# 6G Communications: A Vision on the Potential Applications

Sabuzima Nayak and Ripon Patgiri,
National Institute of Technology Silchar Sabuzima Nayak and Ripon Patgiri,
Department of Computer Science & Engineering, National Institute of Technology
Silchar, Cachar-788010, Assam, India, Email<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
###### Abstract

6G communication technology is a revolutionary technology that will
revolutionize many technologies and applications. Furthermore, it will be
truly AI-driven and will carry on intelligent space. Hence, it will enable
Internet of Everything (IoE) which will also impact many technologies and
applications. 6G communication technology promises high Quality of Services
(QoS) and high Quality of Experiences (QoE). With the combination of IoE and
6G communication technology, number of applications will be exploded in the
coming future, particularly, vehicles, drones, homes, cities, hospitals, and
so on, and there will be no untouched area. Thence, it is expected that many
existing technologies will fully depend on 6G communication technology and
enhance their performances. 6G communication technology will prove as game
changer communication technology in many fields and will be capable to
influence many applications. Therefore, we envision the potential applications
of 6G communication technology in the near future.

###### Index Terms:

6G Communications, Networking, Wireless Communication, Healthcare, Vehicular
Technology, Virtual Communications, Robotics Communications, Internet of
Things, Internet of Everything, Industrial Internet of Things, Industrial
Internet of Everything.

6G mobile communication technology is one of the most prominent emerging
research area and radical technology, which will change our perception on
lifestyle, society and business. With the advent of communication technology,
5G is maturing, and thusly, it demands next generation mobile communication
technology. Since, we have already evidence that every decade there is a
mobile generation, so, it is expected that 5G will serve from 2020 to 2030, 6G
will serve from 2030 to 2040, and 7G will serve from 2040 to 2050. Currently,
5G is deployed in few countries. 5G communication technology is also able
impact on the global economy. It is anticipated that the next generation of 5G
will be able to create more impact on the global economy. Withal, the
requirements of 6G are extremely different from 5G communications [1].
Consequently, the parameters of 6G will enable tremendous new applications and
technologies [2] as depicted in Figure 1. For instance, 6G will be a prominent
player in vehicular technology, healthcare and commercial enterprise. Also, 6G
communication technology will be able to influence many researchers,
practitioners and industrialists. Hence, there are diverse research articles
already discussed on parameters of 6G [3, 4, 5, 6, 4]. Moreover, Q. Bi
highlighted ten trends of 6G communications [7]. Kato et. al. exposes ten
great challenges on integration AI with 6G communication technology. Letaief
et. al. envisioned the truly AI-driven 6G communication technology [8].

Figure 1: 6G communication technology enables Internet of Everything (IoE).
The landscape of Applications of IoE using 6G communication technology. It
depicts future dependability of intelligent vehicles, drones, transportation,
hospital, agriculture, trains, ships, and industry on 6G communication
technology. 6G will not only a smart phone communication anymore, but also a
default communication system for many devices and technologies.

The promises of 6G communication technology create numerous issues and
challenges [9, 10], albeit, worldwide deployment is expected from 2030.
Moreover, rural deployment of such technology is a grand challenge [11].
Currently, 5G communication technology focuses on campus solution, and
thereby, the mobility and coverage are issues. Therefore, 5G is unable to
support many applications. Nonetheless, the 6G communication technology
promises increase coverage and mobility using satellite communication.
Presently, we are lacking $\geq 1$ Tbps data rates and extremely reliable and
low latency, and this triggers compromising in QoS. As per the requirements
and promises of 6G communications, there are numerous research possibilities
and opportunities which will endure many new applications to be conceived.
Thus, 6G will bloom the global economy, society and lives.

In the 6G communication era, many existing technologies will be redefined and
restructured. Reshaping existing technology will shift our lifestyle, society
and business. Prominently, Internet of Things (IoT) will be redefined as
Internet of Everything (IoE) which will be the dawn of many novel
technologies. IoE will be a key player in the 6G communication era, and it
will enable intelligent city, intelligent healthcare (Intelligent Wearable
devices and Intelligent Internet of Medical Things), intelligent Industrial
Internet of Everything (IIIoE), intelligent grid, and Intelligent Robots. We
also foresee that the 6G communication technology will multiply in spawning
many new applications. Therefore, in this article, we envision the impact of
6G communication technology in various fields. Thus, Section I briefs on 6G
technology to highlight the key features. Besides, Section II explores the
enabling technology of 6G communications. Sections III, IV, V, VI and VII
exposes the impact of 6G communications technology on Vehicular Technology,
Robotics, Virtual-reality, Healthcare and City. Likewise, Section VIII exposes
on how 6G communication technology will be able to enable an Industrial
revolution. Furthermore, Sections IX and X exposes that the nano-things and
datacenters depends on the 6G communication technology. Finally, Section XI
draws a suitable conclusion.

##  I 6G Technology

6G communication technology has a potential to change the definition of our
perception of life, lifestyle, society and business. 6G communication is all
about sixth sense communication [12]. Therefore, many countries have already
begun a project on 6G communication technology, particularly, Finland, USA,
South Korea, China [9] and Japan [13]. Its prerequisites are very challenging
and numerous requirements have been already discussed through various articles
[3, 14, 15, 16, 17]. Particularly, 6G communication technology will operate at
(THz) frequency to gain 1 Tbps data rate [18, 19, 20, 21]. However, 1 Tbps
data rate is not enough for holographic communications. It demands more data
rate to provide full holographic communication support. The wavelength
frequency is expected to be $300\mu m$ to $100\mu m$. Researchers are
exploring to increase the efficiency of THz signal by spectrum reusing and
sharing. Some techniques already exist for spectrum reuse such as cognitive
radio (CR). It helps many wireless systems to access the same spectrum through
a spectrum sensing and interference management mechanisms [6]. In case of
spectrum sharing, temporally underutilized or unlicensed spectrum is utilized
to maintain availability and reliability. Symbiotic radio (SR) is a new
technique to support intelligent, heterogeneous wireless networks. It will aid
in efficient spectrum sharing. Even so, deploying these techniques in the 6G
wireless network are still big challenges. Also, 6G is expected to deliver
truly AI-driven communication technology [15, 18, 22, 23]. In addition, 6G
promises to provide high security, secrecy and privacy [9]. Likewise, it will
be three dimensional communication technology, particularly, time, space and
frequency.

###  I-A Transition from Smart to Intelligent

The IoE will be intelligent, thereby, 6G will also be intelligent.
Consequently, all smart devices will be converted to intelligent devices and
these intelligent devices will be truly AI-driven devices [24]. Thusly, the
intelligent device (may be tiny device) will be able to predict, make a
decision and share their experience with other intelligent devices. So, there
is a paradigm shift from smart to intelligent era using 6G communication
technology and AI.

###  I-B Quality of Services

6G Technology promises to provide high Quality of Services (QoS) and the
parameters includes high data rate of at least 1 Tbps, extremely reliable,
further-enhanced mobile broadband (FeMBB), low latency communication (ERLLC),
long-distance and high-mobility communications (LDHMC) ultra-massive machine-
type communications (umMTC) and extremely low-power communications (ELPC)
[25]. Moreover, QoS includes massive broad bandwidth machine type (mBBMT),
mobile broad bandwidth and low latency (MBBLL), and massive low latency
machine type (mLLMT) [5]. These parameters will serve diverse applications to
revolutionize. Thus, the 6G technology will be a revolutionary technology in
many domains. In addition, high QoS will enable many new applications, for
instance, telesurgery. Also, 6G technology promises high QoE along with high
QoS which defines user-centric communications. It will be achieved by
holographic communications, augmented and virtual reality, five sense
communications and tactile Internet. Moreover, QoE will bring revolution in
intelligent devices, intelligent cars, intelligent drones, intelligent
ambulances, and many more [15]. Achieving high QoE depends on the
implementation of all desired parameters by 6G technology. Similarly, Quality
of Life (QoL) is determined to enhance the lifestyle with QoS and QoE in
healthcare. 6G technology will enable the high QoL using communication
technology. The key parameters of QoL are intelligent ambulance services,
remote health monitoring of patients, including elderly persons, intelligent
accident detection, Hospital-to-Home (H2H) service, telesurgery and precision
medicine.

##  II 6G Enabling Technology

###  II-A Internet of Everything (IoE)

6G communication follows 6C’s, namely, capture, communicate, cache, cognition,
compute, and control [5]. Cognition helps in formulating feasible
determinations based on input digital data. These are intelligent
determinations that make the computing easy. Then, computed data are
transmitted to intelligent devices to control the action taken by the devices
[5]. For example, raising an alarm. IoE will use the core service of combined
and enhanced eMBB and mMTC. Requirement of IoE from 6G are huge capacity to
connect millions of intelligent devices and high data rates to support touch
experiences in those devices. Industrial IoT (IIoT) uses the core service of
combined and enhanced URLLC and mMTC. Huge capacity to connect the sensors,
actuators and staff for communication. And, low latency to maintain seamless
integration among them [5]. When 6G will be commercial, it will be the era of
Big data 2.0 that requires a supercomputer to compute and analyze data [24].

###  II-B Edge Intelligence

6G will rely on Edge computing [26] to bring the Cloud features closer to
intelligent devices. Edge technology will offer uninterrupted and high speed
Internet services to the intelligent devices. It collects, computes and
analyses the data in real time in Edge nodes. Edge node also filters data and
transmits only important information to the Cloud for storage. Edge technology
reduces communication and computation cost. Some other advantages are low
latency, reliability, adaptability, scalability and privacy. All the
advantages of Edge technology will greatly help 6G to meet its prerequisites
to provide high QoS. Edge Intelligence is combining Edge computing and AI
[27]. Edge analytics, implement AI algorithms for analysis in Edge nodes [28].
With the help of AI, Edge nodes are capable of developing image, data and
video Edge analytics. Execution of AI algorithms requires high computational
resources, and power consumption, which is limited in the Edge nodes. One of
the network nodes in 6G will be Edge nodes. Moreover, 6G will have real-time
intelligent Edge that computes and analyze on live data [29].

