pnasresearcharticle Li Cell-generated forces are indispensable determinants of
fundamental cell functions such as motility and cell division. As such,
quantifying how the forces change upon perturbations to the cells such as gene
mutations and drug administration is of profound importance. Here we present a
novel machine learning based system that allows for efficient estimations of
the forces that are determined only by “observing” microscope images. Given
that the cellular traction forces are regulated downstream of diverse
signaling pathways, our system – that helps significantly improve the
throughput of the measurements – presents a new, high throughput platform for
real time analysis of the effects of a massive number of genetic and molecular
perturbations on the forces and resulting cell mechanics. H.L. and D.M
developed the machine learning system, and K.I. supported the implementation.
H.A. and T.S.M. designed and worked on the cell experiments. H.L., H.A. and
D.M. analyzed the experimental data. D.M., A.D. and S.D. conceived the idea of
simultaneous TFM with wrinkle extraction. D.M. and A.D. analyzed the physics.
H.L., D.M., A.D. and S.D. wrote the article and designed the research. The
authors declare no competing interests. 1To whom correspondence may be
addressed. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Wrinkle force microscopy: a new machine learning based approach to predict
cell mechanics from images

Honghan Li  Division of Bioengineering, Graduate School of Engineering
Science, Osaka University, 1-3 Machikaneyama Toyonaka, Osaka, 5608531, Japan
Daiki Matsunaga  Division of Bioengineering, Graduate School of Engineering
Science, Osaka University, 1-3 Machikaneyama Toyonaka, Osaka, 5608531, Japan
Tsubasa S. Matsui  Division of Bioengineering, Graduate School of Engineering
Science, Osaka University, 1-3 Machikaneyama Toyonaka, Osaka, 5608531, Japan
Hiroki Aosaki  Division of Bioengineering, Graduate School of Engineering
Science, Osaka University, 1-3 Machikaneyama Toyonaka, Osaka, 5608531, Japan
Koki Inoue  Division of Bioengineering, Graduate School of Engineering
Science, Osaka University, 1-3 Machikaneyama Toyonaka, Osaka, 5608531, Japan
Amin Doostmohammadi  Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, 2100 Copenhagen, Denmark  Division of Bioengineering, Graduate
School of Engineering Science, Osaka University, 1-3 Machikaneyama Toyonaka,
Osaka, 5608531, Japan  Shinji Deguchi  Division of Bioengineering, Graduate
School of Engineering Science, Osaka University, 1-3 Machikaneyama Toyonaka,
Osaka, 5608531, Japan

###### Abstract

Combining experiments with artificial intelligence algorithms, we propose a
new machine learning based approach to extract the cellular force
distributions from the microscope images. The full process can be divided into
three steps. First, we culture the cells on a special substrate allowing to
measure both the cellular traction force on the substrate and the
corresponding substrate wrinkles simultaneously. The cellular forces are
obtained using the traction force microscopy (TFM), at the same time that
cell-generated contractile forces wrinkle their underlying substrate. Second,
the wrinkle positions are extracted from the microscope images. Third, we
train the machine learning system with GAN (generative adversarial network) by
using sets of corresponding two images, the traction field and the input
images (raw microscope images or extracted wrinkle images), as the training
data. The network understands the way to convert the input images of the
substrate wrinkles to the traction distribution from the training. After
sufficient training, the network is utilized to predict the cellular forces
just from the input images. Our system provides a powerful tool to evaluate
the cellular forces efficiently because the forces can be predicted just by
observing the cells under the microscope, which is a way simpler method
compared to the TFM experiment. Additionally, the machine learning based
approach presented here has the profound potential for being applied to
diverse cellular assays for studying mechanobiology of cells.

###### keywords:

cell mechanics $|$ traction force microscopy $|$ GAN

This manuscript was compiled on

There is now growing evidence showing that cells sense mechanical cues in the
surrounding microenvironment to regulate their functions such as
proliferation, differentiation, apoptosis, and pro-inflammation (1, 2, 3, 4,
5, 6). In response to the mechanical cues, cells often adjust their
cytoskeletonal tension and as such many of the mechanical information are
translated into a level of inherent cellular traction forces, and in turn into
intracellular signals regulating the related functions (7, 8, 3, 9). Traction
forces, thus related to various cell functions, are generated by the activity
of nonmuscle myosin II and actin filaments that determine cellular
contractility (10, 11, 2, 12). Because these proteins work downstream of
diverse signaling pathways, it is often difficult to predict how the force may
change upon perturbations to particular molecules such as gene mutations and
drug administration. Thus, technologies allowing for efficiently evaluating
the cellular traction force are expected to appear to enhance comprehensive
understanding of the force-related pathways.

We previously developed a wrinkle assay, a modified version of the method
originally reported by Harris and colleagues (13, 14), in which the silicone
substrates are spatially treated with uniform oxygen plasma to allow them to
buckle upon the forces exerted by cells (15, 16, 17). As the individual
wrinkles are lengthened with the increase in the forces (18), the wrinkle
length, detected for example by a machine learning approach (19), can be used
as a measure of the relative change in the force caused by perturbations such
as specific gene mutations. While this technology is promising in that these
experiments are performed easily to potentially enable a high-throughput
analysis on the force-related pathways, the interpretation of the wrinkle
length was not necessarily straightforward in terms of quantitatively
measuring the magnitude and direction of traction forces.

To overcome this limitation in quantification, here we describe a new machine
learning system that converts the wrinkle information taken by a microscope
into the actual cellular force distributions. For the initial training data,
the cellular traction forces are obtained using the traction force microscopy
(TFM), and we train the machine learning system with GAN (generative
adversarial network) so that the network understands the way to convert the
input microscope images to the force distributions from the training data.
After sufficient training, the network can be utilized to predict the cellular
forces just from the input images. The system would be a powerful tool to
evaluate the cellular forces efficiently because the forces can be predicted
just by observing the cells, which is a way simpler method compared to the TFM
experiment.

Figure 1:  Overview of the methods and procedures that are utilized in the
wrinkle-force microscopy (WFM). (a) Schematic of our experimental setup. A
silicone membrane, which can evaluate the cellular force distribution
(obtained by TFM) and the surface wrinkles simultaneously, is utilized in this
work. (b) The surface wrinkles are extracted from the microscope images by
using our machine learning system (SW-UNet). (c) The machine learning system
(GAN) is trained to understand the relation between the input images (raw
microscope image, or extracted wrinkle images) and corresponding output images
(cellular force distribution). (d) After sufficient training, the system can
predict the force distributions only from the microscope images.

## Full picture of the system

Our goal is to construct a machine learning system that can predict the
cellular force distributions from the microscope image or the extracted
substrate wrinkles. The full process can be divided into three steps as shown
in Fig. 1. First, we culture the cells (A7r5; embryonic rat vascular smooth
muscle cells) on a silicone membrane substrate and measure both the cellular
traction force and the substrate wrinkles simultaneously. As shown in Fig.
1(a), the cellular traction forces are obtained using TFM (20, 21), and cells
generate wrinkles because the surface of PDMS (polydimethyl siloxane) layer is
hardened by the plasma irradiation (22, 16, 23, 19, 24). Second, the wrinkle
positions are extracted from the microscope images as shown in Fig. 1(b) by
using our SW-UNet (small world U-Net) (19), which is a convolutional neural
network (CNN) that reflects the concept of the small world networks (25, 26).
Third, the machine learning system utilizing GAN (27) is trained to understand
a way to convert the microscope image, or the extracted wrinkle image, to the
cellular force distributions as shown in Fig. 1(c). After the training, the
network can be utilized to predict the cellular forces just from the
microscope images.

Figure 2: Three examples of the simultaneous measurement of wrinkles and
traction forces. Each column describes (from left to right) raw image,
wrinkles (red lines), displacement field and traction force field. The white
scale bar in the first column images is $20$ ${\rm\mu m}$. The blue and green
lines inside the second column images describe the principal direction of the
wrinkle and the traction, respectively.

Figure 3:  Quantitative analysis of the traction forces and wrinkles. (a)
Probability distribution function (PDF) of the traction magnitude. (b) The
wrinkle length as the function of the mean traction $\bar{f}$. Note that the
wrinkle length is evaluated by counting the number of pixels after
skeletonizing the wrinkle images. (c) The wrinkle length as the function of
the principal traction $f_{p}$. (d) Probability distribution function of the
angle differences between the wrinkle direction $\phi_{w}$ and the traction
$\phi_{s}$. The figure suggests that the direction of the wrinkles is
predominantly perpendicular to the principal direction of the force.

## Simultaneous measurement of wrinkles and traction forces

Before applying the machine learning system, we begin by considering the
results of the simultaneous force and wrinkle characterization. Figure 2
summarizes representative results obtained by the experiment and analysis. Due
to the pairwise inward pulling generated by cellular traction force (fourth
column), the substrates exhibit displacements toward the cell center (third
column). As the result of the contraction, the wrinkles emerge mostly
underneath the cells (second column). When the cell size is small, the
majority of wrinkles are aligned in a same direction as in Fig. 2(a), while
they tend to point in different directions when the cell size is large and the
traction is strong as in Fig. 2(c).

Figure 3(a) shows the probability distribution function (PDF) of the traction
magnitude of $N\times M$ samples, where $N=103$ is the number of the images
and $M=26\times 26$ is the number of the force observation points. The average
traction is 50.3 $\pm$ 57.1 [Pa] (mean $\pm$ standard deviation). Figure 3(b)
shows that the wrinkle length has a positive correlation with the mean
traction of the images, which is in agreement with our previous experimental
measurements (22), where the relationship between the wrinkle length and
applied force was experimentally investigated using microneedles. The mean
traction is simply obtained by averaging the norm of the traction of the image
as

$\bar{f}=\frac{1}{M}\sum_{m}^{M}|\bm{f}_{m}|$ (1)

where $m$ is the index of the observation points. The wrinkle length is
measured by counting the number of pixels after skeletonizing the wrinkle
images (22). The wrinkle extincts when the mean traction in a image is less
than 10 Pa, which is comparable to the noise level or the resolution of the
current TFM.

Figure 4:  Prediction of the traction forces from microscope images. Each
column shows the result of the traction force predictions by using different
methods (from left to right): ground truth, GAN prediction (input: microscope
image), GAN prediction (input: extracted wrinkle images), and CNN prediction
(input: microscope image). See further examples in Movies S1-S4.

