}\\#_{S^{1}}M_{1}$ is unique if $n$ equals $3$ mod $4$.

##  4 Proof of Theorem 1.1

Before focusing on the Kustarev sum of two copies of $S^{6}$, and proving
Theorem 1.1, we state a standard lemma for convenience regarding how fiber
connect sum affects the fundamental group and homology of manifolds. A
manifold with a circle action is called $\mathbb{Q}$ resp.
$\mathbb{Z}$-equivariantly formal if its equivariant cohomology (with
coeffecients in $\mathbb{Q}$ resp. $\mathbb{Z}$) is a free $H^{*}(BS^{1})$
module. It is known that if a manifold has a circle action with isolated, and
non-empty fixed point set then $M$ being $\mathbb{Q}$ resp. $\mathbb{Z}$
equivariantly formal is equivelant to $H^{i}(M,\mathbb{Q})=0$ resp.
$H^{i}(M,\mathbb{Z})=0$ for all odd $i$ [14, Lemma 2.1].

######  Lemma 4.1.

Suppose that $M_{1},M_{2}$ are simply connected $6$-manifolds, having almost
complex circle actions with isolated, and non-empty fixed point set. Let $M$
be the Kustarev sum of $M_{1}$ and $M_{2}$. Then, $M$ is simply connected and,

$H_{3}(M,\mathbb{Z})=H_{3}(M_{1},\mathbb{Z})\oplus H_{3}(M_{2},\mathbb{Z}).$
$H_{2}(M,\mathbb{Z})=H_{2}(M_{1},\mathbb{Z})\oplus
H_{2}(M_{2},\mathbb{Z})\oplus\mathbb{Z}.$

In particular,

  1. (a)

$M$ is $\mathbb{Q}$ resp. $\mathbb{Z}$ equivariatly formal $\iff$ both $M_{i}$
are $\mathbb{Q}$ resp. $\mathbb{Z}$ equivariatly formal.

  2. (b)

$\chi(M)=4$ and $M$ is $\mathbb{Q}$-equivariatly formal $\iff$ both $M_{i}$
are $\mathbb{Q}$-homology $6$-spheres.

  3. (c)

$\chi(M)=4$ and $M$ is $\mathbb{Z}$-equivariatly formal $\iff$ both $M_{i}$
are diffeomorphic to $S^{6}$.

###### Proof.

The proof of the homology formulas, and a)-b) are an elementary applications
of Mayer-Vietoris and Seifert-Van-Kampen. To prove c), firstly by Poincaré
duality and the universal coefficient theorem, if $N$ is a simply connected
rational homology $S^{6}$, and $0<i<6$ then $H_{i}(N,\mathbb{Z})\neq 0$ can
only happen for $i=2,3$, and furthermore $H_{2}(N,\mathbb{Z})\cong
H_{3}(N,\mathbb{Z})\cong H^{3}(N,\mathbb{Z})$. Since each of $M_{i}$ must be a
$\mathbb{Q}$-homology $6$-sphere by b) and $\mathbb{Z}$-equivariantly formal
by a), both of these two homology groups vanish. That is, each $M_{i}$ is a
simply connected integral homology $S^{6}$, therefore diffeomorphic to
$S^{6}$, for example by the Wall-Jupp-Zubr theorem. ∎

###  4.1 Kustarev sum of two copies of $S^{6}$

In the remainder of this section we prove Theorem 1.1, the general overview of
the argument is as follows. The Kustarev sum of two manifolds is a fiber
connect sum of two manifolds along embedded circles. The fiber connect sum
depends on the embedding of the circles up to ambient isotopy and a choice of
framing of the embedded circles on both sides.

In the first subsection, we recall the notion of a framing of an embedded
submanifold and the definition of the trivial framing for a sphere embedded in
a higher dimensional sphere, following [10]. Next, we observe that free orbits
for a smooth circle action have a canonical framing which is unique up to
homotopy.

In the following section, we recall the definition of the standard almost
complex circle actions on $S^{6}$, and determine that the framing associated
to the free orbits of the actions is the non-trivial one (since
$\pi_{1}(GL_{5}(\mathbb{R}))=\mathbb{Z}_{2}$ it follows that there are two
classes of framings, see the first subsection).