###  II-C Artificial Intelligence

6G will be a truly AI-driven communication network, causing the system self-
aware, self-compute and self-decide on a situation [15, 30]. 6G will provide
global coverage, i.e., space-air-water. 6G will achieve this goal by making
different aspects of communication “intelligent”, and, intelligence means AI.
At the physical layer of the network, AI will help channel state estimation
and prediction, automatic modulation classification, adaptive encoding and
decoding, and intelligent beamforming. Likewise, physical layer security [31]
is key important features of 6G communication technology. At the data link
layer, resource allocation that implements deep reinforcement learning [18].
Likewise, at the transport layer, route computing and intelligent traffic
prediction algorithms will be deployed [5]. 6G will explore dynamic spectrum
access. 6G will depend on AI for bringing the heavy computation responsibility
to provide next generation services.

##  III Vehicular Technology

Many vehicles can be driverless and intelligent [32]. The intelligent vehicles
will optimize fuel consumption, route and work efficiency, because 6G
communication technology can provide real-time services. Intelligent vehicles
are more economical, because they can predict future and optimize the problem
economically. The future vehicle will be featured with truly AI-driven and
high speed wireless communications. The future vehicles will also be
integrated with diverse sensors. The vehicle will be intelligent with the help
of sensor devices and communication technology.

###  III-A Intelligent Cars

It is expected that 6G will be able to deliver self-driven vehicle, i.e.,
steering-less vehicles. The vehicles will be truly AI-driven learning from
practical experiences [33]. The cars will monitor passenger health, for
example, blood pressures, heartbeat rate, body temperature, brain wave
capturing, and emotion detection due to truly real-time communication
capability of 6G. Also, the intelligent cars will turn into an intelligent
ambulance in case the passenger need treatment during the ride. It is achieved
by consulting the remote doctors using mix-virtual reality/Holograph through
6G communication. Intelligent cars will avoid accidents by exchanging
information with nearby vehicles. Batteries will be charged without the wire
in a mobile state by UAVs [34]. Achieving a complete intelligent driving is
really difficult because its prerequisite is a combination of many complex
algorithms. Instances of such complex algorithms are automatic driving, path
planning, obstacle detection, and vehicle monitoring [5]. It will require the
support of high data rate, low latency, seamless connectivity and integration
of AI. It will produce a huge amount of data and also upload these on the
Internet in real-time.

###  III-B Unmanned Aerial Vehicles

UAV will be used for many purposes [35], for instance, fronthaul and backhaul
services in rural area. The intelligent Drones will be capable of
communication in Drone-to-Drone (D2D) and Drone-to-Infrastructures (D2I).
Intelligent Drones can share their knowledge. Also, it will be engaged for
faster deliver of online products. Moreover, it will be utilized in critical
operations where human cannot reach. Police will use Drones to catch a thief.
Also, the surveillance intelligent Drone will continuously monitor the
populated location. These types of intelligent Drones will detect fire,
accidents, traffics, etc., in real-time so that police will be informed in
real-time. Police can also employ the intelligent Drones to spray the tear gas
to control the mobs. Also, drones can be useful in agriculture.

Currently, Closed Circuit Televisions (CCTV) are used for security and
surveillance which makes us very secure. Moreover, the CCTV is also utilized
as a witness in court in many countries. However, CCTVs are not efficacious in
securing many things and identifying the threats. For instance, CCTV is
installed in a fixed pole or wall and they cannot chase any thief. Therefore,
the Drone will be added to fortify the security and surveillance systems. For
illustration, a thief or terrorist can be chased. The Drone can stream the
videos continuously to the monitoring rooms. This requires high speed data
link to provide high definition videos in real-time. Also, intelligent Drone
can alert possible threat to the security force. The Drone will play a
prominent role not only in the city, but also in border areas. It is very
dangerous in Line of Control (LOC) to monitor the enemies for 24$\times$7
mode. The Drone can provide the service of security and surveillance in
extremely dangerous border locations. Thus, Drone can enhance border security
dramatically in defense sector such dangerous areas. It requires 6G
communication technology surveillance, however, 5G communication technology
can also solve, but QoS will be lower.

###  III-C Intelligent Transportation

Achieving a complete intelligent driving is very difficult because its
prerequisite is a combination of many complex algorithms. For example,
automatic driving, path planning, obstacle detection, vehicle monitoring and
emergency rescue operations. It requires high data rate, low latency, seamless
connectivity, etc., and it can be provided by 6G communication technology. It
will produce a huge amount of data and also upload these data on the Internet
in real-time.

####  III-C1 Domestic transportation

Intelligent domestic transportation will remain connected to the 6G Internet
during the whole journey. It will even provide real time information. 6G with
its high speed internet, low latency and global coverage will provide services
smoothly without interruption. Therefore, reliable and real time information
will be provided to the passenger and reducing their waiting time. The real
time data will also help the driver to remain updated about the traffic or any
blockage in the route. In case of heavy traffic it will help drivers to drive
safely. And, blockage in the route due to an accident or natural disaster will
be known earlier. Based on the information the driver can reroute the path and
reduce harmful gas emission. Another key point is maintaining security. Police
can use the camera present in public transportation to track a suspected
vehicle. In case of small public transportation, eg., cab, taxi the passenger
can detect the route taken by the driver. Moreover, in case of any threat the
passenger can inform the police without any direct call. The intelligent
vehicles also continuously monitor its own components and parameters. Upon
detection of any damage or abnormality, then immediate action is taken.

####  III-C2 International Transportation

6G will help in keeping track of vehicles crossing the border of a country by
the security personnel. This will help in reducing crimes such as human
trafficking and smuggling. 6G will provide Internet services during airplane
and ship journey. Airplanes and ships will also remain connected to 6G IoT.
The 6G provides global coverage, thus, predicting any sudden changes in
weather such as rain or thundering. This real time information will help to
take necessary actions. With intelligent transportation the airplane and ships
can be made fully automated. Moreover, global coverage will help tracking the
airplane and ships in case of any accident. And, immediate medical and rescue
services can be provided. In addition, 6G will provide services in underwater.
Therefore, a ship or airplane can be tracked in case it has drowned and got
displaced due to strong water current.

##  IV Intelligent Robotic communication

Robotic communication refers to communication between robots or between humans
and robots. Ad-hoc networks are always used for robotic communication because
robots are independent nodes. Robots depend on the AI for communication.
Robots are greatly helpful in home services. It is used as personal assistance
for physically challenged people and elderly services. Intelligent Robot can
learn to serve efficiently as personal assistance. To achieve learning Robot,
AI and Machine Learning has to be integrated. Robots can be a great teacher as
well as a learner, which will require mobile communication connection to the
Internet. Intelligent Robots can share their experience with other intelligent
Robots. Therefore, Robot-to-Robot (R2R) communications can improve intelligent
robots in service. Moreover, intelligent Robots can be used in the industry
for economical, efficient and faster manufacturing of products.

###  IV-A Aerospace Robotic communication

There is also a plan to develop an intelligence space. An intelligence space
is an intelligence environment to monitor and control activities in space and
smooth communicate with people. The maintenance will mostly depend on robots
[36]. Robots will observe the environment and compute the data to take action
in case of extreme situations. They also have to constantly transmit data to
Earth. Therefore, to achieve such communication 6G will be required to have a
smooth and high speed communication. Moreover, in the intelligence space, the
robots will be distributed. Thus, they require a reliable and dynamic
communication network. Robots will also transmit images and videos.
Transmitting such high sized data compared to files requires high speed and
high data rate of 6G. Moreover, the features of 6G such as high data rate,
speed, high QoS will help to provide a 3D image of the space for a better
understanding of the space activities.

###  IV-B Underwater Robotic communication

Robotics is also explored for underwater tasks such as search and rescue,
imaging and security. The terrorists also use waterways for entering the
country. So, constant surveillance is essential. In such cases sometimes,
robots will be more efficient. However, the underwater robots have to
constantly interact with the ground station. 6G will be very helpful for such
communication. A smooth and reliable communication channel needs to be
established for real time analysis and action planning. 6G high speed and low
latency will help in obtaining images and video data. It will also help in
constantly locating the robot on the map and providing navigation
instructions.

##  V Virtual communication

The QoS provided by 6G communication technology will satisfy many requirements
of virtual communication such as holographic communication, augmented reality
and virtual reality. It requires uninterrupted high speed, low latency to
constantly maintain virtual presence, touch, experience etc.

###  V-A Holographic communication

Holographic communication requires high data rates to provide good QoS and
also streaming videos of high definition. It also requires very low latency
for real-time voices and spontaneous control responses [4, 37]. 6G will be
capable of satisfying all the requirements. Holographic communication will use
the core service of combined and enhanced eMBB and URLLC. Holographic
communication will be a major breakthrough during 6G. Due to high speed and
uninterrupted communication, communicating with a person through holographic
communication will be similar to face-to-face interaction. The person will be
able to move around without anyone’s intervention. Physical presence at a
formal meeting, or interview will not be compulsory.

###  V-B Augmented Reality and Virtual Reality

The requirements of augmented reality (AR) and virtual reality (VR) is similar
to holographic communication. The core service used by AR and VR is combined
and enhanced eMBB and URLLC. High data rates to provide good QoS and high
definition videos. Very low latency is essential for real-time voices and
immediate control responses [5]. Peak data rate requirement of AR and VR is 1
Tbps and user experience of $>$10 Gbps with $>$0.1 ms latency which can be
provided by MBBLL [5]. AR provides visibility inside the object without
separating the different components of the object. For example, various
components of a machine are visible without removing the machine parts.
Moreover, the depth of the visibility can be changed, i.e. user can view
deeper and deeper inside the object (physical depth). Using AR the viewed part
is also zoomed for better visibility. VR helps in creating one’s own
artificial environment. It helps in adding details related to games, movies,
or animation. Moreover, it will also help in better understanding of the
documents. Moreover, due to 6G the holographic communication, AR and VR will
be combined for better collaborative works.

###  V-C Tactile/Haptic Internet

Tactile Internet requires ERLLC and high speed communication to grab the
tactile in real-time. Usually the remote user sends the signal to another
human or robot. This technology is applicable where a physical presence is
compulsory to perform a task. Usually this task requires precise work and
robots cannot always be trained to perform such tasks. For example, surgery or
defusing the bomb. During telegery, the doctor can use tactile technology to
perform a surgery using a human or a robot. As per the movement of the remote
doctor the robot will move. Such detailed work required uninterrupted and high
speed Internet. 6G will provide great support in achieving these goals.