In order to analyze the principal direction of the traction, we construct a
symmetric stress tensor for each image as

$S_{ij}=\frac{1}{2M}\sum^{M}_{m}\left\\{n_{j}(\bm{x}_{m},\bm{x}_{0})f_{i}(\bm{x}_{m})+n_{i}(\bm{x}_{m},\bm{x}_{0})f_{j}(\bm{x}_{m})\right\\}$
(2)

where $\bm{r}=\bm{x}_{m}-\bm{x}_{0}$ is the relative vector from the image
center $\bm{x}_{0}$ and $\bm{n}=\bm{r}/|\bm{r}|$ is the normal vector. By
diagonalizing the tensor, we obtain the principal direction of the traction
$\phi_{s}$ (shown in Fig. 2, second column with green lines) together with the
corresponding principal traction magnitude $f_{p}$, from the eigenvalue that
has the largest norm. At the same time, we obtain the principal direction of
the wrinkles $\phi_{w}$ (also shown in Fig. 2, second column with blue lines)
from the 2D-FFT (fast Fourier transform) image of the wrinkles: $\phi_{w}$ is
an angle that is perpendicular to the direction that has a largest power
spectrum. Figure 3(c) shows that the traction force is contractile ($f_{p}<0$)
and is almost linearly related to the wrinkle length (correlation $R=-0.82$ in
a range $f_{p}<-5$ Pa). Using the relation between the traction and the
wrinkles, the wrinkles can be used for one qualitative marker or indicator for
rough estimation of the cellular traction magnitude. Figure 3(d) shows that
the two angles $\phi_{s}$ and $\phi_{w}$ are perpendicular most of the time.
Since the wrinkle direction is perpendicular to that of the force dipoles, the
wrinkle would be also practical to qualitatively predict the force directions,
as previously done elsewhere (18, 16). Therefore, the length and direction of
wrinkles provide a qualitative measure of the magnitude and direction of
forces exerted by cells on the substrate, respectively. Next, we employ the
machine learning based approach to provide a quantitative measure of the
forces from the microscope images of wrinkles.

Figure 5:  (a) Comparison of the ground truth $f_{x,y}^{\rm true}$ and the
predicted traction $f_{x,y}^{\rm predict}$. Dashed line shows a condition
$f^{\rm predict}=f^{\rm true}$. (b) Correlation coefficient $R$ between
$f_{x,y}^{\rm true}$ and $f_{x,y}^{\rm predict}$. (c)-(d) Errors of the
predicted traction compared to the ground truth data: (c) error in the
traction magnitude $\varepsilon_{f}$ and (d) the traction direction
$\varepsilon_{\theta}$. Blue squares are the average value and black circles
denote the outliers. Note that m.s. denotes the microscope images.

## Traction force prediction using GAN

Finally, we train the network and evaluate the performance of the force
estimation using our GAN network. Figure 4 compares the predicted force
distributions which were estimated by the three different methods. As also
shown in Fig. S1, we trained the network with two different input images, the
raw microscope images (second column in Fig. 4) and the extracted wrinkle
images (third column), to compare the performances. We also evaluated the
force distribution using a standard encoder-decoder type CNN and show the
results in the fourth column. The figure shows that all the three methods
reproduce approximately the same force direction as the ground truth (first
column), and the forces are perpendicular to the wrinkles.

Figure 5(a) compares the traction of ground truth $f_{x,y}^{\rm true}$ and GAN
prediction $f_{x,y}^{\rm predict}$ (input image: microscope images), and it
shows that the prediction is highly correlated with the experimental data. The
correlation coefficient $R$ is evaluated for all 15 test images and averaged
correlations, 0.86-0.88 for GAN and 0.83-0.84 for CNN as shown in Fig. 5(b),
suggest that there are striking agreements. In order to further quantify the
error in the force estimation, we introduce two errors: the error in the force
magnitude $\varepsilon_{f}$ and the force direction $\varepsilon_{\theta}$.
The error $\varepsilon_{f}$ is defined to evaluate the difference in the force
magnitude between the ground truth $f^{\rm true}$ and the prediction $f^{\rm
predict}$ as:

$\varepsilon_{f}=\frac{1}{M}\sum_{m}^{M}\frac{|f_{m}^{\rm predict}-f_{m}^{\rm
true}|}{f_{m}^{\rm true}}\cdot\omega_{m}$ (3)

where $M=26\times 26$ is the number of observation points,
$\omega_{m}=f_{m}^{\rm true}/\bar{f}$ is the weight function and $\bar{f}$ is
the average force in a image which is defined in Eq. (1). Note that we
introduce this weight function in order to put weight on the evaluation of
large vectors rather than small vectors, which give huge errors even for small
differences. We used $N=332$ training image sets and 3 test images for the
evaluation. The total error is calculated by averaging the error of 15 test
images, which are obtained by repeating the evaluation 5 times with randomly
selected different test images. Figure 5(c) shows that the error is 33-35% for
GAN, and it has better performance compared to the encoder-decoder type simple
CNN, which has an error 46%. There is no significant difference by the two
input images (microscope images and wrinkle images), and this result indicates
that performance of the force estimation would not improve drastically by
explicitly teaching the wrinkle position to the machine learning system. Next,
we evaluate the angle difference between the predicted force and the ground
truth as

$\varepsilon_{\theta}=\frac{1}{M}\sum_{m}^{M}|\theta_{m}^{\rm
predict}-\theta_{m}^{\rm true}|\cdot\omega_{m}$ (4)

where $\theta=\arctan(f_{y}/f_{x})$ is the force direction. Figure 5(d) shows
that the errors are 19-20∘ for GAN, and again shows better performance
compared to conventional CNN ($\varepsilon_{\theta}$ = 23-24∘). As for a
traditional CNN, the loss function is designed to measure the error between
predicted results and ground truth, and this criteria of the error are fixed
during the training. While in the case of GAN, the loss function can adapt to
the specific problem dynamically because of the discriminator network, and
this difference brings GAN a better score as shown in the figure. It is
important to note that we have so far acquired a minimal required amount of
training (original data: $\sim$100) to demonstrate the novel concept of
cellular force detection from microscope images. These errors will be
minimized by increasing the number of the training data. As demonstrated
above, our system succeeded in estimating the force distribution just from the
input images with limited levels of errors, in real time. Movies S1-S4 further
show the application of the proposed system in providing high throughout, real
time measure of the traction force distributions during dynamic cell
locomotion.

## Discussion

We proposed a new machine learning based system that can predict the cellular
force distributions from the microscope images. The full process can be
divided into three steps. First, we culture the cells on a plasma-irradiated
silocone substrate and measure both the cellular traction force and the
substrate wrinkles simultaneously. The cellular traction forces are obtained
using the TFM, while cells generate wrinkles on the underlying substrates.
Second, the wrinkle positions are extracted from the microscope images by
using SW-UNet. Third, we train the GAN system by using sets of corresponding
two images, the force distributions and the input images (raw microscope
images or extracted wrinkle images), as the training data. The network
understands the way to convert the input images to the force distributions
from the training. After sufficient training, the network can be utilized to
predict the cellular forces just from the input images. Comparing with the TFM
experiment (test data), the prediction using our system is highly correlated
with the experimental data, with the averaged correlation coefficient of
0.86-0.88 and with 33-35% errors in the force magnitude prediction and angle
errors 19-20∘ in the force direction. We expect that this error would decrease
further by increasing the number of training images. The system would be a
powerful tool to evaluate the cellular forces efficiently because the forces
can be predicted just by observing the cells, which is a way simpler method
compared to performing the TFM experiment every time needed.

TFM is one of the most used methods to evaluate the cellular forces in
mechanobiology study, but as the accuracy of the measurement depends on
successful acquisition of the reference positions of the micro-beads that are
obtained by removing the cells after each of the experiments in conventional
TFM, this method is limited in throughput. The novel GAN-based system proposed
here overcomes this limitation as it provides the nearly same information,
with the high levels of the correlations with the experimental data and the
limited levels of the errors, on the cell mechanics only from the still images
that are acquired just by plating the cells on the silicone substrate without
taking care of the reference as the substrate surface is known to become
planar again upon the absence of the cellular forces in a reversible manner.
Given that early stages of drug screening require testing a massive number of
candidate compounds (23), our system with the potentially high-throughput data
analysis capability will be useful particularly in such screening studies. It
is important to note that our new system is not the one that essentially
competes with TFM, but the huge advantage of the proposed system is focused on
its capability to provide data equivalent to the TFM (with a level of the
errors) and thereby circumvent performing the TFM that needs considerable
technical care. Rather, because the machine learning system depends on the
training data, further innovations in TFM such as super-resolution imaging
(28, 29) are potentially introduced to our system to synergetically output
more sophisticated data. Thus, our approach presents a versatile framework
that integrates the sophisticated experimental techniques and the efficient
measurements.

### Step 1: Simultaneous measurement of traction forces and wrinkles

Based on our previous studies (22, 16, 23, 19, 24), we prepared the substrate
that can reversibly generate wrinkles upon application of cellular forces.
Firstly, a circular cover glass is treated with oxygen plasma (SEDE-GE,
Meiwafosis) to hydrophilize the surface and is desiccated after fluorescent
micro-beads (0.2 $\mu$m in diameter, carboxylate yellow-green fluorescent
beads; Invitrogen) in water solution are distributed on the surface. Secondly,
parts A and B of CY 52-276 (Dow Corning Toray) are mixed at a weight ratio of
1.2:1 and poured onto the cover glass to create a PDMS layer with a height of
30-40 $\mu$m. Thirdly, the cover glass is placed in a 60°C oven for 20 hours
to cure the PDMS. Fourthly, oxygen plasma is applied uniformly along the
surface of the PDMS layer to create an oxide layer that works as the substrate
for cell culture. Finally, the substrate is coated with 10 $\mu$g/mL collagen
type I solution for 3 hours.

For the TFM measurement, fluorescent micro-beads are attached to the substrate
surface as position markers to measure the substrate deformations. The beads
need to be firmly adhered to the surface so that cells would not move the
beads due to endocytosis. In this work, the covalent bonding between the
surface and the beads of 0.001% v/v are performed by following two steps: (i)
silane coupling of the substrate surface using 3-Aminopropyltrimethoxysilane
and (ii) the covalent bonding formation due to carbodiimide. The beads adhered
on the glass surface are monitored to keep the reference position even after
removing the cell using 0.25% Trypsin (Trypsin + 1mm mmol/I EDTA-4Na solution;
Fuji Wako Pure Chemical Corporation).

### Cell culture and microscope setup

A7r5 cells were maintained at 37°C in a stage incubator (INUF-IX3W; Tokai Hit)
under a humidified 5% $\rm{CO_{2}}$ incubator. An inverted microscope (1X73;
Olympus) with a conforcal unit (CSU10; Yokokawa Electric) and oil immersion
lens (phase contrast, UPlanFLN 60x/1.25 Oil Iris Ph3, Olympus Corporation) are
used to capture the cells and fluorescent beads. During the experiment,
DMEM(L)+10% FBS+Penicillin-Streptomycin (Fuji Wako Pure Chemical Corporation)
is used as the culture medium.