Then, we show that the fiber connect sum of two copies of $S^{6}$ along two
subcircles with the non-trivial framing is diffeomorphic to $S^{4}\times
S^{2}$. This involves expressing the fiber connect sum as the result of
surgery on the $6$-sphere, which is in turn diffeomorphic to the boundary of a
certain handle attachment of the $7$-disk which may be determined explicitly.

###  4.2 Framings and fiber connect sum

We follow the definition of fiber connect sum and its notation presented in
[10, Page 99] throughout the remainder of the article. In the case that
$N=S^{1}$ and $E$ is the trivial vector bundle, the definition of [10, Page
99] is as follows:

######  Definition 4.2.

Let $M_{0}$ and $M_{1}$ be two orientable, closed $n$-manifolds and
$\varphi_{i}\colon S^{1}\times\mathbb{R}^{n-1}\to M_{i}$ two embeddings.
Identify $S^{1}\times\mathbb{R}^{n-1}\setminus(S^{1}\times 0)$ with
$S^{1}\times S^{n-2}\times(0,\infty)$ via the standard diffeomorphism. Then we
define $M_{0}\\#_{S^{1}}M_{1}$ as

$(M_{0}\setminus\varphi_{0}(S^{1}\times 0)\cup
M_{1}\setminus\varphi_{1}(S^{1}\times 0))/\sim$

where $\varphi_{0}(z,p,t)\sim\varphi_{1}(z,p,\frac{1}{t})$ for $(z,p,t)\in
S^{1}\times S^{n-2}\times(0,\infty)$.

The choice of an embedding is known as a framing and is equivalent to a
framing of the normal bundle of $\varphi_{i}(S^{1}\times\\{0\\})$. Recall that
a framing of the normal bundle is equivalent to a trivialisation of the normal
bundle. Because, for $n\geq 3$, $\pi_{1}(GL_{n}(\mathbb{R}))=\mathbb{Z}_{2}$,
when the manifolds have dimension at least $4$ for a given embedded circle
there are exactly two choices of framing.

######  Definition 4.3.

Consider the embedded $S^{1}\subset S^{6}$ given by
$S_{0}=\\{(u,v,0,0,0,0,0)\in S^{6}|u^{2}+v^{2}=1\\}$. Consider the vector
subbundle $E$ of $T\mathbb{R}^{7}$ whose fiber over each point is the linear
subspace $\\{x\in\mathbb{R}^{7}|x_{1}=x_{2}=0\\}$. Then, note that the normal
bundle $N$ of $S_{0}$ is the restriction of $E$ to $S_{0}$. The trivial
framing of $S_{0}$ is defined to be the framing of $N$ given by
$e_{3},\ldots,e_{7}$ over every point, where $e_{i}$ denotes the standard
basis of $\mathbb{R}^{7}$.

Because, any two embeddings of $S^{1}$ to $S^{6}$ are ambient isotopic, and
ambient isotopies respect homotopy types of framings, a framing of an embedded
$S^{1}$ in $S^{6}$ is called trivial when it is ambient isotopic to the one
from Definition 4.3.

###  4.3 Uniqueness of equivariant framings

In this section we consider a natural framing of free orbits a manifold with a
circle action, and show that it is unique up to homotopy. In the coming
section, we will determine this homotopy type of the framing for almost
complex circle action on $S^{6}$.

######  Definition 4.4.

Let $M$ be a manifold with a circle action, and $O$ a free orbit. Then, a
framing $F$ of $O$ is called equivariant if it is of the from

$g_{*}(v_{1}),\ldots g_{*}(v_{k}),$

where $v_{1},\ldots,v_{k}$ is a basis of the normal bundle of $O$ at a point
in $O$.

In the following lemma, we show that equivariant framings yield a unique
homotopy class:

######  Lemma 4.5.

Suppose that $M$ is an $n$-manifold with a smooth $S^{1}$-action. Then, for
each free orbit $O$ any two equivariant framings of $O$ are homotopic.