##  VI Intelligent Healthcare

QoL will be the greatest player in ensuring intelligent healthcare. It is
expected that 6G communication will revolutionize healthcare. 6G will be able
to overcome the time and space barriers of healthcare in the near future.
Intelligent healthcare will implements Hospital-to-Home (H2H) service which is
implemented upon intelligent vehicle. This mobile hospital will replace
ambulance services in the near future. H2H will also implement real-time
accident detection and automatic emergency detection. Diverse intelligent
wearable device and sensors will help in the detection of accident
automatically in real-time. Also, needle-free Blood Sample Reader (BSR) sensor
will greatly help in the detection of diseases automatically [38]. BSR sensor
will read the blood sample and send it to the pathology laboratory. Therefore,
Intelligent Wearable Devices (IWD) reduce the risk of medical staff in
contacting with viruses. Also, the patient is not required to travel to the
hospital for blood tests.

###  VI-A Intelligent Internet of medical Things (IIoMT)

Moreover, Intelligent Internet of Medical Things (IIoMT) will help in avoiding
the time and space barriers [38]. For instance, remote doctor can perform
surgical operations using telesurgery which requires high speed communication.
The doctor can supervise the telesurgery through verbal, telestration or tele-
assist [39]. For verbal, doctor(s) will use holographic communication.
Hologram helps the doctor in the movement to have a better visual of the
surgery area. For telestration, doctor(s) will use AR and VR. And, the
doctor(s) can tele-assist the surgery using tactile/haptic technology. The
requirements of telesurgery can be fulfilled by 6G communication technology
and 6G will prove that surgery can be performed beyond boundaries.

##  VII Intelligent City

The key requirement of intelligent city is a real-time communication system.
Intelligent City manages intelligently in real-time the traffic, waste, home,
grid and environment which requires high QoS from wireless communication
technology.

###  VII-A Intelligent Traffic

Smart cities, mainly focus on enhancing theQoL. Intelligent cities will also
focus on the environment. Therefore, intelligent traffic is required because
longer the vehicle will halt in traffic more harmful gases will be released
[40]. Hence, in future the traffic signals will also be connected to IoE.The
Intelligent traffic signal will analyze the data to determine the average
traffic opening duration for smooth movement of vehicles. This duration will
be different for different roads based on real-time data. In case an ambulance
is traveling for an emergency then it can send notification to intelligent
traffic signals. Upon receiving the notification the intelligent traffic
signal will track the ambulance using GPS. At appropriate times the traffic
will open for uninterrupted ambulance movement. For such applications, high
speed and low latency 6G Internet will be essential. Similarly, the police or
army vehicles will be provided an uninterrupted movement. Moreover,
intelligent traffic signals can also be used to interrupt any suspected
vehicle. They will also monitor the vehicles for any traffic rule violation.
In addition, whenever any dispute happens among people the intelligent traffic
signal can immediately alert the police. Moreover, camera present in traffic
signals can monitor for any accident that happens in an isolated and remote
area. In such situations, the live data have to be analyzed quickly for
informing appropriate authority. And, 6G Edge nodes will be capable of
performing those tasks.

###  VII-B Intelligent Waste Management

Intelligent waste management is also a part of an intelligent city to reduce
pollution and health risk. The dustbins will have sensors to determine whether
it is full. In case it is full, the waste collection vehicle will be notified.
It will collect the waste quickly. In some areas the dustbins may get full
quickly, hence, it requires waste pickup twice or more in a day. And, in some
localities the dustbin may be getting full after two days or more. Therefore,
intelligent waste collection will help in optimizing utilization of resources.
All the dustbin sensors will be connected to IoE. 6G global coverage will help
to connect every corner of the city.

###  VII-C Intelligent Home

An intelligent home assistant will connect all intelligent home appliances
such as television, home security, washing machine, music system to IoE. Human
presence and some commands will activate the appliances and allow the people
of the house to remotely access the appliance. An intelligent light will
switch off in the absence of any person in the room or maintain the light
intensity based on sunlight intensity. This is very useful for small children
and elderly people. During an emergency, an intelligent home assistant will
detect the issue and inform the appropriate authority. For example, a fire at
a dangerous level the water sprinkler is automatically opened or informs the
fire station in case fire is not under control. The intelligent home will also
have an intelligent environment monitoring system. It will keep track of the
environmental condition of the home. For example, if the weather is hot then
the central air conditioner will start to keep the home cool. This helps in
keeping the people of the house healthy. If a family member is ill, then that
room’s environmental condition will be different from the rest of the rooms. A
home will have many intelligent home appliances, all these will be connected
to IoE. The data generated will be huge and these need to be processed by 6G
Edge nodes quickly. The people of the home can use intelligent phones or some
wearable to physically control the intelligent home appliances. During
security related situations such as any forced entry or fire, 6G Edge nodes
have to compute the data and inform the appropriate authority quickly even
without the permission of the owner.

###  VII-D Intelligent Grid

Intelligent grid is important because 6G nodes will perform heavy computation
and will require periodic or constant energy supply. The most important point
is to provide global coverage, 6G will have high density, therefore, an
intelligent grid is very essential for the 6G communication system. The
intelligent grid is AI based, fully automated, remote control and has self-
healing features. With the exhaustion of natural resources, intelligent grid
will rely completely on renewable energy resources. It will intelligently
integrate the distributed and renewable energy resources. 6G nodes will both
depend and help in the power grid. The intelligent 6G nodes placed at
different locations can determine the power consumption during different times
of the year or season. These data will be used to determine the power
production requirement for an intelligent city. After sunset the 6G nodes will
automatically ON the street light. One important point is that an intelligent
home will try to reduce electricity consumption, thus, optimizing the power
supply. Intelligent grid will also store energy in case of emergencies. 6G
nodes will intelligently switch to reserve energy in case of power cuts.
Another concept is micro grid. Using renewable energy resources, the consumers
will also produce electricity. Such consumers are called prosumers. Prosumers
will sell excess electricity to the grid. Such a small grid is called a micro
grid [41]. Moreover, in case of a power cut the intelligent nodes will take
the electricity from the micro grid. With 6G connectivity, the monitoring and
maintenance of the micro grid are easy. Real time electricity reserve in the
micro grid is recorded and saved. Any illegal sale of electricity, such as
industries or illegal institutions doing illegal activity is monitored. The
electricity billing will also be intelligent, the intelligent home will send
the electricity consumption details to the power grid datacenters and a bill
will be automatically forwarded to the consumer.

##  VIII Industrial revolution

The Industry 4.0 was revolutionized as digitalization [42]. Industry 5.0 is
about personalization and human-centric manufacturing. Industry 5.0 will be
able to integrate AI and 5G communication to increase productivity. Moreover,
the industrial Robots will become more smart due to the advent of AI and
communication technology. Ultra-massive Machine-Type communications take place
in the industry due to the huge number of sensors and robots. The robots and
sensors require extremely reliable and low latency communication for
precision. Also, communications help in faster production. Therefore, it is
expected that productivity will increase in Industry 5.0. Moreover,
intelligent Robots can optimize their works efficiently and economically. The
Industry focuses on all the processes from procuring the raw materials to
delivery of finished product to customers. Therefore, an industry has to solve
many complex problems to reduce overall profit. The communication system has
to support industrial IoT (IIoT). IIoT application focuses on power system
monitoring, demand-side energy management, integration of renewable energy
generators and coordination of distributed power storage [43]. Moreover, the
sensors and robots will produce huge volumes of data. These data will be
processed in Edge nodes [44]. However, Industry 5.0 will not be capable of
supporting Industries completely due to the presence of many issues. But, it
is expected that Industry 6.0 will be about truly AI-driven industry. 6G Edge
nodes will be able to handle heavy computation, hence, providing spontaneous
responses. 6G with high density will be able to manage a massive number of
robots and sensors. Industry 6.0 will deliver Industrial Internet of
Everything (IIoE) [15]. Thus, Industry 6.0 will be the Intelligentization of
Industry.

##  IX Internet of Nano-Things

Nano-things have components of nano size, hence, they are capable of
performing simple tasks. It has small memory, thus, data storage capacity is
low. A communication network consisting of nano-things is called the Internet
of Nano-Things (IoNT) [45]. Nano-things communicate within short range.
Therefore, within a small range more nano-nodes need to be deployed.
Communication of IoNT can be performed by using molecular or THz
communication. However, THz communication will be secure, reliable and faster
compared to molecular communication [46]. For THz communication, 6G will be an
excellent choice. Therefore, 6G will support IoNT and will make it possible
for deployment. Nano-things are small and simple, hence, speed will be more.
In this regard, 6G will $>$1Tbps speed will perfectly complement IoNT. 6G
technology will have high density. Hence, controlling the IoNT network with a
massive number of nano-things will be easy. Another important point is in some
situations the nano-things have to continuously generate valuable information,
but due to small memory these data need to be transmitted to the datacenter.
Therefore, with high speed 6G communication network nano-things will transmit
the data smoothly.

##  X Datacenter connectivity

6G with its high QoS will provide great support for the development of many
technologies. For instance, holographic communication, IoNT, telesurgery. 6G
will deploy a massive number of network nodes such as Edge nodes, satellites,
drones, etc. to provide high density and global coverage. These 6G network
nodes will be capable of computation and analysis. However, these nodes have
smaller power supplies and memory. The data generated by the 6G network nodes
are high in both volume and quality. For example, underwater rescue operations
stream a real time high quality video. This high sized data is not possible to
store in the 6G network nodes. Thus, the data are transmitted to the
datacenter for storage. Datacenter requires high data rate and extremely low
latency [47]. Due to the high speed of 6G the data generation and data
transmission to the datacenter can be done simultaneously. Moreover, the data
can be saved in multiple locations in parallel. Moreover, with data storage in
the datacenter, other operations can also be performed. 6G provides also
provides security during data transmission.

##  XI Conclusion

In this article, we have discussed all potential applications of 6G
communication technology, and we envision and identify the potential of 6G
communication technology in many fields, for instance, transportation, city.
It has the potential to revolutionize many technologies and applications. We
also explore the impact of 6G communication technology in diverse
applications. There are tremendous applications that will fully depend on 6G
communication technology, particularly, vehicular technology, healthcare
sectors, modern cities and industries. Many new technologies and applications
are yet to be conceived due to lack of 6G communication technology. Moreover,
we will enter into an intelligent era from smart era and this transition will
change our perception on lifestyle, society and business. Furthermore, 6G
communication technology will also impact the global economy. Therefore, we
will evidence that 6G communication technology will be a game changer
technology.