### Traction force microscopy (TFM)

The software ImageJ/Fiji and its plugin FTTC (Fourier transform traction
cytometry) (30, 31) are used to evaluate the force field from the displacement
field. The substrate is considered as a soft elastic isotropic material that
follows the linear elastic theory. First, the displacement of the substrate
surface $\bm{u}$ is measured by tracking the movement of the fluorescent beads
using PIV (particle image velocimetry). Second, the traction field is obtained
from the displacement field by solving the governing equation for the elastic
halfspace (32, 33) given by

$\bm{u}(\bm{x})=\int_{S}\bm{G}(\bm{x},\bm{y})\bm{t}(\bm{y})dS(\bm{y})$ (5)

where $\bm{t}$ is the traction force, $\bm{x}$ and $\bm{y}$ are the positions
of the displacement and the traction force, respectively. $\bm{G}$ is the
Green’s function that is given by

$\bm{G}(\bm{x})=\frac{1+\nu}{\pi Er^{3}}\begin{pmatrix}(1-\nu)r^{2}+\nu
r_{x}^{2}&\nu r_{x}r_{y}\\\ \nu r_{x}r_{y}&(1-\nu)r^{2}+\nu
r_{y}^{2}\end{pmatrix}$ (6)

where $E$ is the Young’s modulus, $\nu$ is the Poisson’s ratio,
$\bm{r}=(r_{x},r_{y})=\bm{x}-\bm{y}$ is the relative position vector and
$r=|\bm{r}|$. The software FFTC solves Eq. (5) in the Fourier space, which is
given by

$\tilde{\bm{t}}=(\bm{G}^{T}\bm{G}+\lambda^{2}\bm{I})^{-1}\bm{G}^{T}\tilde{\bm{u}}$
(7)

where tilde symbols denote the variables in Fourier space, $\lambda$ is the
regularization parameter (33) and $\bm{I}$ is the unit tensor. In order to
evaluate the optimal parameter $\lambda$ for the Tikhonov regularization, the
L-curve criterion (34, 33) is applied. Note that $E$ is experimentally
determined (16) to be 5400 Pa and $\nu$ is assumed 0.5 (incompressible) that
is a typical value for PDMS material.

### Step2: Wrinkle extraction

We use a new method SW-UNet (19), which is a CNN based on U-Net (35) to
extract wrinkle patterns from the microscope image as shown in Fig. 1(b). As
the training data, we prepare 236 sets of corresponding two images (microscope
image and manually labeled wrinkle image). The number of data is increased to
2596 by using the image augmentation techniques. We used NVIDIA Titan RTX to
accelerate the training process, and the Adam optimizer is utilized.

### Step 3: Prediction of traction force based on GAN-based system

Assume that we have $N_{o}$ sets of corresponding images and data; the input
images $x$ (microscope images, or extracted wrinkle images) and the force
distributions $y$ as shown in Fig. S1. We effectively have the number of
training data set $2N_{o}$ because the wrinkle image has only 1D information
at each pixel (intensity $I(x,y)$; see also images in Fig. S1) while the force
distributions have 2D information (2D force,
$\bm{f}(x,y)=\\{f_{x}(x,y),f_{y}(x,y)\\}$). We designed the network to
evaluate the cellular force only for a single axis at one time and focus only
on the $x$-directional force at each evaluation. An input image $I_{i}$ is
used as the training data set ($I_{i}$, $f_{x}$) and ($I^{\prime}_{i}$,
$f_{y}$) where $I^{\prime}_{i}$ is an image that rotates $I_{i}$ by 90
degrees.

The force distributions are converted to gray scale images that have
intensities

$I(f_{d})=a\arctan\left(\frac{f_{d}}{b}\right)+I_{\rm mid}$ (8)

where $a=81.2$, $b=50.0$, $I_{\rm mid}=255/2$ are the coefficients for the
conversions, and $f_{d}$ is the components of the force $d=x,y$. The force
distributions in gray scale, which are generated from the test images, can be
converted back to the force using this equation. As the training data, we
prepared $N=332$ sets (83 original images) of corresponding two images. Note
that we increased the number of training data by rotating the images,
$83\times 4=332$.

### GAN structure

Our goal is to convert a physical quantity (wrinkle geometry) to another
physical quantity (force distribution). Even though the mechanical formulation
between the two quantities is given, it is not necessarily straightforward to
solve this inverse problem because of its complex nonlinear dynamics (36, 37).
Instead, we achieve this purpose by training our machine learning system to
understand the underlying mechanical rules. Considering this conversion of the
physical quantity as an image "translation", we utilize GAN (generative
adversarial network) (38) in this work.

GAN mainly consists of two networks, generator $G$ and discriminator $D$ as
shown in Fig. S1, and the network is trained by a competition of two networks.
Goal of the generator $G$ is to generate fake images (fake force distributions
$G(x)$) from the input images $x$ (microscope images or extracted wrinkle
images) and tries to mimic the real images $y$ (real force distributions),
while the discriminator $D$ tries to distinguish true and fake images from the
group of images. As the training proceeds, the generator learns how to produce
fake images that are difficult to be distinguished by the discriminator from
the real images, and the discriminator learns the rules to distinguish
true/fake images. Once the training is completed, the trained generator $G$
can now be used as the translator to predict the force distribution from the
input images $x$ even for test images, which were not included in the training
process.

In present work, we design the generator $G$ with a form of encoder-decoder
which is based on U-Net (35) but without the copy-and-crop path, and Markovian
discriminator (PatchGAN) (39) is utilized as the discriminator $D$. The
generator $G$ converts the input images $x$ to the fake force distribution
images $G(x)$. The images of force distribution, $y$ or $G(x)$, and the input
images $x$ are concatenated into a single image as the input of discriminator.
The two networks $G$ and $D$ are trained based on the labels of real/fake, and
we utilize the loss function $\mathcal{L}$ that is used in pix2pix (27):

$\mathcal{L}(G,D)=\mathbb{E}_{x,y}{[\log
D(x,y)]}+\mathbb{E}_{x}{[\log(1-D(x,G(x)))]}+\lambda L_{1}(y,G(x))$ (9)

where $\mathbb{E}$ is the expected value, $L_{1}(G)$ is the L1 distance
between generated images $G(x)$ and ground truths $y$ and $\lambda=100$ is the
weight for the L1 term. The first term $\mathbb{E}[\log D]$ denotes the
expected probability that the discriminator categorizes $y$ as the real data,
while the second term $\mathbb{E}[\log(1-D(x,G(x)))]$ denotes the probability
that the discriminator categorizes the generated image $G(x)$ as the fake
data. The goal of the generator $G$ is to minimize $\mathcal{L}$ while the
discriminator $D$ tries to maximize it.

We use the training parameters as follows: 100 training epochs (batch size =
1), $\varepsilon=0.0002$ learning rate for generator and discriminator, the
parameters $\beta_{1}=0.5$ and $\beta_{2}=0.9$ are used for the Adam
optimizer. The whole learning process is again accelerated by Nvidia Titan
RTX.

This work was supported by JSPS KAKENHI Grant Number 18H03518 and 20K14649,
ACT-X JST (Grant No. JPMJAX190S), and Multidisciplinary Research Laboratory
System for Future Developments (MIRAI LAB). AD acknowledges support from the
Novo Nordisk Foundation (grant no. NNF18SA0035142), Villum Fonden (grant no.
29476), Danish Council for Independent Research, Natural Sciences
(DFF-117155-1001), and funding from the European Union’s Horizon 2020 research
and innovation program under the Marie Sklodowska-Curie grant agreement no.
847523 (INTERACTIONS).

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Figure S1: Schematic showing the structure of GAN (generative adversarial
network) utilized in the present work.

## Supplementary Information

  * •

Figure S1

  * •

Movies S1-4

## Supplemental video captions

  * •

Movie S1: Sample of the force estimation using our system. The cell is MEF
(mouse embryonic fibroblast). The substrate is prepared by mixing parts A and
B of CY 52-276 with a weight ratio of 1.1:1.

  * •

Movie S2: Sample of the force estimation using our system. The conditions are
same as Movie 1.

  * •

Movie S3: Sample of the force estimation using our system. The conditions are
same as Movie 1.

  * •

Movie S4: Sample of the force estimation using our system. The conditions are
same as Movie 1.


# On a Van Kampen Theorem for Hawaiian Groups

Ameneh Babaee<EMAIL_ADDRESS>Behrooz Mashayekhy<EMAIL_ADDRESS>Hanieh
Mirebrahimi<EMAIL_ADDRESS>Hamid Torabi<EMAIL_ADDRESS>Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic
Structures, Ferdowsi University of Mashhad,
P.O.Box 1159-91775, Mashhad, Iran.

###### Abstract

The paper is devoted to study the $n$th Hawaiian group $\mathcal{H}_{n}$,
$n\geq 1$, of the wedge sum of two spaces
$(X,x_{*})=(X_{1},x_{1})\vee(X_{2},x_{2})$. Indeed, we are going to give some
versions of the van Kampen theorem for Hawaiian groups of the wedge sum of
spaces. First, among some results on Hawaiian groups of semilocally strongly
contractible spaces, we present a structure for the $n$th Hawaiian group of
the wedge sum of CW-complexes. Second, we give more informative structures for
the $n$th Hawaiian group of the wedge sum $X$, when $X$ is semilocally
$n$-simply connected at $x_{*}$. Finally, as a consequence, by generalizing
the well-known Griffiths space for dimension $n\geq 1$, we give some
information about the structure of Hawaiian groups of Griffiths spaces at any
points.

###### keywords:

Hawaiian group, Hawaiian earring, Van Kampen Theorem, Griffiths space.

###### MSC:

[2010]55Q05, 55Q20, 55P65, 55Q52.

††journal:

##  1 Introduction and Motivation

In 2000, K. Eda and K. Kawamura [4] generalized the well-known Hawaiian
earring space to higher dimensions $n\in\mathbb{N}$ as the following subspace
of $(n+1)$-dimensional Euclidean space ${\mathbb{R}}^{(n+1)}$

$\mathbb{HE}^{n}=\\{(r_{0},r_{1},...,r_{n})\in{\mathbb{R}}^{(n+1)}\ |\
(r_{0}-1/k)^{2}+{\sum}_{i=1}^{n}r_{i}^{2}=(1/k)^{2},k\in{\mathbb{N}}\\}.$

Here $\theta=(0,0,...,0)$ is regarded as the base point of the $n$-dimensional
Hawaiian earring $\mathbb{HE}^{n}$, and $\mathbb{S}_{k}^{n}$ denotes the
$n$-sphere in $\mathbb{HE}^{n}$ with radius 1/k.

In 2006, U.H. Karimov and D. Repovš [6] defined a new notion, the $n$th
Hawaiian group of a pointed space $(X,x_{0})$, denoted by
${\mathcal{H}}_{n}(X,x_{0})$, to be the set of all pointed homotopy classes
$[f]$, where $f:({\mathbb{HE}^{n}},\theta)\to(X,x_{0})$ is a pointed map. The
operation of the $n$th Hawaiian group comes naturally from the operation of
the $n$th homotopy group so that the following map

${\varphi}:{\mathcal{H}}_{n}(X,x_{0})\to\prod_{\mathbb{N}}{\pi}_{n}(X,x_{0}),$
(I)

defined by
$\varphi([f])=([f{\mid}_{\mathbb{S}_{1}^{n}}],[f{\mid}_{\mathbb{S}_{2}^{n}}],...)$
is a homomorphism, for all $n\in\mathbb{N}$. For every pointed space
$(X,x_{0})$, the image of $\varphi$ and also $\mathcal{H}_{n}(X,x_{0})$
contain $\prod^{W}_{\mathbb{N}}{\pi}_{n}(X,x_{0})$ as a normal subgroup (see
the proof of [1, Theorem 2.13]).