###### Proof.

Suppose for a contradiction that there were two such non-homotopic framings.
Then, by the equivariant slice theorem there would exist an
$S^{1}$-equivariant self-diffeomorphism $\phi$ of $D^{n-1}\times S^{1}$, with
action $z.(p,w)=(p,zw)$ which maps the class of trivial framing of
$\\{0\\}\times S^{1}$ to the non-trivial one. But, choosing $F$ to be a
constant framing, and applying the $S^{1}$-equivariance of $\phi$ directly
shows that $\phi_{*}F$ is a constant framing, hence represents the same
homotopy class as $F$, which is the desired contradiction.

∎

###  4.4 Computation of equivariant framing

In this subsection, we compute the class of the equivariant framing for the
standard almost complex $S^{1}$-actions on $S^{6}$. We will use this is in the
next subsection to determine the diffeomorphism type of the Kustarev sum of
two copies of $S^{6}$.

######  Theorem 4.6.

Consider the standard almost complex $S^{1}$ actions on $S^{6}$. Then, the
homotopy class of the equivariant framing of a free orbit is the non-trivial
one.

###### Proof.

We recall the definition of almost complex $S^{1}$-actions on $S^{6}$
following [6, Example 1.9.1]. It holds that $S^{6}=G_{2}/\mathrm{SU}(3)$ and
$S^{6}$ admits an invariant almost complex structure. Let $T^{2}\subset G_{2}$
be a Cartan subgroup, then the induced action on $S^{6}$ may be realized as
the restriction of a linear action on $\mathbb{R}^{7}$ as follows: For
$(t,s)\in T^{2}=S^{1}\times S^{1}$ and $(z_{1},z_{2},z_{3},x)\in S^{6}$ we
have

$(t,s)\cdot(z_{1},z_{2},z_{3},x)=(t^{-1}z_{1},sz_{2},(ts)z_{3},x).$

This action has two isolated fixed points $(0,0,0,\pm 1)$. Consider the
subcircle $S^{1}\to T^{2}$, $t\mapsto(t^{a},t^{b})$, $a,b\in\mathbb{Z}$,
$a,b\neq 0$, which gives an almost complex circle action on $S^{6}$:

$t\cdot(z_{1},z_{2},z_{3},x)=(t^{-a}z_{1},t^{b}z_{2},t^{a+b}z_{3},x).$

To determine the class of the equivariant (normal) framing we note that for a
given embedded circle $C$ in $S^{6}$, there is a natural map $\Psi$ sending
normal framings $F$ of $C$ in $S^{6}$ to framings of $T\mathbb{R}^{7}|_{C}$,
given by

$\Psi(F(t))=[c(t),\dot{c}(t),F(t)].$

It is not hard to check that this map is invariant with respect to ambient
isotopy, by extending the ambient isotopy of $S^{6}$ radially on a
neighborhood of $S^{6}\subset\mathbb{R}^{7}$.

Note that a tangent framing of a circle in $\mathbb{R}^{7}$ is the same as a
map $S^{1}\rightarrow GL_{7}(\mathbb{R})$, after choosing a point on the
circle and a basis of the tangent space at that point. Moreover the class in
$\pi_{1}(GL_{7}(\mathbb{R}))$ is independent of such choices. Such a tangent
framing is called trivial if it represents the trivial element in
$\pi_{1}(GL_{7}(\mathbb{R}))$.

Determining the class of the equivariant normal framing will be done by
computing the image of this class by $\Psi$. First, it is necessary to
determine how $\Psi$ acts on homotopy classes of framings. Because, $\Psi$ is
invariant with respect to ambient isotopy, it is necessary just to do this for
a given choice of embedded circle and framing. Taking $C$ to be a standard
coordinate subcircle in the first two co-ordinates with trivial framing
$e_{3},\ldots,e_{7}$.

Then, the image under $\Psi$ under this framing, is the framing associated to
the map $\phi:S^{1}\rightarrow GL_{7}(\mathbb{R})$,

$\phi(t)=\begin{pmatrix}R(t)&0\\\ 0&I_{5}\\\ \end{pmatrix}$

Where $R(t)$ is the standard 2 by 2 rotation matrix associated to $t\in
S^{1}$, and $I_{5}$ is the 5 by 5 identity matrix. Clearly, this represents
the non-trivial element in $\pi_{1}(GL_{7}(\mathbb{R}))$. Therefore $\Psi$
maps the class of the trivial normal framing in $S^{6}$ to the class of the
non-trivial tangent framing in $\mathbb{R}^{7}$, an almost identical argument
shows it sends the non-trivial class to the trivial one.