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|  Sabuzima Nayak<EMAIL_ADDRESS>is a research scholar in the
Department of Computer Science & Engineering, National Institute of Technology
Silchar, Assam-788010, India. Her research interest is Bioinformatics, Bloom
Filter, Communication and Networking, and Big Data. She has published numerous
papers in reputed journal, conferences and books.
---|---
|  Ripon Patgiri<EMAIL_ADDRESS><EMAIL_ADDRESS>is currently working
as Assistant Professor in the Department of Computer Science & Engineering,
National Institute of Technology Silchar, Assam-788010, India. His research
interests are Bloom Filter, Communication and Networks, Big Data, and File
System. He has published numerous papers in reputed journal, conferences and
books. He is a senior member of IEEE and member of ACM, EAI, and IETE He has
also chaired many conferences.
---|---


# Homotopic Nerve Complexes with
Free Group Presentations

J.F. Peters  Computational Intelligence Laboratory, University of Manitoba,
WPG, MB, R3T 5V6, Canada and Department of Mathematics, Faculty of Arts and
Sciences, Adi̇yaman University, 02040 Adi̇yaman, Turkey,
<EMAIL_ADDRESS>Dedicated to Yuriy Trokhymchuk & Saroja V. Banavar

###### Abstract.

This paper introduces homotopic nerve complexes in a planar Whitehead CW space
and their Rotman free group presentations. Nerve complexes were introduced by
P.S. Alexandrov during the 1930s and recently given a formal structure from a
computational topology perspective by H. Edelsbrunner and J.L. Harer in 2010.
A homotopic nerve results from the nonvoid intersection of a collection of
homotopic 1-cycles. Briefly, a 1-cycle is a finite sequence of path-connected
vertexes with no end vertex and with a nonvoid interior. A homotopic 1-cycle
has the structure of a 1-cycle in a CW space in which cycle edges are replaced
by homotopic maps. A group $G(V,+)$ containing a basis $\mathcal{B}$ is free,
provided every member of $V$ can be written as a linear combination of
elements (generators) of the basis $\mathcal{B}\subset V$. Let
$\bigtriangleup$ be the members $v$ of $V$, each written as a linear
combination of the basis elements of $\mathcal{B}$. A presentation of $G(V,+)$
is a mapping $\mathcal{B}\times\bigtriangleup\to G(\left\\{v\in
V\mathrel{\mathop{:}}=\sum_{k\in\mathbb{Z}\atop
g\in\mathcal{B}}{kg}\right\\},+)$. The main results in this paper are (1)
Every homotopic vortex nerve has a free group presentation and (2) For a
vortex nerve that consists of a finite collection of closed, convex sets in
Euclidean space, the nerve and union of sets in the nerve have the same
homotopy type.

###### Key words and phrases:

1-Cycle, Free Group, Homotopy, Nerve, Presentation

###### 2010 Mathematics Subject Classification:

55P57 (homotopy theory); 55U10 (Simplicial sets and complexes); 57M05 (Free
Group Presentations)

The research has been supported by the Natural Sciences & Engineering Research
Council of Canada (NSERC) discovery grant 185986 and Instituto Nazionale di
Alta Matematica (INdAM) Francesco Severi, Gruppo Nazionale per le Strutture
Algebriche, Geometriche e Loro Applicazioni grant 9 920160 000362, n.prot U
2016/000036 and Scientific and Technological Research Council of Turkey
(TÜBİTAK) Scientific Human Resources Development (BIDEB) under grant no:
2221-1059B211301223.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Rotman Free Group Presentation
  4. 4 Main Results
  5. 5 Application
  6. A Cell Complexes

$\boldsymbol{v_{1}}$$\boldsymbol{v_{0}}$$\boldsymbol{K}$$\boldsymbol{v_{3}}$$\boldsymbol{v_{i}}$$\boldsymbol{v_{(n-1)[n]}}$$\boldsymbol{v_{i+1}}$$\boldsymbol{\mbox{cyc}E}$$\boldsymbol{\to}$

-v(-1.0,2)(-0.5,3) -v(-0.5,3)(0.5,3.5) -v(0.5,3.5)(1.5,3.0) -v(1.5,3.0)(2.0,2.0) -v(2.0,1.0)(1.5,0.0) -v(1.5,0.0)(0.5,-0.5) -v(-0.5,0.0)(-1.0,1.0) -v(-1.0,1.0)(-1.0,2) $\boldsymbol{v_{1}}$$\boldsymbol{v_{0}}$$\boldsymbol{h_{i}(0)=v_{i}}$$\boldsymbol{h_{i}(1)=v_{i+1}}$$\boldsymbol{h_{(n-1)[n]}(0)}$$\boldsymbol{h_{0}(0)=h_{n-1[n]}(1)}$$\boldsymbol{=\ v_{0}}$$\boldsymbol{h_{1}(0)\ =h_{0}(1)}$$\boldsymbol{h_{2}(0)=h_{1}(1)}$$\boldsymbol{h_{3}(0)=v_{3}}$$\boldsymbol{\mbox{hCyc}E}$

Figure 1. 1-Cycle $\boldsymbol{\mbox{cyc}E\ \to}$ homotopic cycle
$\boldsymbol{\mbox{hCyc}E}$ on space $\boldsymbol{K}$.

##  1\. Introduction

This paper is based on [16], which introduces homotopic nerve complexes in a
planar Whitehead CW space [23, §4-5] and their Rotman free group presentations
[18, §11,p.239]. A CW complex in a space $K$ is a closure-finite cell complex
that is Hausdorff (distinct points in $K$ are in disjoint neighborhoods),
satisfying the containment property (closure of every cell complex is in $K$)
and intersection property (common parts of cell complexes in $K$ are also in
$K$). A complex $K$ is locally finite, i.e., every point $p\in K$ is a member
of some finite subcomplex of $K$ and every complex has a finite number of
faces [19, §5.2,p.65]. A planar CW complex $K$ is a collection of 0-cells
(vertexes), 1-cells (edges) and 2-cells (filled triangles). Collections of
planar cells attached to each other are sub-complexes in $K$. For the details,
see Appendix A.

##  2\. Preliminaries

The foundations for homotopic nerves are briefly introduced in this section,
starting with the definition of a 1-cycle.

######  Definition 1.

1-Cycle. In a CW space $K$ [23], a 1-cycle $E$ (denoted by $\mbox{cyc}E$) in a
CW space $K$ is a collection of path-connected vertexes (0-cells) on edges
(1-cells) attached to each other with no end vertex and $\mbox{cyc}E$ has a
nonvoid interior. ◼

######  Definition 2.

Closure. Let $A\in 2^{X}$ (nonvoid subset $A$ in a space $X$) and
$D(x,A)=\inf\left\\{a\in A:\left|x-a\right|\right\\}$ be the Hausdorff
distance between a point $x\in A$ and subset $A$ [6, §22,p. 128]. The closure
of $A$ [12, §1.18,p. 40] is defined by

$\mbox{c$\ell$}(A)=\left\\{x\in X:D(x,A)=0\right\\}.\
\mbox{\qquad{\color[rgb]{0,0,1}\SquareSteel}}$

######  Definition 3.

For a space $X,A\in 2^{X}$, let $\mbox{c$\ell$}A$ be the closure of $A$. Then
the boundary of $A$ (denoted by $\mbox{bdy}A$) is the set of all points on the
border of $\mbox{c$\ell$}A$ and not in the complement of $\mbox{c$\ell$}A$
(denoted by $\partial\mbox{c$\ell$}A$). Also, the interior of $A$ (denoted by
$\mbox{int}A$) is the set of all points in $\mbox{c$\ell$}A$ and not on the
boundary of $A$111This version of $\mbox{int}A$ comes from T. Vergili., i.e,

$\displaystyle\partial(\mbox{c$\ell$}A)$
$\displaystyle=X\setminus\mbox{c$\ell$}A,\ \mbox{all points in $X$ and not in
$\mbox{c$\ell$}A$}.$ $\displaystyle\mbox{int}(A)$ $\displaystyle=\left\\{E\in
2^{X}:E\subset\mbox{c$\ell$}A\ \mbox{and}\
E\cap\mbox{bdy}A=\emptyset\right\\}.$ $\displaystyle\mbox{bdy}(A)$
$\displaystyle=X\setminus(\mbox{int}A\cup\partial\mbox{c$\ell$}A).$
$\displaystyle\mbox{c$\ell$}A$ $\displaystyle=\mbox{bdy}(A)\cup\mbox{int}(A).\
\mbox{\qquad{\color[rgb]{0,0,1}\SquareSteel}}$

######  Example 1.

A sample 1-cycle $\mbox{cyc}E$ with vertices $v_{0},\dots,v_{(n-1)[n]}$ with
$[n]=\mbox{mod}\ n,n\in\mathbb{Z}^{0,+}$, is shown in Fig. 1. This 1-cycle
$\mbox{cyc}E$ has a finite number of $i\ \mbox{mod}\ n$ vertices, with the
initial vertex $v_{0}=v_{n\ \mbox{mod}\ n}$. $\mbox{cyc}E$ also has a nonvoid
interior (denoted by $\mbox{int}(\mbox{cyc}E)$) containing at least one filled
triangle with one of its vertices equal to centroid of the cycle and its other
two vertices on the boundary of $\mbox{cyc}E$ (denoted by
$\mbox{bdy}(\mbox{cyc}E)$). In other words,

$\mbox{c$\ell$}(\mbox{cyc}E)=\mbox{bdy}(\mbox{cyc}E)\cup\mbox{int}(\mbox{cyc}E).\mbox{\qquad{\color[rgb]{0,0,1}\SquareSteel}}$

The edges in a 1-cycle cell complex in a CW space are replaced by homotopic
maps to obtain a homotopic cycle.