One can see that ${\mathcal{H}}_{n}:hTop_{\ast}\to Groups$ is a covariant
functor from the pointed homotopy category, $hTop_{\ast}$, to the category of
all groups, $Groups$, for any $n\geq 1$. If $\beta:(X,x_{0})\to(Y,y_{0})$ is a
pointed map, then
$\mathcal{H}_{n}(\beta)={\beta}_{\ast}:{\mathcal{H}}_{n}(X,x_{0})\to{\mathcal{H}}_{n}(Y,y_{0})$
defined by ${\beta}_{\ast}([f])=[\beta\circ f]$ is a homomorphism (see [6]).

Although the $n$th Hawaiian group functor is a pointed homotopy invariant
functor on the category of all pointed topological spaces, it is not freely
homotopy invariant. Because unlike other homotopy invariant functors, Hawaiian
groups of contractible spaces are not necessarily trivial. Karimov and Repovš
[6] gave a contractible space, the cone over $\mathbb{HE}^{1}$, with
nontrivial $1$th Hawaiian group at some points (consisting of the points at
which $C\mathbb{HE}^{1}$ is not locally $1$-simply connected), but with
trivial homotopy, homology and cohomology groups. More precisely, it can be
shown that ${\mathcal{H}}_{1}(C({\mathbb{HE}}^{1}),\theta)$ is uncountable,
using [6, Theorem 2]: “if $X$ is first countable at $x_{0}$, then countability
of $n$th Hawaiian group ${\mathcal{H}}_{n}(X,x_{0})$ implies locally
$n$-simply connectedness of $X$ at $x_{0}$.” Furthermore, a converse of the
above statement can be found in [1, Corollaries 2.16 and 2.17]: “let $X$ be
first countable at $x_{0}$. Then $\mathcal{H}_{n}(CX,\tilde{x})$ is trivial if
and only if $X$ is locally $n$-simply connected at $x_{0}$ and it is
uncountable otherwise”. Accordingly, this functor can help us to obtain some
local properties of spaces. In addition, unlike homotopy groups, Hawaiian
groups of pointed space $(X,x_{0})$ depend on the behaviour of $X$ at $x_{0}$,
and then their structures depend on the choice of the base point. In this
regard, there exist some examples of path connected spaces with non-isomorphic
Hawaiian groups at different points, such as the $n$-dimensional Hawaiian
earring, where $n\geq 2$ (see [1, Corollary 2.11]).

Despite the above different behaviors between Hawaiian groups and homotopy
groups, they have some similar behaviors. For instance, it was proved that
similar to the $n$th homotopy group, the $n$th Hawaiian group of any pointed
space is abelian, for $n\geq 2$ [1, Theorem 2.3]. Also, the Hawaiian groups
preserve products in the category $hTop_{*}$ [1, theorem 2.12].

In this paper, we investigate the Hawaiian groups of the coproduct in the
category $hTop_{*}$ which is the wedge sum of a given family of pointed
spaces. In fact, we are going to give some versions of the van Kampen theorem
for Hawaiian groups of the wedge sum of spaces. In Section 2, among some
results on Hawaiian groups of semilocally strongly contractible spaces, we
intend to present a structure for the $n$th Hawaiian group of the wedge sum of
CW-complexes. A space $X$ is called semilocally strongly contractible at
$x_{0}$ if there exists some open neighbourhood $U$ of $x_{0}$ such that the
inclusion $i:U\hookrightarrow X$ is nullhomotopic in $X$ relative to
$\\{x_{0}\\}$, in other words, $i:(U,x_{0})\hookrightarrow(X,x_{0})$ is
nullhomotopic (see [4]).

In Section 3, we present the $n$th Hawaiian group of the wedge sum
$(X,x_{*})=(X_{1},x_{1})\vee(X_{2},x_{2})$ as the semidirect product of two
its subgroups that are more perceptible, when $X$ is semilocally $n$-simply
connected at $x_{*}$. Also, we prove that the Hawaiian group of a pointed
space equals the Hawaiian group of every neighbourhood of the base point if
all $n$-loops are small. An $n$-loop $\alpha:(\mathbb{S}^{n},1)\to(X,x)$ is
called small, if for each neighbourhood $U$ of $x$, $\alpha$ has a homotopic
representative in $U$ (see [7, 10]).

In Section 4, generalizing the well-known Griffiths space, we define the $n$th
Griffiths space for $n\geq 2$, as the wedge sum of two copies of the cone over
the $n$-dimensional Hawaiian earring. For the sake of clarity, we call the
well known Griffiths space as the $1$th Griffiths space. Then, using results
of Sections 2 and 3, we intend to give some information about the structure of
Hawaiian groups of Griffiths spaces at any points.

In this paper, all homotopies are relative to the base point, unless stated
otherwise.

##  2 Hawaiian Groups of Semilocally Strongly Contractible Spaces

In this section, we investigate Hawaiian groups of wedge sum of pointed spaces
which are semilocally strongly contractible at the base points. The property
semilocally strongly contractibility was defined by Eda and Kawamura [4].
First, we compare semilocally strongly contractible property with some
familiar properties, such as locally contractible, locally strongly
contractible and semilocally contractible properties.

######  Recall 2.1.

Let $(X,x)$ be a pointed space, then $X$ is called locally contractible at $x$
if for each open neighbourhood $U$ of $x$ in $X$, there exists some open
neighbourhood $V$ of $x$ contained in $U$ such that the inclusion
$V\hookrightarrow U$ is freely nullhomotopic.

We say that $X$ is locally strongly contractible at $x$ if for each open
neighbourhood $U$ of $x$ in $X$, there exists some open neighbourhood $V$ of
$x$ contained in $U$ such that the inclusion $V\hookrightarrow U$ is
nullhomotopic relative to $\\{x\\}$, or briefly, $(V,x)\hookrightarrow(U,x)$
is nullhomotopic.

Moreover, $X$ is called semilocally contractible at $x$, if there exists some
open neighbourhood $U$ of $x$ such that the inclusion $i:U\hookrightarrow X$
is freely nullhomotopic in $X$.

One can see that there are some relations between these properties. Locally
strongly contractible property implies locally contractibility, and locally
contractible property implies semilocally contractibility, but the converse
statements do not necessarily hold. Moreover, locally strongly contractible
property implies semilocally strongly contractibility, and semilocally
strongly contractible implies semilocally contractibility, but converse
statements do not necessarily hold. The following example shows that a given
space can behave differently at different points.

######  Example 2.2.

The $1$th Griffiths space $\mathcal{G}_{1}$ is locally strongly contractible
at two vertices $v$ and $v^{\prime}$. Therefore, it is locally contractible,
semilocally strongly contractible and semilocally contractible at two vertices
$v$ and $v^{\prime}$. Moreover, $\mathcal{G}_{1}$ is semilocally contractible
at all points except at the common point $x_{*}$. Also, $\mathcal{G}_{1}$ is
semilocally contractible at any point of $A$ and $A^{\prime}$, but not
semilocally strongly contractible. Finally, $\mathcal{G}_{1}$ is neither
semilocally strongly contractible, semilocally contractible nor even
semilocally $1$-simply connected at $x_{*}$ (see [5]).

Figure 1: The $1$th Griffiths space.

The $1$-dimensional Hawaiian earring $\mathbb{HE}^{1}$ is not semilocally
contractible at the origin. However, it is locally strongly contractible at
other points. There exist many spaces which are semilocally contractible at
each point, but not strongly at some points. For example, consider $C\Delta$
as the cone over the Seirpeinski gasket. It is semilocally contractible at any
points, but it is semilocally strongly contractible just at the vertex.

The following lemma describes the Hawaiian group of the wedge sum of two
spaces by its subgroups. This construction helps us to obtain next
isomorphisms.

######  Lemma 2.3.

Let $(X_{1},x_{1})$ and $(X_{2},x_{2})$ be two pointed spaces, and
$(X,x_{*})=(X_{1},x_{1})\vee(X_{2},x_{2})$.

  1. (i)

If $U_{1}$ and $U_{2}$ are two open neighbourhoods of $x_{1}$ and $x_{2}$ in
$X_{1}$ and $X_{2}$, respectively, and $j:U_{1}\vee U_{2}\to X$ is the
inclusion map, then

$\mathcal{H}_{n}(X,x_{*})=j_{*}\mathcal{H}_{n}(U_{1}\vee
U_{2},x_{*}){\prod_{\mathbb{N}}}^{W}\pi_{n}(X,x_{*}).$ (1)

  2. (ii)

If $x\in X_{1}\setminus\\{x_{1}\\}$ and $\\{x_{1}\\}$ is closed in $X_{1}$,
then

$\mathcal{H}_{n}(X,x)=\mathcal{H}_{n}(X_{1},x){\prod_{\mathbb{N}}}^{W}\pi_{n}(X,x).$
(2)

###### Proof.

  1. (i)

Let $U_{1}$ and $U_{2}$ be two open neighbourhoods of $x_{1}$ and $x_{2}$ in
$X_{1}$ and $X_{2}$, respectively, $j:U_{1}\vee U_{2}\to X$ be the inclusion
map, and $f:(\mathbb{HE}^{n},\theta)\to(X,x_{*})$ be a pointed map. Since
$U_{1}\vee U_{2}$ is open in $X$, there exists $K\in\mathbb{N}$ such that if
$k\geq K$, then $im(f|_{\widetilde{\bigvee}_{k\geq
K}\mathbb{S}_{k}^{n}})\subseteq U_{1}\vee U_{2}$. We can define
$\underline{f},\overline{f}:\mathbb{HE}^{n}\to(X,x)$ by
$\underline{f}|_{\bigvee_{k<K}\mathbb{S}_{k}^{n}}=f|_{\bigvee_{k<K}\mathbb{S}_{k}^{n}}$,
$\underline{f}|_{\widetilde{\bigvee}_{k\geq
K}\mathbb{S}_{k}^{n}}=C|_{\widetilde{\bigvee}_{k\geq K}\mathbb{S}_{k}^{n}}$,
$\overline{f}|_{\bigvee_{k<K}\mathbb{S}_{k}^{n}}=C|_{\bigvee_{k<K}\mathbb{S}_{k}^{n}}$,
and $\overline{f}|_{\widetilde{\bigvee}_{k\geq
K}\mathbb{S}_{k}^{n}}=f|_{\widetilde{\bigvee}_{k\geq K}\mathbb{S}_{k}^{n}}$,
where $C$ is the constant map. Obviously,
$[f]=[\underline{f}][\overline{f}]=[\overline{f}][\underline{f}]$. Moreover,
$[\underline{f}]$ is an element of $\prod_{\mathbb{N}}^{W}\pi_{n}(X,x_{*})$
and also, $[\overline{f}]$ is an element of $j_{*}\mathcal{H}_{n}(U_{1}\vee
U_{2},x_{*})$. Therefore, $\mathcal{H}_{n}(X,x_{*})$ is generated by
$j_{*}\mathcal{H}_{n}(U_{1}\vee
U_{2},x_{*})\cup\prod_{\mathbb{N}}^{W}\pi_{n}(X,x_{*})$. The equality (1)
holds because
$\prod_{\mathbb{N}}^{W}\pi_{n}(X,x_{*})\unlhd\mathcal{H}_{n}(X,x_{*})$ (see
the proof of [1, Theorem 2.13]).