Finally, it remains to determine the class of the image of the equivariant
normal framing of this circle action under $\Psi$. To do this, note that
because the circle action extends to a linear circle action on
$\mathbb{R}^{7}$, the way that this action acts on tangent vectors is exactly
by the linear action itself.

Picking a point $p=c(0)$ on $C$ and defining a continuous path of
transformations of $T\mathbb{R}^{7}_{p}$ which deform a basis of the form
$c(0),\dot{c}(0),F(0)$ to the standard basis of $\mathbb{R}^{7}$, shows that
this class of the image of the equivariant normal framing under $\Psi$, is the
class of the map $F:S^{1}\rightarrow GL_{7}(\mathbb{R})$,

$F(t)=\begin{pmatrix}R_{-a}(t)&0&0&0\\\ 0&R_{b}(t)&0&0\\\ 0&0&R_{a+b}(t)&0\\\
0&0&0&1\end{pmatrix}$

Finally, because $-a+b+a+b=2b$ is even, the class of this loop is trivial in
$\pi_{1}(GL_{7}(\mathbb{R})).$ That is the image under $\Psi$ of the
equivariant normal framing is the trivial one, showing by the above
considerations about $\Psi$, that the class of the equivariant normal framing
is the non-trivial one.

∎

###  4.5 Determining the diffeomorphism type: Surgery and Handle attachments

The following Lemma shows that the fiber connect sum of two copies of $S^{6}$
with the non-trivial framing and the fiber connect sum of two copies of
$S^{6}$ with the trivial framing are diffeomorphic.

######  Lemma 4.7.

Consider an embedded circle $C$ in $S^{6}$. Let $h_{1}$ denote the trivial
framing of $C$ and $h_{2}$ the non-trivial framing. Then, the fiber connect
sum $(S^{6},C,h_{1})\\#(S^{6},C,h_{1})$ is diffeomorphic to the fiber connect
sum $(S^{6},C,h_{2})\\#(S^{6},C,h_{2})$.

###### Proof.

Fix an embedding $\phi:S^{1}\times\mathbb{R}^{5}\rightarrow N_{c}$,
corresponding to the trivial framing of $C$, where $N_{C}$ is a neighbourhood
of $C$ in $S^{6}$.

Set coordinates of $S^{1}\times\mathbb{R}^{5}$ by identifying is as a subset
of $\mathbb{C}^{3}\times\mathbb{R}$, given by
$\\{(z_{1},z_{2},z_{3},t)\in\mathbb{C}^{3}\times\mathbb{R}||z_{1}|=1\\}.$
Define a map $h:S^{1}\times\mathbb{R}^{5}\rightarrow
S^{1}\times\mathbb{R}^{5}$ given by the formula

$h(z_{1},z_{2},z_{3},t)=(z_{1},z_{1}z_{2},z_{3},t).$

Note that $h$ is a diffeomorphism with inverse
$h^{-1}(z_{1},z_{2},z_{3},t)=(z_{1},z_{1}^{-1}z_{2},z_{3},t).$

Then, $\phi\circ h:S^{1}\times\mathbb{R}^{5}\rightarrow N_{c}$ is an embedding
corresponding to the non-trivial framing of $C$. Following the definition of
fiber connect sum, set $\alpha:(0,\infty)\rightarrow(0,\infty)$ an orientation
reversing diffeomorphism, and set
$\alpha_{E}:(S^{1}\times\mathbb{R}^{5})\setminus(S^{1}\times\\{0\\})\rightarrow(S^{1}\times\mathbb{R}^{5})\setminus(S^{1}\times\\{0\\})$;

$\alpha_{E}(z_{1},z_{2},z_{3},t)=(z_{1},0,0,0)+\frac{\alpha(|(z_{2},z_{3},t)|)}{|(z_{2},z_{3},t)|}(0,z_{2},z_{3},t).$