######  Definition 4.

Homotopic-Map.
Let $I=[0,1]$, the unit interval. A homotopic map(aka  _path_) in a space $X$
is a continuous map $h:I\to X$ with endpoints
$h(0)=x_{0},h(t)\in[h(0),h(1)],t\in I$ and $h(1)=x_{1}$ [19, §2.1,p.11]. ◼

######  Definition 5.

Homotopic-Cycle [17, App. A].
A homotopic cycle $\mbox{hCyc}E$ in a CW space $K$ is a sequence of continuous
maps $\left\\{h_{i}\right\\}_{i=0}^{(n-1)[n]}$, $h_{i}:I\to
K,i\in\mathbb{Z},I=[0,1],[n]=\ \mbox{mod}\ n$, with

$\displaystyle h_{0}(0)$ $\displaystyle=h_{n[n]}(0)=v_{0}\in
K,h_{n-1[n]}(1)=v_{n-1[n]}\in K.$ $\displaystyle\mbox{Given}\ g$
$\displaystyle\in\left\\{h_{i}\right\\}_{i=0}^{(n-1)[n]},$ $\displaystyle g$
$\displaystyle=h_{0}(0),\ \mbox{a basis element in $\mbox{hCyc}E$, so that}$
$\displaystyle v$
$\displaystyle\overbrace{\mathrel{\mathop{:}}=kg=kh_{0}(0)=h_{0+k}(0)=h_{k}(0)\in
K,k\in\mathbb{Z}.}^{\mbox{{\color[rgb]{0,0,1}\bf$h_{i+k}(0)$\ \mbox{walks
forward}\ $k$ vertexes\ \mbox{from $g$ to reach}\ $v$}}}$
$\displaystyle\bar{h}_{i+k}(0)$
$\displaystyle\overbrace{\mathrel{\mathop{:}}=h_{(i+k)-k}(0)=h_{i}(0).}^{\mbox{{\color[rgb]{0,0,1}\bf$\bar{h}_{i+k}(0)$\
\mbox{walks back}\ $k$ vertexes from $h_{i+k}(0)$ to reach $h_{i}(0)$}}}$
$\displaystyle+:K\times K$ $\displaystyle\to K,\ \mbox{is defined by}$
$\displaystyle+(v,v^{\prime})$
$\displaystyle=\overbrace{+(kg,k^{\prime}g)=kg+k^{\prime}g.}^{\mbox{{\color[rgb]{0,0,1}\bf
move to $\boldsymbol{v^{\prime\prime}}$ via $\boldsymbol{v,v^{\prime}}$}}}$
$\displaystyle=kh_{0}(0)+k^{\prime}h_{0}(0)=h_{(k+k^{\prime})[n]}(0)=v^{\prime\prime}\in
K.\mbox{\qquad{\color[rgb]{0,0,1}\SquareSteel}}$

In a homotopic cycle $\mbox{hCyc}E$, every vertex $v_{i}$ is reachable by $k$
maps from a distinguished vertex $h_{0}(0)=v_{0}$ (a generator in the free
group presentation of the homotopic cycle). An inverse homotopic map
$\bar{h}_{i+k}(0)$ is defined in terms of reverse movement (traversal from the
$h_{i+k}(0)$ cycle vertex to the previous $h_{i}(0)$ cycle vertex)222Here,
inverses are defined in terms of homotopic map indices, instead of
$\bar{h}_{i}(t)=h_{i}(1-t)$ [5, §1.1, p. 27]. Many thanks to Tane Vergili for
pointing this out.. For simplicity, we consider only $i\in\mathbb{Z}$ for
movements between cycle vertexes.

######  Example 2.

A planar homotopic cycle over a collection of vertexes in a CW space is shown
in Fig. 1. From Def. 5, the expression $3v_{0}=3h_{0}(0)=h_{0+3}(0)=v_{3}\in
K$ represents a move from vertex $v_{0}$ to vertex $v_{3}$. What we observed
about the structure of the 1-cycle $\mbox{cyc}E$ in Example 1 carries over in
delineating the structure of a homotopic cycle. That is, $\mbox{hCyc}E$ also
has a nonvoid interior (denoted by $\mbox{int}(\mbox{hCyc}E)$) that it
inherits from 1-cycle $\mbox{cyc}E$. The boundary of $\mbox{hCyc}E$ (denoted
by $\mbox{bdy}(\mbox{hCyc}E)$) has been constructed with its sequence of
continuous homotopic maps. In other words,

$\mbox{c$\ell$}(\mbox{hCyc}E)=\mbox{bdy}(\mbox{hCyc}E)\cup\mbox{int}(\mbox{hCyc}E).\mbox{\qquad{\color[rgb]{0,0,1}\SquareSteel}}$

######  Theorem 1.

[Jordan Curve Theorem [8]].
A simple closed curve lying on the plane divides the plane into two regions
and forms their common boundary.

###### Proof.

For the first complete proof, see O. Veblen [20]. For a simplified proof via
the Brouwer Fixed Point Theorem, see R. Maehara [9]. For an elaborate proof,
see J.R. Munkres [11, §63, 390-391, Theorem 63.4]. ∎

The boundary region of any planar homotopic cycle $\mbox{hCyc}E$ (denoted by
$\partial(\mbox{hCyc}E)$) contains all cell complexes that are not part of the
cycle in a CW space.

######  Theorem 2.

Every homotopic cycle in a CW space

  1. 1o

is a finite collection of path-connected vertices and

  2. 2o

the boundary of a homotopic cycle partitions the CW space into two regions.

###### Proof.

$1^{o}$: Let $\mbox{hCyc}E$ be a homotopic cycle in a CW space $K$. From Def.
5, the end-points $h_{i[n]}(0),h_{i[n]}(1)$ are points in $K$, which are
0-cells (vertices) in $K$. The 0-cells are path-connected, since each
$h_{i+1[n]}(0)=h_{i[n]}(1)$ and $h_{0[n]}(0)=h_{n-1[n]}(1)$. The number of
vertices in $\mbox{hCyc}E$ is restricted by the choice of the modulus $n$.
Hence, $\mbox{hCyc}E$ has a finite collection of path-connected vertices.
$2^{o}$: From Def. 4, every homotopic map $h:I\to K$ is continuous. Also, from
Def. 4, the sequence of homotopic maps in a homotopic cycle $\mbox{hCyc}E$ on
$K$ constructs a planar, simple, closed curve. Then, from Theorem 1 (Jordan
Curve Theorem), $\mbox{hCyc}E$ separates the space $K$ into two regions,
namely,

  1. Region outside $\mbox{hCyc}E$

: $\partial(\mbox{hCyc}E)=K\setminus\mbox{hCyc}E$, and

  2. Region inside $\mbox{hCyc}E$

:

$\mbox{int}(\mbox{hCyc}E)=K\setminus(\mbox{bdy}(\mbox{hCyc}E)\cup\partial(\mbox{hCyc}E)).$

∎

$\boldsymbol{v_{1}}$$\boldsymbol{v_{0}}$$\boldsymbol{v^{\prime}_{0}=v_{5}}$$\boldsymbol{v^{\prime}_{j}}$$\boldsymbol{\mbox{cyc}E}$$\boldsymbol{\mbox{cyc}E^{\prime}}$$\boldsymbol{v_{i}}$$\boldsymbol{v_{2}}$$\boldsymbol{v_{3}}$$\boldsymbol{v_{4}}$$\boldsymbol{v^{\prime}_{1}}$
Figure 2. Nerve
$\boldsymbol{\mbox{Nrv}E=\left\\{\mbox{cyc}E,\mbox{cyc}E^{\prime}:\mbox{cyc}E\cap\mbox{cyc}E^{\prime}\neq\emptyset\right\\}}$.

######  Definition 6.

Nerve Complex.
A nerve complex $\mbox{Nrv}E$ in a CW space is a collection of nonempty cell
complexes with nonvoid intersection. This is an example of an Alexandrov nerve
[2, §4.3,p. 39]. ◼

A vertex with a collection of triangles attached to it, is called the nucleus
of the attached triangles (also called the nucleus of an Alexandrov-Hopf nerve
complex or Nerv [3]).

######  Lemma 1.

Every vertex in the triangulation of the vertices in a CW space is the nucleus
of an Alexandrov nerve complex.

###### Proof.

Let $K$ be a CW space containing $n$ trianulated vertexes. Each vertex $v\in
K$ has 1 or more triangles attached to it, forming a cell complex $E$. Then
$E$ is a collection of cell complexes (filled triangles) with nonvoid
intersection. By Def. 6, $E$ is an Alexandrov nerve complex. Hence, each $E\in
2^{K}$ is the nucleus of an Alexandrov nerve. ∎

######  Theorem 3.

A CW space containing $n$ triangulated vertexes contains $n$ Alexandrov-Hopf
nerve complexes.

###### Proof.

Immediate from Lemma 1. ∎

Alexandrov nerves are also found in collections of intersecting 1-cycles.

######  Theorem 4.

A pair of pair of 1-cycles with a common vertex in a CW space is a nerve
complex.