  2. (ii)

Since $X_{1}$ is a retract of $X$, one can consider $\mathcal{H}_{n}(X_{1},x)$
as a subgroup of $\mathcal{H}_{n}(X,x)$. Let
$f:(\mathbb{HE}^{n},\theta)\to(X,x)$ be a pointed map. Since $\\{x_{1}\\}$ is
closed in $X_{1}$, $X_{1}\setminus\\{x_{1}\\}$ is open. Let
$i:X_{1}\setminus\\{x_{1}\\}\to X$ be the inclusion map. Similar to the
previous part, $[f]$ can be factorized as
$[f]=[\underline{f}][\overline{f}]=[\overline{f}][\underline{f}]$, where
$[\underline{f}]$ is an element of $\prod_{\mathbb{N}}^{W}\pi_{n}(X,x)$ and
also $[\overline{f}]$ is an element of
$i_{*}\mathcal{H}_{n}(X_{1}\setminus\\{x_{1}\\},x)$ which is a subgroup of
$\mathcal{H}_{n}(X_{1},x)$. Thus, $\mathcal{H}_{n}(X,x)$ is generated by
$\mathcal{H}_{n}(X_{1},x)\cup\prod_{\mathbb{N}}^{W}\pi_{n}(X,x)$. Again,
equality (2) holds by normality of $\prod_{\mathbb{N}}^{W}\pi_{n}(X,x)$.

∎

Note that equalities (1) and (2) may not be the same. For instance, in Example
3.6 one group is trivial, but not the other one, in general.

######  Note 2.4.

Using the property that every open neighbourhood in the wedge sum is a wedge
sum of open neighbourhoods, one can rewrite the proof of Lemma 2.3 for
arbitrary spaces stated as follows. Let $(X,x_{0})$ be a pointed space. Then
$\mathcal{H}_{n}(X,x_{0})=j_{*}\mathcal{H}_{n}(U,x_{0})\prod_{\mathbb{N}}^{W}\pi_{n}(X,x_{0})$,
for each open neighbourhood $U$ with $x_{0}\in U\subseteq X$ and $j:U\to X$ as
the inclusion map.

A result similar to the following theorem was proved in [1, Theorem 2.5] by a
slightly different argument.

######  Theorem 2.5.

Let $(X_{1},x_{1})$ and $(X_{2},x_{2})$ be two pointed spaces,
$(X,x_{*})=(X_{1},x_{1})\vee(X_{2},x_{2})$, and $n\geq 1$. If $X_{1}$ and
$X_{2}$ are semilocally strongly contractible at $x_{1}$ and $x_{2}$,
respectively, then

$\mathcal{H}_{n}(X,x_{*})={\prod_{\mathbb{N}}}^{W}\pi_{n}(X,x_{*}).$

###### Proof.

Since $X_{1}$ and $X_{2}$ are semilocally strongly contractible at $x_{1}$ and
$x_{2}$, respectively, there exist open neighbourhoods $U_{1}$ of $x_{1}$ and
$U_{2}$ of $x_{2}$ such that inclusion maps
$(U_{1},x_{1})\hookrightarrow(X_{1},x_{1})$ and
$(U_{2},x_{2})\hookrightarrow(X_{2},x_{2})$ are nullhomotopic. By joining
these homotopies, one can see that $j:U_{1}\vee U_{2}\to X_{1}\vee X_{2}$ is
nullhomotopic and hence $j_{*}\mathcal{H}_{n}(U_{1}\vee U_{2},x_{*})$ is
trivial. The result holds by Lemma 2.3. ∎

The following example shows that Theorem 2.5 does not hold without condition
semilocally strongly contractible on both of spaces.

######  Example 2.6.

Consider $x_{*}$ as the common point of the $1$th Griffiths space. We show
that if Theorem 2.5 holds for the $1$th Griffiths space at $x_{*}$, then
$\pi_{1}(\mathcal{G}_{1},x_{*})$ is trivial which is a contradiction (see
[5]). Assume that
$\mathcal{H}_{1}(\mathcal{G}_{1},x_{*})=\prod_{\mathbb{N}}^{W}\pi_{1}(\mathcal{G}_{1},x_{*})$,
then the homomorphism
$\varphi:{\mathcal{H}}_{n}(X,x_{0})\to\prod_{\mathbb{N}}{\pi}_{n}(X,x_{0})$
(see (I)) can be considered as the natural injection. If we show that the
homomorphism $\varphi$ is surjective, then the natural injection
$\prod^{W}_{\mathbb{N}}\pi_{1}(\mathcal{G}_{1},x_{*})\to\prod_{\mathbb{N}}\pi_{1}(\mathcal{G}_{1},x_{*})$
is surjective which is impossible in a nontrivial way.

To prove surjectivity of $\varphi$, let
$\\{[f_{k}]\\}\in\prod_{\mathbb{N}}\pi_{1}(\mathcal{G}_{1},x_{*})$. Since any
$n$-loop at $x_{*}$ is small, we can find a homotopic representative
$f^{\prime}_{k}$ of $[f_{k}]$ in $U_{k}$, for all $k\in\mathbb{N}$. Now we can
define $f:\mathbb{HE}^{1}\to\mathcal{G}_{1}$ by
$f|_{\mathbb{S}_{k}^{1}}=f^{\prime}_{k}$, satisfying
$\varphi([f])=\\{[f_{k}]\\}$.

CW-complexes are locally homeomorphic to some cells. Thus, they are
semilocally strongly contractible at any point. The following result presents
an isomorphism for the Hawaiian group of the wedge sum of CW-complexes.

######  Corollary 2.7.

Let $X_{1}$ and $X_{2}$ be two locally finite $(n-1)$-connected CW-complexes,
$(X,x_{*})=(X_{1},x_{1})\vee(X_{2},x_{2})$, and $n\geq 2$. Then

$\mathcal{H}_{n}(X,x_{*})\cong\mathcal{H}_{n}(X_{1},x_{1})\oplus\mathcal{H}_{n}(X_{2},x_{2}).$
(3)

###### Proof.

CW-complexes are semilocally strongly contractible. Thus by Theorem 2.5,
$\mathcal{H}_{n}(X,x)\cong\bigoplus_{\mathbb{N}}\pi_{n}(X,x)$, when $n\geq 2$.
Now by [9, Proposition 6.36],
$\pi_{n}(X,x)\cong\pi_{n}(X_{1})\oplus\pi_{n}(X_{2})$, and after a
rearrangement,
$\mathcal{H}_{n}(X,x)\cong\bigoplus_{\mathbb{N}}\pi_{n}(X_{1})\oplus\bigoplus_{\mathbb{N}}\pi_{n}(X_{2})$.
We obtain the result, using [1, Theorem 2.5]. ∎

An analogous isomorphism for (3) does not hold, when $n=1$. To obtain such an
isomorphism on the $1$th Hawaiian group, we must replace direct sum by free
product, because the fundamental group and the $1$th Hawaiian group are not
abelian groups, in general. Using Theorem 2.5 and the van Kampen Theorem for
wedge sum we have the following result.

######  Corollary 2.8.

Let $X_{1}$ and $X_{2}$ be two semilocally strongly contractible spaces at
$x_{1}$ and $x_{2}$, respectively, and
$(X,x_{*})=(X_{1},x_{1})\vee(X_{2},x_{2})$. Then

$\mathcal{H}_{1}(X,x_{*})\cong{\prod_{\mathbb{N}}}^{W}\big{(}\pi_{1}(X_{1},x_{1})*\pi_{1}(X_{2},x_{2})\big{)}.$
(4)

Note that the isomorphism (4) is not similar to the case $n\geq 2$, even if
$X$ is a special CW-complex. Because if we consider
$\mathcal{H}_{1}(X,x_{*})\cong\mathcal{H}_{1}(X_{1},x_{1})*\mathcal{H}_{1}(X_{2},x_{2})$,
then by Theorem 2.5,
$\mathcal{H}_{1}(X,x_{*})\cong\prod^{W}_{\mathbb{N}}\pi_{1}(X_{1},x_{1})*\prod^{W}_{\mathbb{N}}\pi_{1}(X_{2},x_{2})$.
Hence, by isomorphism (4) we must have

${\prod_{\mathbb{N}}}^{W}\big{(}\pi_{1}(X_{1},x_{1})*\pi_{1}(X_{2},x_{2})\big{)}\cong{\prod_{\mathbb{N}}}^{W}\pi_{1}(X_{1},x_{1})*{\prod_{\mathbb{N}}}^{W}\pi_{1}(X_{2},x_{2}),$

which is impossible by [8, Page 183, 6.3.10], in a nontrivial way. Therefore,
$\mathcal{H}_{1}(X,x_{*})\not\cong\mathcal{H}_{1}(X_{1},x_{1})*\mathcal{H}_{1}(X_{2},x_{2})$,
in a nontrivial way.

##  3 Hawaiian Groups of Semilocally $n$-Simply Connected Spaces

In this section, we study more on Hawaiian groups of the wedge sum in
semilocally $n$-simply connected spaces. We present results for $n=1$ and
$n\geq 2$, separately, due to the difference in group structures.

Recall that for $n\geq 1$, a space $X$ is called $n$-simply connected at $x$
if $\pi_{n}(X,x)$ is trivial and it is called $n$-connected at $x$ if
$\pi_{j}(X,x)$ is trivial, for $1\leq j\leq n$. Also, $X$ is called
semilocally $n$-simply connected at $x$ if there exists a neighbourhood $U$ of
$x$ such that the homomorphism $\pi_{n}(j):\pi_{n}(U,x)\to\pi_{n}(X,x)$,
induced by the inclusion, is trivial.