By definition $(S^{6},C,h_{1})\\#(S^{6},C,h_{1})$ is diffeomorphic to two
copies of $S^{6}\setminus C$ glued along $N_{C}\setminus C$ by
$\phi\circ\alpha_{E}\circ\phi^{-1}$. Similarly
$(S^{6},C,h_{2})\\#(S^{6},C,h_{2})$ is diffeomorphic to two copies of
$S^{6}\setminus C$ glued along $N_{C}\setminus C$ by $\phi\circ
h\circ\alpha_{E}\circ h^{-1}\circ\phi^{-1}$. Then, one may check directly from
the formulas that

$h\circ\alpha_{E}\circ h^{-1}=\alpha_{E}.$

The gluing maps are equal and the two spaces are diffeomorphic.

∎

Since, we have proven in the previous section, that the Kustarev sum $M$ is
the fiber connect sum of two copies of $S^{6}$ with the non-trivial framing.
By Lemma 4.7 it is diffeomorphic to the fiber connect sum of two copies of
$S^{6}$ with the trivial framing.

Next, the following fact asserted in [10, Page 112] shows that $M$ is
diffeomorphic to the surgery on $S^{6}$ along an embedded circle with the
trivial framing.

######  Lemma 4.8.

[10, Page 112] Let M be an $n$\- manifold with an embbedded sphere $S$ and
framing $h$. The fiber connect sum of $M$ with $S^{n}$, where $S\subset M$ has
framing $h$ and the co-ordinate subsphere of $S^{n}$ has the trivial framing,
is diffeomorphic to the surgery on M along $(S,h)$.

Therefore $M$ is diffeomorphic to the result of the surgery on $S^{6}$ along
$M$ with the trivial framing.

######  Lemma 4.9.

The surgery on $S^{6}$ along $S^{1}$ with the trivial framing, is
diffeomorphic to $S^{4}\times S^{2}$.

###### Proof.

Consider the disk $D^{7}$, and attach a $2$-handle to $\partial D^{7}=S^{6}$
via the trivial framing. Then, by [5, Example 4.1.4 (d)], the resulting
manifold with boundary is $D^{5}\times S^{2}$. It is known that the boundary
of the handle attachment along an embedded sphere in the boundary is
diffeomorphic to the surgery on the same sphere in the boundary, with the same
framing ([5, Page 154], [10, page 112]). Therefore, the surgery on $S^{6}$
along $S^{1}$ with trivial framing is diffeomorphic to the boundary of
$D^{5}\times S^{2}$, namely $S^{4}\times S^{2}$. ∎

###### Proof of Theorem 1.1.

The Kustarev sum of two copies of $S^{6}$, is by definition the fiber connect
sum of two copies of $S^{6}$ along circles with the equivariant framing on
both sides, denote this manifold by $M$. By Theorem 4.6 the equivariant
framing is the non-trivial one. By Lemma 4.7 $M$ is diffeomorphic to the the
fiber connect sum of two copies of $S^{6}$ along circles with the trivial
framing on both sides. By Lemma 4.8 $M$ is diffeomorphic to the result of
sugery of $S^{6}$ along an embedded circle with the trivial framing. By Lemma
4.9, $M$ is diffeomorphic to $S^{4}\times S^{2}$. ∎

###  4.6 Proof of Corollary 1.2

In the following, we consider the Kustarev sum of two copies of $S^{6}$, with
almost complex circle actions with weights

$\\{a,b,-a-b\\},\\{-a,-b,a+b\\}$

and

$\\{c,d,-c-d\\},\\{-c,-d,c+d\\},$

respectively (see [6, Example 1.9.1] for the construction of circle actions on
$S^{6}$.). In the following we choose the parameters so that we have $b>a>1$,
and $d>c>1$.

###### Proof of Corollary 1.2.

Suppose for a contradiction that the Kustarev sum is equivariantly
diffeomorphic to a linear action. Let $\Phi$ denote the equivariant
diffeomorphism.