###### Proof.

Let $\mbox{cyc}E,\mbox{cyc}E^{\prime}\in 2^{k}$ be 1-cycles in a CW space K.
By definition, $\mbox{cyc}E,\mbox{cyc}E^{\prime}$ are cell complexes in $K$.
Also, let $v\in K$ be a vertex and let
$\mbox{cyc}E\cap\mbox{cyc}E^{\prime}=\left\\{v\right\\}$. Hence, by Def. 6,
$\mbox{Nrv}=\mbox{cyc}E\cup\mbox{cyc}E^{\prime}$ is a nerve complex, which is
also an Alexandroff nerve or _Nerv_ [3]. ∎

######  Example 3.

A pair of intersecting 1-cycles $\mbox{cyc}E,\mbox{cyc}E^{\prime}$ in a nerve
complex is shown in Fig. 2. In this nerve complex,

$\displaystyle\mbox{cyc}E\cap\mbox{cyc}E^{\prime}$
$\displaystyle=\left\\{v_{5}\right\\}\in\mbox{cyc}E$
$\displaystyle=\left\\{v^{\prime}_{0}\right\\}\in\mbox{cyc}E^{\prime}.\mbox{\qquad{\color[rgb]{0,0,1}\SquareSteel}}$

-v(-1.0,2)(-0.5,3) -v(-0.5,3)(0.5,3.5) -v(0.5,3.5)(1.5,3.0) -v(1.5,3.0)(2.0,2.0) -v(2.0,1.0)(1.5,0.0) -v(1.5,0.0)(0.5,-0.5) -v(-0.5,0.0)(-1.0,1.0) -v(-1.0,1.0)(-1.0,2) $\boldsymbol{v_{1}}$$\boldsymbol{v_{0}}$-v(2,2)(2.5,3) -v(2.5,3)(3.5,3.5) -v(3.5,3.5)(4.5,3.5) -v(4.5,3.5)(5.5,3.0) -v(5.5,3.0)(6.0,2.0) -v(6.0,2.0)(5.0,1.5) -v(5.0,1.5)(4.0,1.5) -v(4.0,1.5)(3.0,1.5) -v(3.0,1.5)(2.0,2) $\boldsymbol{h^{\prime}_{0}(0)=h_{5}(0)}$$\boldsymbol{h^{\prime}_{j}(0)}$$\boldsymbol{\mbox{hCyc}E}$$\boldsymbol{\mbox{hCyc}E^{\prime}}$$\boldsymbol{h_{i}(0)=h_{i-1}(1)}$$\boldsymbol{h_{i}(1)}$$\boldsymbol{h_{0}(0)=h_{(n-1)[n]}(1)}$$\boldsymbol{=\ v_{0}}$$\boldsymbol{h_{1}(0)\ =h_{0}(1)}$$\boldsymbol{h_{2}(0)=h_{1}(1)}$$\boldsymbol{h_{3}(0)}$$\boldsymbol{h_{4}(0)}$ Figure 3. homotopic nerve $\boldsymbol{\mbox{hNrv}E=\left\\{\mbox{hCyc}E,\mbox{hCyc}E^{\prime}:\mbox{hCyc}E\cap\mbox{hCyc}E^{\prime}\neq\emptyset\right\\}}$.

######  Definition 7.

Homotopic Nerve. A homotopic nerve $E$ (denoted by $\mbox{hNrv}E$) is a
collection of homotoptic cycles with nonvoid intersection. ◼

######  Theorem 5.

Every collection of homotopic cycles with a common vertex in a CW space is a
homotopic nerve complex.

###### Proof.

Symmetric with the proof of Theorem 4. ∎

######  Example 4.

In Fig. 3, homotopic cycles $\mbox{hCyc}E,\mbox{hCyc}E^{\prime}$ have vertex
$h_{5}(0)$ in common. Hence, by Theorem 5,
$\mbox{hCyc}E\cup\mbox{hCyc}E^{\prime}$ is a homotopic nerve complex. ◼

######  Lemma 2.

Every vertex in the triangulation of the vertices in a CW space is the nucleus
of an homotopic nerve.

###### Proof.

Symmetric with the proof of Lemma 1. ∎

######  Theorem 6.

A CW space containing $n$ triangulated vertexes contains $n$ homotopic nerves.

###### Proof.

Immediate from Lemma 2. ∎

-v(-1.0,2)(-0.5,3) -v(-0.5,3)(0.5,3.5) -v(0.5,3.5)(1.5,3.0) -v(1.5,3.0)(2.0,2.0) -v(1.5,3.0)(0.5,1.5) -v(0.5,1.5)(2.0,2.0) -v(0.5,1.5)(2.0,1.0) -v(1.5,0.0)(0.5,1.5) -v(0.5,1.5)(0.5,-0.5) -v(1.5,0.0)(0.5,1.5) -v(0.5,1.5)(-1.0,1.0) -v(-1.0,2.0)(0.5,1.5) -v(0.5,1.5)(-0.5,3.0) -v(0.5,1.5)(0.5,3.5) -v(-0.5,0.0)(0.5,1.5) -v(2.0,1.0)(1.5,0.0) -v(1.5,0.0)(0.5,-0.5) -v(-0.5,0.0)(-1.0,1.0) -v(-1.0,1.0)(-1.0,2) $\boldsymbol{v_{1}}$$\boldsymbol{v_{0}}$$\boldsymbol{K}$$\boldsymbol{h_{i}(0)=h_{i-1}(1)}$$\boldsymbol{h_{i}(1)}$$\boldsymbol{h_{0}(0)=h_{(n-1)[n]}(1)}$$\boldsymbol{=\ v_{0}}$$\boldsymbol{h_{1}(0)\ =h_{0}(1)}$$\boldsymbol{h_{2}(0)=h_{1}(1)}$$\boldsymbol{h_{3}(0)=v_{3}}$$\boldsymbol{h_{(n-1)[n]}(0)}$$\boldsymbol{\mbox{hCyc}E}$$\boldsymbol{h_{4}(0)}$$\boldsymbol{h_{5}(0)}$$\boldsymbol{c}$ Figure 4. Paths on a triangulated $\boldsymbol{\mbox{hCyc}E}$.

######  Definition 8.

Homotopic triangle. Given a set $V$ of 3 vertexes in a CW space, a sequence of
homotopic maps, one for each pair of vertices in $V$, constructs a homotopic
triangle $E$ (denoted by $\bigtriangleup_{h}E$). ◼

An obvious place to look for homotopic triangles is in terms of the centroid
of a homotopic cycle.

######  Example 5.

Let

$\displaystyle h_{c}(0)$ $\displaystyle=h_{4}(0)\ \mbox{vertex on
$\mbox{hCyc}E$},$ $\displaystyle h_{c}(1)$ $\displaystyle=c\ \mbox{centroid of
$\bigtriangleup_{h}$},$ $\displaystyle h_{c^{\prime}}(0)$ $\displaystyle=c,$
$\displaystyle h_{c^{\prime}}(1)$ $\displaystyle=h_{5}(0).$

for homotopic maps $h_{c},h_{c^{\prime}}$ from vertex $h_{4}(0)=h_{c}(0)$ to
centroid $c$ of a homotopic cycle and from $h_{c^{\prime}}(0)=c$ to
$h_{c^{\prime}}(1)=h_{5}(0)$ as shown in Fig. 4. This results in the homotopic
triangle

$\bigtriangleup_{h}h_{4}(0)ch_{5}(0)=\bigtriangleup_{h}h_{c}(0)ch_{c^{\prime}}(1).\mbox{\qquad{\color[rgb]{0,0,1}\SquareSteel}}$

######  Theorem 7.

Every triangulated homotopic cycle constructs an Alexandrov-Hopf nerve.

###### Proof.

Let $\mbox{hCyc}E$ be a homotopic cycle derived from the set of vertices $V$
along the boundary of a 1-cycle $\mbox{cyc}E$. Also let $c$ be centroid of
$\mbox{cyc}E$ and introduce a homotopic map from $h:V\times I\to c$, defined
by $h_{v}(0)=v\in V$ and $h_{v}(1)=c$. From Def. 8, the collection of
homotopic maps in $\mbox{hCyc}E$ combined with the homotopic $h:I\to c$
construct a collection of homotopic triangles $T$, which have the centroid
$c\in\mbox{int}(\mbox{hCyc}E)$ in common. Hence, by Def. 6, $T$ is an
Alexandroff nerve. ∎

######  Theorem 8.

Every collection of homotopic triangles with nonempty intersection is an
Alexandrov nerve.

###### Proof.

Given a collection $E$ of homotopic triangles with common vertex $h_{i}(0)$,
then by Def. 6, $T$ is an Alexandroff nerve. ∎

######  Example 6.

A sample collection of homotopic triangles in an Alexandroff nerve is shown in
Fig. 4. ◼

-v(-1.0,2)(-0.5,3) -v(-0.5,3)(0.5,3.5) -v(0.5,3.5)(1.5,3.0) -v(-0.3,0.85)(-0.5,1.5) -v(-0.5,1.5)(-0.4,2.25) -v(-0.4,2.25)(0.1,2.95) -v(0.1,2.95)(0.85,2.95) -v(0.85,2.95)(1.5,2.25) -v(1.5,0.75)(0.9,0.25) -v(1.5,2.25)(1.5,0.85) -v(0.9,0.25)(-0.3,0.85) -v(1.5,3.0)(2.0,2.0) -v(1.5,3.0)(0.5,1.5) -v(0.5,1.5)(2.0,2.0) -v(0.5,1.5)(2.0,1.0) -v(1.5,0.0)(0.5,1.5) -v(0.5,1.5)(0.5,-0.5) -v(1.5,0.0)(0.5,1.5) -v(0.5,1.5)(-1.0,1.0) -v(-1.0,2.0)(0.5,1.5) -v(0.5,1.5)(-0.5,3.0) -v(0.5,1.5)(0.5,3.5) -v(-0.5,0.0)(0.5,1.5) -v(2.0,1.0)(1.5,0.0) -v(1.5,0.0)(0.5,-0.5) -v(-0.5,0.0)(-1.0,1.0) -v(-1.0,1.0)(-1.0,2) $\boldsymbol{v_{1}}$$\boldsymbol{v_{0}}$$\boldsymbol{K}$$\boldsymbol{h_{i}(0)=h_{i-1}(1)}$$\boldsymbol{h_{i}(1)}$$\boldsymbol{h_{0}(0)=h_{(n-1)[n]}(1)}$$\boldsymbol{=\ v_{0}}$$\boldsymbol{h_{1}(0)\ =h_{0}(1)}$$\boldsymbol{h_{3}(0)=v_{3}}$$\boldsymbol{h_{(n-1)[n]}(0)}$$\boldsymbol{\mbox{bhCyc}E^{\prime}}$-v(-2.5,3.0)(-0.35,2.5) $\boldsymbol{\mbox{hCyc}E}$-v(-2.6,1.5)(-1.0,1.5) $\boldsymbol{h_{4}(0)}$$\boldsymbol{h_{5}(0)}$$\boldsymbol{c}$ Figure 5. Barycentric homotopic cycle $\boldsymbol{\mbox{bhCyc}E}$.

######  Definition 9.

Barycentric 1-Cycle.
Let $B$ be the set of barycenters on the triangles on a homotopic cycle in a
planar CW space $K$. The edges between each pair of adjacent barycenters in
$B$ define a barycentric 1-cycle. cycle. ◼

######  Theorem 9.

A complete sequence of paths over the vertexes in a barycentric 1-cycle
constructs a barycentric homotopic cycle.