######  Theorem 3.1.

Let $(X_{1},x_{1})$ and $(X_{2},x_{2})$ be two pointed spaces and
$(X,x_{*})=(X_{1},x_{1})\vee(X_{2},x_{2})$. If $X$ is semilocally $1$-simply
connected at $x_{*}$, then

$\mathcal{H}_{1}(X,x_{*})\cong j_{*}\mathcal{H}_{1}(U_{1}\vee
U_{2})\ltimes{\prod_{\mathbb{N}}}^{W}\pi_{1}(X,x_{*}),$

for some neighbourhood $U_{1}\vee U_{2}$ of $x_{*}$ with the inclusion map
$j:U_{1}\vee U_{2}\to X$.

###### Proof.

By Lemma 2.3 part 1, $\mathcal{H}_{1}(X,x_{*})=j_{*}\mathcal{H}_{1}(U_{1}\vee
U_{2},x_{*})\prod_{\mathbb{N}}^{W}\pi_{1}(X,x_{*})$ for each open
neighbourhood $U_{1}\vee U_{2}$ of $x_{*}$. Let $U_{1}\vee U_{2}$ be the
neighbourhood for which $\pi_{1}(j):\pi_{1}(U_{1}\vee
U_{2},x_{*})\to\pi_{1}(X,x_{*})$ is the trivial homomorphism and let $[f]\in
j_{*}\mathcal{H}_{1}(U_{1}\vee
U_{2},x_{*})\cap\prod_{\mathbb{N}}^{W}\pi_{1}(X,x_{*})$. Since $[f]\in
j_{*}\mathcal{H}_{1}(U_{1}\vee U_{2},x_{*})$, there exists
$\tilde{f}:(\mathbb{HE}^{1},\theta)\to(U_{1}\vee U_{2},x_{*})$ such that
$j_{*}[\tilde{f}]=[f]$, or equivalently, $j\circ\tilde{f}\simeq f$. Also,
since $[f]\in\prod_{\mathbb{N}}^{W}\pi_{1}(X,x_{*})$, $f$ can be assumed as a
map with $f|_{\widetilde{\bigvee}_{k\geq
K}\mathbb{S}_{k}^{1}}=C|_{\widetilde{\bigvee}_{k\geq K}\mathbb{S}_{k}^{1}}$
for some $K\in\mathbb{N}$. Hence $j\circ\tilde{f}|_{\widetilde{\bigvee}_{k\geq
K}\mathbb{S}_{k}^{1}}\simeq C|_{\widetilde{\bigvee}_{k\geq
K}\mathbb{S}_{k}^{1}}$. Using [1, Lemma 2.2], one can replace $\tilde{f}$ with
a map $\hat{f}$ such that $\hat{f}|_{\widetilde{\bigvee}_{k\geq
K}\mathbb{S}_{k}^{1}}\simeq C|_{\widetilde{\bigvee}_{k\geq
K}\mathbb{S}_{k}^{1}}$. Thus
$[\hat{f}]\in\prod_{\mathbb{N}}^{W}\pi_{1}(U_{1}\vee U_{2},x_{*})$ and then
$[f]\in\prod_{\mathbb{N}}^{W}\pi_{1}(j)\pi_{1}(U_{1}\vee U_{2},x_{*})$ which
is trivial, because of the choice of $U_{1}\vee U_{2}$. Hence
$j_{*}\mathcal{H}_{1}(U_{1}\vee
U_{2},x_{*})\cap\prod_{\mathbb{N}}^{W}\pi_{1}(X,x_{*})=\langle e\rangle$, and
the result holds by normality of $\prod_{\mathbb{N}}^{W}\pi_{1}(X,x_{*})$ in
$\mathcal{H}_{1}(X,x_{*})$. ∎

By a similar argument, we can conclude the following result for the wedge sum
of $1$-simply connected spaces.

######  Theorem 3.2.

Let $(X_{1},x_{1})$ and $(X_{2},x_{2})$ be two pointed spaces and
$(X,x_{*})=(X_{1},x_{1})\vee(X_{2},x_{2})$. If $\pi_{1}(X_{1},x)=\langle
e\rangle$ for some $x\in X_{1}\setminus\\{x_{1}\\}$, then

$\mathcal{H}_{1}(X,x)\cong\mathcal{H}_{1}(X_{1},x)\ltimes{\prod_{\mathbb{N}}}^{W}\pi_{1}(X,x).$

In the following two theorems, we reconstruct isomorphisms in Theorems 3.1 and
3.2 for $n\geq 2$. In this case, since all groups are abelian and all
subgroups are normal, semidirect product $\ltimes$ must be replaced by direct
sum $\oplus$.

######  Theorem 3.3.

Let $(X_{1},x_{1})$ and $(X_{2},x_{2})$ be two pointed spaces,
$(X,x_{*})=(X_{1},x_{1})\vee(X_{2},x_{2})$, and $n\geq 2$. If $X$ is
semilocally $n$-simply connected at $x_{*}$, then

$\mathcal{H}_{n}(X,x_{*})\cong j_{*}\mathcal{H}_{n}(U_{1}\vee
U_{2})\oplus\bigoplus_{\mathbb{N}}\pi_{n}(X,x_{*}),$

for some neighbourhood $U_{1}\vee U_{2}$ of $x_{*}$ with the inclusion map
$j:U_{1}\vee U_{2}\to X$.

######  Theorem 3.4.

Let $(X_{1},x_{1})$ and $(X_{2},x_{2})$ be two pointed spaces,
$(X,x_{*})=(X_{1},x_{1})\vee(X_{2},x_{2})$, and $n\geq 2$. If
$\pi_{n}(X_{1},x)=\langle e\rangle$ for some $x_{1}\neq x\in X_{1}$, then

$\mathcal{H}_{n}(X,x)\cong\mathcal{H}_{n}(X_{1},x)\oplus\bigoplus_{\mathbb{N}}\pi_{n}(X,x).$

Ž. Virk [10] defined small $1$-loop and studied small loop spaces. Note that a
nullhomotopic loop is a small loop. H. Passandideh and F.H. Ghane [7] defined
and studied the notions of $n$-homotopically Hausdorffness and small
$n$-loops, for $n\geq 2$. An $n$-loop $\alpha:(\mathbb{S}^{n},1)\to(X,x)$ is
called small if it has a homotopic representative in every open neighbourhood
of $x$.

######  Theorem 3.5.

Let $(X_{1},x_{1})$ and $(X_{2},x_{2})$ be two pointed spaces,
$(X,x_{*})=(X_{1},x_{1})\vee(X_{2},x_{2})$, and $n\geq 1$. Also, let
$\\{x_{1}\\}$ be a closed subset in $X_{1}$, and $x_{1}\neq x\in X_{1}$.

  1. (i)

All $n$-loops in $X$ at $x_{*}$ are small if and only if

$\mathcal{H}_{n}(X,x_{*})=j_{*}\mathcal{H}_{n}(U_{1}\vee U_{2},x_{*}),$ (5)

for any neighbourhoods $U_{1}$ and $U_{2}$ of $x_{1}$ and $x_{2}$ in $X_{1}$
and $X_{2}$, respectively, when $j:U_{1}\vee U_{2}\to X$ is the inclusion map.

  2. (ii)

If all $n$-loops in $X$ at $x$ are small, then

$\mathcal{H}_{n}(X,x)=\mathcal{H}_{n}(X_{1},x).$ (6)

###### Proof.

  1. (i)

Let $U_{1}$ and $U_{2}$ be two arbitrary neighbourhoods of $x_{1}$ and
$x_{2}$, respectively, and let $[f]\in\prod_{\mathbb{N}}^{W}\pi_{n}(X,x_{*})$.
We show that $[f]\in j_{*}\mathcal{H}_{n}(U_{1}\vee U_{2},x_{*})$, and then
the equality (5) is obtained by Lemma 2.3. Since
$[f]\in\prod_{\mathbb{N}}^{W}\pi_{n}(X,x_{*})$, one can consider $f$ as
$f|_{\widetilde{\bigvee}_{k\geq
K}\mathbb{S}_{k}^{n}}=C|_{\widetilde{\bigvee}_{k\geq K}\mathbb{S}_{k}^{n}}$
for some $K\in\mathbb{N}$. Moreover, since each $n$-loop in $X$ is small at
$x_{*}$, any $n$-loop $\alpha$ is homotopic to some $n$-loop in $U_{1}\vee
U_{2}$, say $\tilde{\alpha}:\mathbb{S}^{n}\to U_{1}\vee U_{2}$, such that
$j\circ\tilde{\alpha}\simeq\alpha$. By induction on finite join of $n$-loops,
$f|_{\widetilde{\bigvee}_{k<K}\mathbb{S}_{k}^{n}}$ has a homotopic
representative $\tilde{f}$ in $U_{1}\vee U_{2}$ with $j\circ\tilde{f}\simeq
f|_{\widetilde{\bigvee}_{k<K}\mathbb{S}_{k}^{n}}$. Therefore, $[f]\in
j_{*}\mathcal{H}_{n}(U_{1}\vee U_{2},x_{*})$.

Conversely, let $\alpha$ be an $n$-loop in $X$ at $x_{*}$. Consider the map
$f:(\mathbb{HE}^{n},\theta)\to(X,x_{*})$ so that
$f|_{\mathbb{S}_{1}^{n}}=\alpha$ and $f|_{\mathbb{S}_{k}^{n}}=c$ for $k>1$.
Then $[f]\in j_{*}\mathcal{H}_{n}(U_{1}\vee U_{2},x_{*})$ by equality (5) for
any neighbourhoods $U_{1}$ and $U_{2}$. Let $[\tilde{f}]$ be the element of
$\mathcal{H}_{n}(U_{1}\vee U_{2},x_{*})$ such that $j\circ\tilde{f}=f$. Hence,
$j\circ\tilde{f}|_{\mathbb{S}_{1}^{n}}=f|_{\mathbb{S}_{1}^{n}}=\alpha$, that
is $\alpha$ is homotopic to some $n$-loop in $U_{1}\vee U_{2}$. Since $U_{1}$
and $U_{2}$ are arbitrary neighbourhoods, $\alpha$ is a small $n$-loop.

  2. (ii)

Since $X_{1}\setminus\\{x_{1}\\}$ is open and all $n$-loops at $x$ in $X$ are
small, similar to the proof of the previous part, one can show that
$\prod_{\mathbb{N}}^{W}\pi_{n}(X,x)\subseteq
i_{*}\mathcal{H}_{n}(X_{1}\setminus\\{x_{1}\\},x)$, where
$i:X_{1}\setminus\\{x_{1}\\}\to X$ is the inclusion map. Moreover,
$i_{*}\mathcal{H}_{n}(X_{1}\setminus\\{x_{1}\\},x)$ is contained in
$\mathcal{H}_{n}(X_{1},x)$ as a subgroup. Thus, by Lemma 2.3, the equality (6)
holds.

∎

Since a nullhomotopic $n$-loop is a special case of small $n$-loop, the
equalities (5) and (6) hold for $n$-simply connected spaces. For example,
Theorem 3.5 holds for two cones joining at their vertices which is not only
$n$-simply connected, but also contractible. Recall that the $1$th Griffiths
space, the wedge sum of two cones, is not contractible, even more Griffiths
[5] proved that it is not $1$-simply connected.