For integers $w_{1},w_{2},w_{3}$ linear $S^{1}$ actions on $S^{4}\times
S^{2}$, are given by $S^{4}\times
S^{2}\subset(\mathbb{C}^{2}\oplus\mathbb{R})\times(\mathbb{C}\oplus\mathbb{R})$,
given by

$z.((u_{1},u_{t},t),(v,s))=((z^{w_{1}}u_{1},z^{w_{2}}u_{2},t),(z^{w_{3}}v,s)).$

The fixed point set of the action consists of the four points of the form
$((0,0,\pm 1),(0,\pm 1))$. Since the action on the Kustarev sum has isolated
fixed points the linear action has isolated fixed points. Therefore, without
loss of generality $w_{i}>0$ for each $i$.

Let $p$ be a fixed point which is mapped to $((0,0,-1),(0,-1))$. The
$\mathbb{R}$ weights of the linear action at this fixed point
$w_{1},w_{2},w_{3}$ and the $\mathbb{R}$ weights of the Kustarev sum are
$\\{a,b,a+b\\}$, where $a,b>1$. Because, $\Phi$ is an equivariant
diffeomorphism on neighbourhoods of these fixed points, the weights are equal.
To see this one can pick an invariant Riemannian metric and consider the
exponential at each of the fixed points respectively. In particular, $w_{i}>1$
for each $i$, and $w_{i}$ are pairwise coprime.

Next, since each $w_{i}$ is greater than one, there are $6$ isotropy spheres
of the linear action, of weight $w_{1},w_{2}$ containing
$\\{((0,0,-1),(0,1)),((0,0,1),(0,1))\\}$ and containing
$\\{((0,0,-1),(0,-1)),((0,0,1),(0,-1))\\}$.

Also, there exist isotropy spheres of weight $w_{3}$ containing
$\\{((0,0,1),(0,-1)),((0,0,1),(0,1))\\}$ and containing
$\\{((0,0,-1),(0,-1)),((0,0,-1),(0,1))\\}$.

In particular, for the linear action the subset of points with non-trivial
stabilizer group is connected. However, since the Kustarev sum is performed on
a neighborhood of a free orbit, it is clear that for the action on the
Kustarev sum the set of points with non-trivial stabilizer group is
disconnected, a contradiction. ∎

##  5 Some Topological consequences of Jang’s Classification

In this section, we prove a topological consequence of Jang’s classification
where the manifold is simply connected and $\mathbb{Z}$-equivariantly formal
and fits into Case 6 of Jang’s classification (also called the $Bl_{C}(S^{6})$
case).

######  Theorem 5.1.

Suppose that $M$ is a simply connected, almost complex $6$-manifold and an
almost complex circle action fitting into case $6$ of Jang’s classification.
Moreover, suppose that it is $\mathbb{Z}$-equivariantly formal. Then, $M$ is
diffeomorphic to a Quadric $Q\subset\mathbb{CP}^{4}$.

###### Proof.

Since the circle action has $4$ fixed points, $\chi(M)=4$. Then, combining
this condition with $\pi_{1}(M)=1$, and $H^{3}(M,\mathbb{Z})=0$,
$H_{2}(M,\mathbb{Z})=\mathbb{Z}$, a routine application of the universal
coefficients theorem shows that $H^{i}(M,\mathbb{Z})=\mathbb{Z}$ for
$i=0,2,4,6$ and $H^{i}(M,\mathbb{Z})=0$ otherwise.

By Lemma 2.1 $c_{1}(J)^{3}=-2$. In particular this implies that $w_{2}(M)$ is
the non-trivial element of $H^{2}(M,\mathbb{Z}_{2})=\mathbb{Z}_{2}$, because
otherwise $c_{1}$ is even so $c_{1}^{3}$ would be divisible by $8$. Also by
Lemma 2.2 $c_{1}c_{2}=0$. Let $\alpha=-c_{1}\in H^{2}(M,\mathbb{Z})$, note
that $\alpha$ is a generator of $H^{2}(M,\mathbb{Z})=\mathbb{Z}$. Then by
Poincare duality $c_{1}c_{2}=0$ implies that $c_{2}=0$. Finally, using the
equation $p_{1}=c_{1}^{2}-2c_{2}$, gives that $p_{1}=\alpha^{2}$. Hence we
have shown that the cohomology ring and Pontryagin, and second Stiefel-Whitney
classes are isomorphic to the Quadric, the result follows from the Wall-Jupp-
Zubr theorem. ∎

##  6 Some questions and further directions

In [11], Kustarev originally utilized his surgery operation to show that there
are infinitely many diffeomorphism types of almost complex $6$-manifolds
having a almost complex circle action with two fixed points. He showed this by
taking the Kustarev sum of $S^{6}$, and an almost complex $6$-manifold of the
form $M^{4}\times T^{2}$, with a free almost complex circle action on the
$T^{2}$-factor. This construction shows that the fundamental group varies
through infinitlely many isomorphism types.