###### Proof.

Let $B$ be the set of barycenters on the triangles in a homotopic cycle and
let $h:I\to B$ be a homotopic map over $B$. From Def. 5, the sequence of maps
$h_{0},h_{1},\dots,h_{i\mbox{mod}n}$ over adjacent pairs of barycenters
$b,b^{\prime}$ in $B$ ($h_{i}(0)=b,h_{i}(1)=b^{\prime}$) constructs a
barycentric homotopic cycle. ∎

######  Example 7.

A sample planar barycentric homotopic cycle is shown in Fig. 5. ◼

##  3\. Rotman Free Group Presentation

A finite group $G$ is free, provided every element $x\in G$ is a linear
combination of its basis elements (called generators) [10, §1.4, p. 21]. We
write $\mathcal{B}$ to denote a nonempty basis set of generators
$\left\\{g_{1},\dots,g_{\left|\mathcal{B}\right|}\right\\}$ and
$G(\mathcal{B},+)$ to denote the free group with binary operation $+$.

######  Definition 10.

Rotman Presentation[18, p.239]
Let $X=\left\\{g_{1},\dots\right\\},\bigtriangleup=\left\\{v=\sum
kg_{i},v\in\mbox{group}G,g_{i}\in X\right\\}$ be a set of generators of
members of a nonempty set $X$ and set of relations between members of $G$ and
the generators in $X$. A mapping of the form
$\left\\{X,\bigtriangleup\right\\}\to G$, a free group, is called a
presentation of $G$. ◼

We write $G(V,+)$ to denote a group $G$ on a nonvoid set $V$ with a binary
operation $+$. For a group $G(V,+)$ presentable as a collection of linear
combinations of members of a basis set $\mathcal{B}\subseteq V$, we write
$G(\mathcal{B},+)$.

######  Definition 11.

Free Group Presentation of a Cell Complex.
Let $2^{K}$ be the collection of cell complexes in a CW space $K$, $E\in
2^{K}$ containing $n$ vertexes, $G(E,+)$ a group on nonvoid set $E$ with
binary operation $+$, $\bigtriangleup=\left\\{v=\sum kg_{i},v\in E,g_{i}\in
E\right\\}$ be a set of generators of members in $E$, set of relations between
members of $E$ and the generators $\mathcal{B}\subset E$,
$g_{i}\in\mathcal{B},v=h_{i\mbox{mod}\ n}(0)\in K$, $k_{i}$ the $i^{th}$
integer coeficient $\mbox{mod}n$ in a linear combination
$\mathop{\sum}\limits_{i,j}k_{i}g_{j}$ of generating elements
$g_{j}=h_{j}(0)\in\mathcal{B}$. A free group presentation of $G$ is a
continuous map $f:2^{K}\times\bigtriangleup\to 2^{K}$ defined by

$\displaystyle f(\mathcal{B},\bigtriangleup)$
$\displaystyle=\left\\{v\mathrel{\mathop{:}}={\mathop{\sum}\limits_{i,j}k_{i}g_{j}\in\bigtriangleup}:v\in
E,g_{j}\in\mathcal{B},k_{i}\in\mathbb{Z}\right\\}$
$\displaystyle=\overbrace{\boldsymbol{G(\left\\{g_{1},\dots,g_{{}_{\left|\tiny\mathcal{B}\right|}}\right\\}},+).}^{\mbox{{\color[rgb]{0,0,1}\bf$\mathcal{B}\times\bigtriangleup\mapsto$
free group $\boldsymbol{G}$ presentation of
$G(E,+)$}}}\mbox{\qquad{\color[rgb]{0,0,1}\SquareSteel}}$

######  Lemma 3.

[15, §4, p. 10] Every 1-cycle in a CW space $K$ has a free group presentation.

###### Proof.

By definition, the vertices in a 1-cycle $\mbox{cyc}E$ are path-connected in a
CW space $K$. From [15], every 1-cycle can represented as a group
$G(\mbox{cyc}E,+)$ with binary (move) operation $+$. Choose a vertex
$g=v_{0}\in\mbox{cyc}E$ and let basis
$\mathcal{B}_{\mbox{cyc}E}=\left\\{g\right\\}$. Every vertex
$v_{i}\in\mbox{cyc}E,i>0$ is reachable in $k$ moves from $g\in\mathcal{B}$.
Then we have $v_{i}=kg=v_{0+k}$. Let $\bigtriangleup=\left\\{v=\sum
kg,g\in\mathcal{B},\mbox{\&}\ v,g\in\mbox{cyc}E\right\\}$. Then let the map
$f:2^{K}\times\bigtriangleup\to 2^{K}$ be defined by

$\displaystyle f(\mathcal{B}_{\mbox{cyc}E},\bigtriangleup)$
$\displaystyle=\left\\{v\mathrel{\mathop{:}}={\mathop{\sum}\limits_{i,j}k_{i}g\in\bigtriangleup}:v\in\mbox{cyc}E,g\in\mathcal{B}_{\mbox{cyc}E},k_{i}\in\mathbb{Z}\right\\}$
$\displaystyle=\overbrace{\boldsymbol{G(\left\\{g\right\\}},+).}^{\mbox{{\color[rgb]{0,0,1}\bf$\boldsymbol{\mathcal{B}_{\mbox{cyc}E}\times\bigtriangleup}\mapsto$
free group $\boldsymbol{G}$ presentation of $G(\mbox{cyc}E,+)$}}}$

∎

##  4\. Main Results

This section introduces main results for homotopic nerves.

######  Lemma 4.

[17] Every homotopic cycle in a CW space has a free group presentation.

###### Proof.

Let $E$ be a set of 1-cycle vertices in the domain and range of the maps
$h:I\to E$ in a homotopic cycle $\mbox{hCyc}E$ in a CW space $K$,
$G(\mbox{hCyc}E,+)$ a group on the vertexes of $\mbox{hCyc}E$ with move
operation $+$, and let

$\displaystyle\mathcal{B}$ $\displaystyle\subset\mbox{hCyc}E,$
$\displaystyle\bigtriangleup$ $\displaystyle=\left\\{v=\sum
kh_{i}(0),v,h_{i}(0)\in\mbox{hCyc}E\right\\}.$

be the basis $\mathcal{B}$ set of members of $\mbox{hCyc}E$ and set of
relations $\bigtriangleup$ between members of $\mbox{hCyc}E$ and the
generators in basis $\mathcal{B}$. Then the free group presentation of $G$ is
a mapping $f:2^{K}\times\bigtriangleup\to 2^{K}$ defined by

$\displaystyle f(\mathcal{B},\bigtriangleup)$
$\displaystyle=\left\\{v\mathrel{\mathop{:}}={\mathop{\sum}\limits_{i,j}k_{i}h_{j}(0)\in\bigtriangleup}:v\in\mbox{hCyc}E,h_{j}\in\mathcal{B},k_{i}\in\mathbb{Z}\right\\}$
$\displaystyle=\overbrace{\boldsymbol{G(\left\\{h_{1},\dots,h_{{}_{\left|\tiny\mathcal{B}\right|}}\right\\}},+).}^{\mbox{{\color[rgb]{0,0,1}\bf$\boldsymbol{\mathcal{B}\times\bigtriangleup}\mapsto$
free group presentation of $G(\mbox{hCyc}E,+)$}}}$

∎

######  Example 8.

From Lemma 4, the barycentric homotopic cycle $\mbox{bhCyc}E^{\prime}$ in
Example 7 has a free group presentation. ◼

######  Theorem 10.

Every homotopic nerve in a CW space has a free group presentation.

###### Proof.

Symmetric with the proof of Lemma 4, after replacing homotopic cycle
$\mbox{hCyc}E$ with homotopic nerve $\mbox{hNrv}E$ and replacing generator
$g_{j}$ with $h_{j}(0)$ in the basis for the free group presentation of the
homotopic nerve in a CW space. ∎

-v(-1.0,2)(-0.5,3) -v(-0.5,3)(0.5,3.5) -v(0.5,3.5)(1.5,3.0) -v(1.5,2.25)(1.5,3.0) -v(-0.3,0.85)(-0.5,1.5) -v(-0.5,1.5)(-0.4,2.25) -v(-0.4,2.25)(0.1,2.95) -v(0.1,2.95)(0.85,2.95) -v(0.85,2.95)(1.5,2.25) -v(1.5,0.75)(0.9,0.25) -v(1.5,2.25)(1.5,0.85) -v(0.9,0.25)(-0.3,0.85) -v(1.5,3.0)(2.0,2.0) -v(1.5,3.0)(0.5,1.5) -v(0.5,1.5)(2.0,2.0) -v(0.5,1.5)(2.0,1.0) -v(1.5,0.0)(0.5,1.5) -v(0.5,1.5)(0.5,-0.5) -v(1.5,0.0)(0.5,1.5) -v(0.5,1.5)(-1.0,1.0) -v(-1.0,2.0)(0.5,1.5) -v(0.5,1.5)(-0.5,3.0) -v(0.5,1.5)(0.5,3.5) -v(-0.5,0.0)(0.5,1.5) -v(2.0,1.0)(1.5,0.0) -v(1.5,0.0)(0.5,-0.5) -v(-0.5,0.0)(-1.0,1.0) -v(-1.0,1.0)(-1.0,2) $\boldsymbol{v_{1}}$$\boldsymbol{v_{0}}$$\boldsymbol{K}$$\boldsymbol{h_{i}(0)=h_{i-1}(1)}$$\boldsymbol{h_{i}(1)}$$\boldsymbol{h_{0}(0)=h_{(n-1)[n]}(1)}$$\boldsymbol{=\ v_{0}}$$\boldsymbol{h_{1}(0)\ =h_{0}(1)}$$\boldsymbol{h_{3}(0)=v_{3}}$$\boldsymbol{h_{(n-1)[n]}(0)}$$\boldsymbol{\mbox{hBr}A}$-v(3.0,3.0)(1.5,2.5) $\boldsymbol{\mbox{hCyc}E^{\prime}}$-v(-2.5,3.0)(-0.35,2.5) $\boldsymbol{\mbox{hCyc}E}$-v(-2.6,1.5)(-1.0,1.5) $\boldsymbol{h_{4}(0)}$$\boldsymbol{h_{5}(0)}$$\boldsymbol{c}$ Figure 6. Homotopic vortex $\boldsymbol{\mbox{vor}E}$.