######  Example 3.6.

For a space $X$, put $Y=\frac{X\times[-1,1]}{X\times\\{0\\}}$ the wedge sum of
two cones over $X$ at their vertices and $\tilde{x}_{t}=[(x_{0},t)]$ for
$t\neq 0$. One can see that $Y$ is contractible at the common point, and
hence, it is $n$-simply connected. By Theorem 3.5,
$\mathcal{H}_{n}(Y,\tilde{x}_{t})=\mathcal{H}_{n}(CX,\tilde{x}_{t})$. Also, by
[1, Theorem 2.13],
$\mathcal{H}_{n}(Y,x_{0})\cong\frac{\mathcal{H}_{n}(X,x_{0})}{\prod^{W}_{k\in\mathbb{N}}\pi_{n}(X,x_{0})}$.

Let $x_{*}$ be the common vertex of the two cones, then
$\mathcal{H}_{n}(Y,x_{*})$ is trivial, for $Y$ is semilocally strongly
contractible at $x_{*}$ and by Theorem 2.5,
$\mathcal{H}_{n}(Y,x_{*})=\prod_{\mathbb{N}}^{W}\pi_{n}(Y,x_{*})$ which is
trivial.

The following example reveals that Theorem 3.5 does not hold, if some non
small loop exists.

######  Example 3.7.

Let $(X,x_{*})=(C(\mathbb{HE}^{1}),\theta)\vee(\mathbb{S}^{1},1)$ (see Figure
2) . If one assumes that
$\mathcal{H}_{1}(X,x_{*})=\mathcal{H}_{1}(C(\mathbb{HE}^{1}),x_{*})$, then the
simple $1$-loop in $\mathbb{S}^{1}$ must be nullhomotopic which is a
contradiction.

Figure 2: The wedge sum of a circle and a cone on the Hawaiian earring

##  4 Hawaiian Groups of Griffiths Spaces

In this section, by generalizing the Griffiths space to higher dimensions and
applying the results of Sections 2 and 3, we study the $n$th Hawaiian group of
the $n$th Griffiths space, for $n\geq 1$.

Eda [3] introduced the free $\sigma$-product
$\times^{\sigma}_{\mathbb{N}}\mathbb{Z}$ as the group consisting of all
reduced $\sigma$-words, and then proved that it is isomorphic to
$\pi_{1}(\mathbb{HE}^{1},\theta)$ [3, Theorem A.1]. To prove, Eda remarked
that each $1$-loop in the $1$-dimensional Hawaiian earring is homotopic to
some proper $1$-loop [3, Lemma A.3]. A $1$-loop
$\alpha:(\mathbb{I},\dot{\mathbb{I}})\to(\mathbb{HE}^{1},\theta)$ is called
proper whenever for each subinterval $[a,b]\subseteq\mathbb{I}$, if
$\alpha|_{[a,b]}$ is nullhomotopic, then it is constant.

Also, O. Bogopolski and A. Zastrow proved that

$\pi_{1}(\mathcal{G}_{1},a)\cong\frac{\times^{\sigma}_{\mathbb{N}}\mathbb{Z}}{\overline{\langle\times^{\sigma}_{\mathbb{N}_{e}}\mathbb{Z},\times^{\sigma}_{\mathbb{N}_{o}}\mathbb{Z}\rangle}^{N}},$

where $\mathbb{N}_{o}$ and $\mathbb{N}_{e}$ denote the set of odd and even
numbers, respectively [2, Theorem 3.4]. This isomorphism is induced by the
natural embedding $\iota:\mathbb{HE}^{1}\to\mathcal{G}_{1}$ causing
epimorphism
$\pi_{1}(\iota):\times^{\sigma}_{\mathbb{N}}\mathbb{Z}\to\pi_{1}(\mathcal{G}_{1},x_{*})$
together with
$\overline{\langle\times^{\sigma}_{\mathbb{N}_{e}}\mathbb{Z},\times^{\sigma}_{\mathbb{N}_{o}}\mathbb{Z}\rangle}^{N}$
as its kernel.

The following lemma gives a useful description of the group
$\mathcal{H}_{1}(\mathbb{HE}^{1},\theta)$ which is used in sequel.

######  Lemma 4.1.

Let $\mathcal{B}$ be the subgroup of
$\prod_{\mathbb{N}}\times^{\sigma}_{\mathbb{N}}\mathbb{Z}$ consisting of all
countably infinite tuples of reduced $\sigma$-words that the number of
components including letter of type $m$ is finite, for all $m\in\mathbb{N}$.
Then

$\mathcal{H}_{1}(\mathbb{HE}^{1},\theta)\cong\mathcal{B}.$

###### Proof.

By [1, Theorem 2.9], $\mathcal{H}_{1}(\mathbb{HE}^{1},\theta)$ is isomorphic
to the subgroup of $\prod_{\mathbb{N}}\pi_{1}(\mathbb{HE}^{1},\theta)$
consisting of all sequences of homotopy classes of $1$-loops with some
representative converging uniformly to the constant $1$-loop. We show that
$\mathcal{B}$ equals this subgroup.

Let $\\{U_{l};\ l\in\mathbb{N}\\}$ be the local basis at $\theta$ defined in
the proof of [1, Theorem 2.9], and also let
$\\{[f_{k}]\\}\in\prod_{\mathbb{N}}\pi_{1}(\mathbb{HE}^{1},\theta)$. Then
$\\{f_{k}\\}$ converges uniformly to the constant $1$-loop if and only if for
each $l\in\mathbb{N}$, there exists $K_{l}\in\mathbb{N}$ such that
$im(f_{k})\subseteq U_{l}$ whenever $k\geq K_{l}$. Assume that $f_{k}$ is the
corresponding proper representative for all $k\in\mathbb{N}$. The image of
$f_{k}$ is contained in $U_{l}$ if and only if
$im(f_{k})\cap\mathbb{S}_{m}^{1}=\\{\theta\\}$, where $(m<l)$, for
$U\cap\mathbb{S}^{1}_{m}$ is contractible and $f_{k}$ has no trivial subpath.
Therefore, $\\{f_{k}\\}$ converges uniformly to the constant $1$-loop if and
only if for each $l\in\mathbb{N}$, there exists $K_{l}\in\mathbb{N}$ such that
$im(f_{k})\cap\mathbb{S}_{m}^{1}=\\{\theta\\}$ whenever $k\geq K_{l}$.

By [3, Theorem A.1],
$\pi_{1}(\mathbb{HE}^{1},\theta)\cong\times^{\sigma}_{\mathbb{N}}\mathbb{Z}$,
the group of reduced $\sigma$-words. Moreover, in a given reduced
$\sigma$-word, the letter of type $m$ exists if and only if the $m$th circle
$\mathbb{S}_{m}^{1}$ of $\mathbb{HE}^{1}$ appears in the corresponding proper
$1$-loop. Therefore $\mathcal{H}_{1}(\mathbb{HE}^{1},\theta)$ is isomorphic to
the subgroup of $\prod_{\mathbb{N}}\times^{\sigma}_{\mathbb{N}}\mathbb{Z}$
consisting of all countably infinite tuples of reduced $\sigma$-words such
that the number of components including letter of type $m$ is finite, for all
$m\in\mathbb{N}$. ∎

The following theorem investigates the structure of the $1$th Hawaiian group
of the $1$th Griffiths space at the common point, the two vertices and the
other points. Let
$i:\mathbb{HE}^{1}\to\mathcal{G}_{1}\setminus\\{v_{1},v_{2}\\}$ be the
embedding which maps $(2m-1)$th circle onto the the horizontal left $m$th
circle and maps $2m$th circle onto the horizontal right $m$th circle, for
$m\in\mathbb{N}$ (see Figure 1). Also, let
$j:\mathcal{G}_{1}\setminus\\{v_{1},v_{2}\\}\to\mathcal{G}$ be the inclusion
map .

######  Theorem 4.2.

Let $\mathcal{G}_{1}$ be the $1$th Griffiths space, $x_{*}$ the common point,
$a\in A\cup A^{\prime}$, and $x\in\mathcal{G}_{1}\setminus(A\cup
A^{\prime}\cup\\{x_{*}\\})$. Then

$\displaystyle\mathrm{(i)}$
$\displaystyle\mathcal{H}_{1}(\mathcal{G}_{1},x_{*})$
$\displaystyle\cong\frac{\mathcal{H}_{1}(\mathbb{HE}^{1},\theta)}{i_{*}^{-1}\ker
j_{*}}\cong\frac{\mathcal{B}}{\prod_{\mathbb{N}}^{W}\overline{\langle\times^{\sigma}_{\mathbb{N}_{e}}\mathbb{Z},\times^{\sigma}_{\mathbb{N}_{o}}\mathbb{Z}\rangle}^{N}},$
(7) $\displaystyle\mathrm{(ii)}$
$\displaystyle\mathcal{H}_{1}(\mathcal{G}_{1},a)$
$\displaystyle\cong\frac{{\mathcal{H}}_{1}(\mathbb{HE}^{1},\theta)}{\prod_{\mathbb{N}}^{W}{\pi}_{1}(\mathbb{HE}^{1},\theta)}\ltimes{\prod_{\mathbb{N}}}^{W}\pi_{1}(\mathcal{G}_{1},a)\cong\frac{\mathcal{B}}{\prod^{W}_{\mathbb{N}}\times^{\sigma}_{\mathbb{N}}\mathbb{Z}}\ltimes{\prod_{\mathbb{N}}}^{W}\frac{\times^{\sigma}_{\mathbb{N}}\mathbb{Z}}{\overline{\langle\times^{\sigma}_{\mathbb{N}_{e}}\mathbb{Z},\times^{\sigma}_{\mathbb{N}_{o}}\mathbb{Z}\rangle}^{N}},$
(8) $\displaystyle\mathrm{(iii)}$
$\displaystyle\mathcal{H}_{1}(\mathcal{G}_{1},x)$
$\displaystyle\cong{\prod_{\mathbb{N}}}^{W}\frac{\times^{\sigma}_{\mathbb{N}}\mathbb{Z}}{\overline{\langle\times^{\sigma}_{\mathbb{N}_{e}}\mathbb{Z},\times^{\sigma}_{\mathbb{N}_{o}}\mathbb{Z}\rangle}^{N}}.$
(9)

###### Proof.

  1. (i)

Since all $1$-loops in $\mathcal{G}_{1}$ at $x_{*}$ are small, Theorem 3.5
implies that
$\mathcal{H}_{1}(\mathcal{G}_{1},x_{*})=j_{*}\mathcal{H}_{1}(U_{1}\vee
U_{2},x_{*})$ when $U_{1}$ and $U_{2}$ are arbitrary neighbourhoods of $x_{*}$
in the two cones. Suppose that $U_{m}$ $(m=1,2)$ is the whole of the
corresponding cone except its vertex. Then $U_{1}\vee
U_{2}=\mathcal{G}_{1}\setminus\\{v_{1},v_{2}\\}$ and thus