It appears however, that the following question about topological types of
almost complex $6$-manifolds with two fixed points remained open.

######  Question 6.1.

Suppose that $M$ is a closed, almost complex, simply connected
$\mathbb{Q}$-homology $S^{6}$ with an almost complex circle action with two
fixed points. Is $M$ diffeomorphic to $S^{6}$?

We note that without the assumption of a circle action there is a wealth of
almost complex, simply connected, rational homology $6$-spheres.

We also pose a question about actions with $4$ fixed points. Similarly to the
above, in each of cases 1-4 of Jang’s classification [9, Theorem 1.1] taking a
Kustarev sum with almost complex $6$-manifolds with manifolds having free
circle actions, shows that there are infinitely diffeomorphism types and
furthermore $\pi_{1}$ varies through infinitlely many isomorphism types.

In the following question we pose a finiteness question for simply connected,
equivariantly formal manifolds appearing in Jang’s classification.

######  Question 6.2.

  1. a)

Are there finitely many diffeomorphism types of, simply connected,
$\mathbb{Z}$-equivariantly formal, almost complex 6-manifolds having a circle
action with 4 fixed points?

  2. b)

Are there finitely many homotopy types of simply connected,
$\mathbb{Z}$-equivariantly formal, almost complex 6-manifolds having a circle
action with 4 fixed points?

The existence of the new example $S^{4}\times S^{2}$ of Theorem 1.1, which
fits into the above class but is not covered by previous classifications [1,
15], perhaps gives additional motivation to consider Question 6.2.

## References

  * [1] K. Ahara: 6-dimensional almost complex $S^{1}$-manifolds with $\chi(M)=4$. J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 38 no 1, 47-72 (1991).
  * [2] M. F. Atiyah and R. Bott. The moment map and equivariant cohomology. Topology, Vol. 23, No.1 pp 1-28 (1984).
  * [3] N. Berline. M. Vergne. Classes caracteristiques equivariantes. Formule de localisation en cohomologie equivariante, C.R.Acad.Sci. Paris 295 (1982) 539–541.
  * [4] J. W.Gray. Some global properties of contact structures .Ann. of Math. (2)69 421–450 (1959).
  * [5] R.E. Gompf, A.I. Stipsicz. $4$-manifolds and Kirby Calculus. Graduate Studies in Mathematics, Volume 20. American Mathematical Society (1999).
  * [6] V. Guillemin, C. Zara. 1-skeleta, Betti numbers, and equivariant cohomology. Duke Math. J. 107(2): 283-349 (2001).
  * [7] A. Hatcher. Algebraic Topology. Cambridge University Press (2002).
  * [8] F. Hirzebruch. T. Berger. R. Jung. Manifolds and Modular Forms. Aspects of Mathematics (1992).
  * [9] D. Jang. Circle actions on almost complex manifolds with 4 fixed points. Mathematische Zeitschrift volume 294, pages287–319 (2020).
  * [10] A.A. Kosinski. Differentiable Manifolds. Academic Press Inc. (1992).
  * [11] A. Kustarev. Almost complex circle actions with few fixed points, Russian Mathematical Surveys, Volume 68, Issue 3, 574–576, (2013).
  * [12] A. Kustarev. Chern numbers of manifolds with torus action. arXiv:1506.05355.
  * [13] W. S. Massey. Obstructions to the existence of almost complex structures. Bull. Amer. Math. Soc.67, 559–564 (1961).
  * [14] M. Masuda, T. Panov On the cohomology of torus manifolds. Osaka J. Math.43 , 711–746 (2006).
  * [15] S. Tolman. A Symplectic Generalization of Petries Conjecture. Transactions of the American Mathematical Society 362 (08): 3963–3996, (2010).

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