$\mathcal{H}_{1}(\mathcal{G}_{1},x_{*})=j_{*}\mathcal{H}_{1}(U_{1}\vee
U_{2},x_{*})=j_{*}\mathcal{H}_{1}(\mathcal{G}_{1}\setminus\\{v_{1},v_{2}\\},x_{*}).$

Moreover, the embedded Hawaiian earring $i(\mathbb{HE}^{1})$ is a deformation
retract of $\mathcal{G}_{1}\setminus\\{v_{1},v_{2}\\}$ with projection
$p:\mathcal{G}_{1}\setminus\\{v_{1},v_{2}\\}\to\mathbb{HE}^{1}$ as the
retraction. Therefore,

$\mathcal{H}_{1}(\mathcal{G}_{1},x_{*})=j_{*}\mathcal{H}_{1}(\mathcal{G}_{1}\setminus\\{v_{1},v_{2}\\},x_{*})\cong
j_{*}i_{*}\mathcal{H}_{1}(\mathbb{HE}^{1},\theta).$

By the First Isomorphism Theorem,

$j_{*}i_{*}\mathcal{H}_{1}(\mathbb{HE}^{1},\theta)\cong\frac{i_{*}\mathcal{H}_{1}(\mathbb{HE}^{1},\theta)}{\ker
j_{*}}\cong\frac{\mathcal{H}_{1}(\mathbb{HE}^{1},\theta)}{i_{*}^{-1}\ker
j_{*}}.$

In the following, we show that $i_{*}^{-1}\ker j_{*}$ is mapped isomorphically
onto
$\prod_{\mathbb{N}}^{W}\overline{\langle\times^{\sigma}_{\mathbb{N}_{e}}\mathbb{Z},\times^{\sigma}_{\mathbb{N}_{o}}\mathbb{Z}\rangle}^{N}$,
by the same isomorphism which maps $\mathcal{H}_{1}(\mathbb{HE}^{1},\theta)$
onto $\mathcal{B}$, in Lemma 4.1.

Let $[g]\in i_{*}^{-1}\ker j_{*}$. Then $i_{*}[g]\in\ker j_{*}$, and
therefore, $j\circ i\circ g\simeq C_{x_{*}}$. Hence, the $1$-loops $(j\circ
i\circ g|_{S_{k}^{1}})$’s are nullhomotopic with some null convergent sequence
of homotopies, say $\\{H_{k}\\}$. Thus, there exists $K\in\mathbb{N}$ such
that $imH_{k}\subseteq\mathcal{G}_{1}\setminus\\{v_{1},v_{2}\\}$ for $k\geq
K$. Therefore, $i\circ g|_{\mathbb{S}^{1}_{k}}$ is nullhomotopic in
$\mathcal{G}_{1}\setminus\\{v_{1},v_{2}\\}$ for $k\geq K$ by null convergent
homotopies $\\{H_{k}\\}_{k\geq K}$. Accordingly, $i\circ
g|_{\widetilde{\bigvee}_{k\geq K}\mathbb{S}^{1}_{k}}$ is nullhomotpic in
$\mathcal{G}\setminus\\{v_{1},v_{2}\\}$. Therefore, $p\circ i\circ
g|_{\widetilde{\bigvee}_{k\geq
K}\mathbb{S}^{1}_{k}}=g|_{\widetilde{\bigvee}_{k\geq K}\mathbb{S}^{1}_{k}}$ is
nullhomotopic in $\mathbb{HE}^{1}$.

Moreover, for $k<K$, $j\circ i\circ g|_{S^{1}_{k}}$ is nullhomotopic in
$\mathcal{G}_{1}$ or equivalently $[g|_{S_{k}^{1}}]\in\ker\pi_{1}(\iota)$.
Note that $\ker\pi_{1}(\iota)$ is mapped isomorphically onto
$\overline{\langle\times^{\sigma}_{\mathbb{N}_{e}}\mathbb{Z},\times^{\sigma}_{\mathbb{N}_{o}}\mathbb{Z}\rangle}^{N}$,
by the same isomorphism mapping $\pi_{1}(\mathbb{HE},\theta)$ onto
$\times^{\sigma}_{\mathbb{N}}\mathbb{Z}$. Thus, $[g]$ is corresponded
injectively to an element of
$\prod_{\mathbb{N}}^{W}\overline{\langle\times^{\sigma}_{\mathbb{N}_{e}}\mathbb{Z},\times^{\sigma}_{\mathbb{N}_{o}}\mathbb{Z}\rangle}^{N}$,
where the corresponding is the same as the isomorphism mapping
$\mathcal{H}_{1}(\mathbb{HE},\theta)$ onto $\mathcal{B}$, in Lemma 4.1. One
can check that this correspondence is also surjective.

  2. (ii)

By Theorem 3.2,

$\mathcal{H}_{1}(\mathcal{G}_{1},a)\cong\mathcal{H}_{1}(C\mathbb{HE}^{1},a)\ltimes{\prod_{\mathbb{N}}}^{W}\pi_{1}(\mathcal{G}_{1},a)\cong\mathcal{H}_{1}(C\mathbb{HE}^{1},a)\ltimes{\prod_{\mathbb{N}}}^{W}\frac{\times^{\sigma}_{\mathbb{N}}\mathbb{Z}}{\overline{\langle\times^{\sigma}_{\mathbb{N}_{e}}\mathbb{Z},\times^{\sigma}_{\mathbb{N}_{o}}\mathbb{Z}\rangle}^{N}}.$

Now by [1, Theorem 2.13]
$\mathcal{H}_{1}(C\mathbb{HE}^{1},a)\cong\frac{{\mathcal{H}}_{1}(\mathbb{HE}^{1},\theta)}{\prod_{\mathbb{N}}^{W}{\pi}_{1}(\mathbb{HE}^{1},\theta)}$.

By Lemma 4.1, $\mathcal{H}_{1}(\mathbb{HE}^{1},\theta)\cong\mathcal{B}$. This
isomorphism maps the subgroup
$\prod_{\mathbb{N}}^{W}{\pi}_{1}(\mathbb{HE}^{1},\theta)$ onto
$\prod^{W}_{\mathbb{N}}\times^{\sigma}_{\mathbb{N}}\mathbb{Z}$. Consequently
$\mathcal{H}_{1}(C\mathbb{HE}^{1},a)\cong\frac{\mathcal{B}}{\prod^{W}_{\mathbb{N}}\times^{\sigma}_{\mathbb{N}}\mathbb{Z}}$,
and hence the isomorphism (8) holds.

  3. (iii)

Obviously, $\mathcal{G}_{1}$ is semilocally strongly contractible at $x$.
Therefore,
$\mathcal{H}_{1}(\mathcal{G}_{1},x)\cong\prod_{\mathbb{N}}^{W}\pi_{1}(\mathcal{G}_{1},x)$.

∎

We define the $n$th Griffiths space by the wedge sum of two copies of cones on
$\mathbb{HE}^{n}$ at the origin for $n\geq 2$. In the following theorem we
give some information on the structure of the $n$th Hawaiian group of the
$n$th Griffiths space. Note that by $\mathbb{Z}^{k}$ we mean the subgroup of
$\bigoplus_{\mathbb{N}}\mathbb{Z}$ consisting of all countably infinite tuples
with all zero components except the first $k$ components and thus
$\mathbb{Z}^{1}\leqslant\mathbb{Z}^{2}\leqslant\mathbb{Z}^{3}\leqslant\cdots$.

######  Theorem 4.3.

Let $n\geq 2$ and $\mathcal{G}_{n}$ be the $n$th Griffiths space, $a\in A\cup
A^{\prime}$, and $x\in\mathcal{G}_{1}\setminus(A\cup
A^{\prime}\cup\\{x_{*}\\})$. Then

$\displaystyle\mathrm{(i)}$ $\displaystyle\mathcal{H}_{n}(\mathcal{G}_{n},a)$
$\displaystyle\cong\frac{\mathcal{H}_{n}(\mathbb{HE}^{n},\theta)}{\bigoplus_{\mathbb{N}}\pi_{n}(\mathbb{HE}^{n},\theta)}\oplus\bigoplus_{\mathbb{N}}\pi_{n}(\mathcal{G}_{n})\cong\frac{\prod_{\mathbb{N}}\bigoplus_{\mathbb{N}}\mathbb{Z}}{\bigcup_{k\in\mathbb{N}}\prod_{\mathbb{N}}\mathbb{Z}^{k}}\oplus\bigoplus_{\mathbb{N}}\pi_{n}(\mathcal{G}_{n})$
$\displaystyle\mathrm{(ii)}$ $\displaystyle\mathcal{H}_{n}(\mathcal{G}_{n},x)$
$\displaystyle\cong\bigoplus_{\mathbb{N}}\pi_{n}(\mathcal{G}_{n}).$

###### Proof.

  1. (i)

By Theorem 3.4,
$\mathcal{H}_{n}(\mathcal{G}_{n},a)\cong\mathcal{H}_{n}(C\mathbb{HE}^{n},a)\oplus\bigoplus_{\mathbb{N}}\pi_{n}(\mathcal{G}_{n},a)$.
Using [1, Theorem 2.13],
$\mathcal{H}_{n}(C\mathbb{HE}^{n},a)\cong\frac{{\mathcal{H}}_{n}(\mathbb{HE}^{n},\theta)}{\bigoplus_{\mathbb{N}}\pi_{n}(\mathbb{HE}^{n},\theta)}$.
Replacing $\mathcal{H}_{n}(\mathbb{HE}^{n},\theta)$ with
$\prod_{\mathbb{N}}\bigoplus_{\mathbb{N}}\mathbb{Z}$ via [1, Theorem 2.11],
which maps $\bigoplus_{\mathbb{N}}\pi_{n}(\mathbb{HE}^{n},\theta)$
isomorphically onto
$\bigcup_{k\in\mathbb{N}}\prod_{\mathbb{N}}\mathbb{Z}^{k}$, the result holds.

  2. (ii)

Since $\mathcal{G}_{n}$ is semilocally strongly contractible at $x$, the
isomorphism holds by Theorem 2.5.

∎

We recall that the $n$th Hawaiian group of the $n$th Griffiths space at the
common point is generated by two its subgroups;
$\mathcal{H}_{n}(\mathcal{G}_{n},x_{*})=j_{*}\mathcal{H}_{n}(U_{1}\vee
U_{2},x_{*})\big{(}\bigoplus_{\mathbb{N}}\pi_{n}(\mathcal{G}_{n},x_{*})\big{)}$,
when $U_{1}$ and $U_{2}$ are arbitrary neighbourhoods of the origin in two
cones, and $j:U_{1}\vee U_{2}\to\mathcal{G}_{n}$ is the inclusion map.

## Acknowledgements

This research was supported by a grant from Ferdowsi University of Mashhad-
Graduate Studies (No. 42705).

## References

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