######  Remark 1.

A collection of _ Nested 1-cycles_ provides the basic structure of a vortex.
Cycles are _nested_ , provided each appear, one inside the other. For example,
a pair of 1-cycles $\mbox{cyc}E,\mbox{cyc}E^{\prime}$ are nested, provided
$\mbox{cyc}E\subset\mbox{int}(\mbox{cyc}E^{\prime})$. In this work,
intersecting, nested cycles are possible, i.e.,

$\mbox{cyc}E\subset\mbox{int}(\mbox{cyc}E^{\prime})\ \mbox{and, possibly,}\
\mbox{cyc}E\cap\mbox{cyc}E^{\prime}\neq\emptyset.$

That is, each pair of nested cycles in a vortex can have at least one vertex
in common. To solve the problem of structuring a vortex $\mbox{vor}E$
containing non-intersecting nested cycles so that $\mbox{vor}E$ can be mapped
to a homotopic vortex, bridge edges are introduced (see Def. 15). ◼

######  Definition 12.

Given a pair of nested 1-cycles $\mbox{cyc}E,\mbox{cyc}E^{\prime}$, a bridge
edge $A$ (denoted by $\mbox{br}A$) is an edge with one vertex on $\mbox{cyc}E$
and one vertex on $\mbox{cyc}E^{\prime}$, or vice versa. ◼

A homotopic bridge is a path between edge vertices of a bridge edge.

######  Definition 13.

Given a pair of nested, homotopic 1-cycles
$\mbox{hCyc}E,\mbox{hCyc}E^{\prime}$, a homotopic bridge $A$ (denoted by
$\mbox{hBr}A$) is a homotopic map $h$ such that $h(0)=v\in\mbox{hCyc}E$ and
$h(1)=v^{\prime}\in\mbox{hCyc}E^{\prime}$, or vice versa. ◼

######  Example 9.

A sample homotopic bridge $\mbox{hBr}A$ between a pair of homotopic cycles is
shown in Fig. 6. ◼

######  Definition 14.

A vortex cell complex $E$ (denoted by $\mbox{vor}E$) is a collection of nested
1-cycles with at least one bridge edge between a pair of vertexes on the
cycles in the vortex, or the 1-cycles in $\mbox{vor}E$ have nonvoid
intersection. ◼

By replacing each of the edges in a vortex cell complex $\mbox{vor}E$ with a
homotopic map and replacing every bridge edge in $\mbox{vor}E$ with a
homotopic bridge, we obtain a homotopic vortex.

######  Definition 15.

A homotopic vortex $E$ (denoted by $\mbox{hVor}E$) is a collection of nested
homotopic 1-cycles with nonvoid intersection or with at least one homotopic
bridge between each pair of non-intersecting cycle vertexes in the vortex. ◼

######  Example 10.

A sample homotopic vortex $\mbox{hVor}E$ with a bridge edge is shown in Fig.
6. ◼

######  Theorem 11.

Every homotopic vortex has a free group presentation.

###### Proof.

Symmetric with the proof of Lemma 4, after replacing homotopic cycle
$\mbox{hCyc}E$ with homotopic vortex $\mbox{hVor}E$ in group
$G(\mbox{hVor}E,+)$ and replacing generator $g_{j}$ with $h_{j}(0)$ in basis
$\mathcal{B}$, the set of generator vertexes either on the end points of the
homotopic bridges or in the intersection of the $\mbox{hVor}E$ homotopic
cycles. Then introduce

$\bigtriangleup=\left\\{v=\sum kh_{i}(0),v\in E,h_{i}(0)\in\mathcal{B}\subset
E\right\\}$

with $E$ equal to the set of vertexes in the domain and range of the maps
$h:I\to E$ in $\mbox{hVor}E$ in a CW space $K$ and introduce a continuous
mapping $f:2^{K}\times\bigtriangleup\to 2^{K}$ defined by

$\displaystyle f(\mathcal{B},\bigtriangleup)$
$\displaystyle=\left\\{v\mathrel{\mathop{:}}={\mathop{\sum}\limits_{i,j}k_{i}h_{j}(0)\in\bigtriangleup}:v\in
E,h_{j}\in\mathcal{B},k_{i}\in\mathbb{Z}\right\\}$
$\displaystyle=\overbrace{\boldsymbol{G(\mathcal{B},+).}}^{\mbox{{\color[rgb]{0,0,1}\bf$\boldsymbol{\mathcal{B}\times\bigtriangleup}\mapsto$
free group presentation of $G(\mbox{hVor}E,+)$}}}$

∎

-v(-1.0,2)(-0.5,3) -v(-0.5,3)(0.5,3.5) -v(0.5,3.5)(1.5,3.0) -v(-0.3,0.85)(-0.5,1.5) -v(-0.5,1.5)(-0.4,2.25) -v(-0.4,2.25)(0.1,2.95) -v(0.1,2.95)(0.85,2.95) -v(0.85,2.95)(1.5,3.0) -v(1.5,3.0)(1.5,2.25) -v(1.5,0.75)(0.9,0.25) -v(1.5,2.25)(1.5,0.85) -v(0.9,0.25)(-0.3,0.85) -v(1.5,3.0)(2.0,2.0) -v(1.5,3.0)(0.5,1.5) -v(0.5,1.5)(2.0,2.0) -v(0.5,1.5)(2.0,1.0) -v(1.5,0.0)(0.5,1.5) -v(0.5,1.5)(0.5,-0.5) -v(1.5,0.0)(0.5,1.5) -v(0.5,1.5)(-1.0,1.0) -v(-1.0,2.0)(0.5,1.5) -v(0.5,1.5)(-0.5,3.0) -v(0.5,1.5)(0.5,3.5) -v(-0.5,0.0)(0.5,1.5) -v(2.0,1.0)(1.5,0.0) -v(1.5,0.0)(0.5,-0.5) -v(-0.5,0.0)(-1.0,1.0) -v(-1.0,1.0)(-1.0,2) $\boldsymbol{v_{1}}$$\boldsymbol{v_{0}}$$\boldsymbol{K}$$\boldsymbol{h_{i}(0)=h_{i-1}(1)}$$\boldsymbol{h_{i}(1)}$$\boldsymbol{h_{0}(0)=h_{(n-1)[n]}(1)}$$\boldsymbol{=\ v_{0}}$$\boldsymbol{h_{1}(0)\ =h_{0}(1)}$$\boldsymbol{h_{(n-1)[n]}(0)}$$\boldsymbol{\mbox{hCyc}E\cap\mbox{bhCyc}E^{\prime}=h_{4}(0)}$-v(3.0,4.5)(2.0,3.3) $\boldsymbol{\mbox{bhCyc}E^{\prime}}$-v(-2.5,3.0)(-0.35,2.5) $\boldsymbol{\mbox{hCyc}E}$-v(-2.6,1.5)(-1.0,1.5) $\boldsymbol{h_{4}(0)}$$\boldsymbol{h_{5}(0)}$$\boldsymbol{c}$ Figure 7. Homotopic vortex nerve $\boldsymbol{\mbox{vorNrv}E}$.

######  Definition 16.

A vortex nerve $E$ (denoted by $\mbox{vorNrv}E$) is a collection of nested
1-cycles that have nonempty intersection. ◼

By replacing each of the edges in a vortex nerve $\mbox{vorNrv}E$ homotopic
paths, we obtain a homotopic vortex nerve.

######  Definition 17.

A homotopic vortex nerve $E$ (denoted by $\mbox{hVorNrv}E$) is a collection of
homotopic vortexes that have nonempty intersection. ◼

######  Example 11.

A sample homotopic vortex nerve $\mbox{hVorNrv}E$ containing a single vortex
is shown in Fig. 7. This is a homotopic vortex nerve, which results from the
intersecting of a pair of nested homotopic cycles
$\mbox{hCyc}E,bhcycE^{\prime}$, i.e.,

$\mbox{hCyc}E\cap\mbox{bhCyc}E^{\prime}=h_{4}(0).$

This is also an example of an Alexandroff nerve, since the homotopic cycles in
$\mbox{hVorNrv}E$ have nonvoid intersection. The minimal basis
$\mathcal{B}=\left\\{h_{4}(0)\right\\}$ in the free group presentation of this
nerve. ◼

-v(0,0)(1,1) -v(1,1)(0,2) -v(0,2)(-1,1.5) -v(-1,1.5)(-1,0.5) -v(-1,0.5)(0,0) -v(0,0)(1,1) -v(1,1)(-.55,1.25) -v(-.55,1.25)(-.55,0.75) -v(-.55,0.75)(0,0.5) -v(0,0.5)(1,1) -v(1,1)(2.0,2.0) -v(2.0,2.0)(3.0,1.5) -v(3.0,1.5)(3.0,0.5) -v(3.0,0.5)(2.0,0.0) -v(2.0,0.0)(1,1) -v(1,1)(2.55,1.25) -v(2.55,1.25)(2.55,0.75) -v(2.55,0.75)(2.0,0.5) -v(2.0,0.5)(1,1) $\boldsymbol{\mbox{hCyc}A}$$\boldsymbol{\mbox{hCyc}B}$$\boldsymbol{\mbox{hCyc}A^{\prime}}$$\boldsymbol{\mbox{hCyc}B^{\prime}}$$\boldsymbol{v_{0}}$$\boldsymbol{K}$$\boldsymbol{\mbox{hVorNrv}E}$$\boldsymbol{\mbox{hVor}E}$$\boldsymbol{\mbox{hVor}E^{\prime}}$ Figure 8. Vigolo Hawaiian earrings pair $\boldsymbol{\to}$ homotopic vortex nerve $\boldsymbol{\mbox{hVorNrv}E}$

######  Example 12.

A second homotopic vortex nerve $\mbox{hVorNrv}E$ is shown in Fig. 8. This is
a homotopic vortex nerve, which results from an intersecting of a pair of
vortexes $\mbox{hVor}E,\mbox{hVor}E^{\prime}$, i.e.,

$\mbox{hVor}E\cap\mbox{hVor}E^{\prime}=\boldsymbol{v_{0}}.$

This nerve has been derived from a homotopic version of a Vigolo Hawaiian
earring [21], which is a wide ribbon for Vigolo. The minimal basis
$\mathcal{B}=\left\\{v_{0}\right\\}$ in the free group presentation of this
nerve. ◼

######  Theorem 12.

Every homotopic vortex nerve has a free group presentation.

###### Proof.

Symmetric with the proof of Theorem 11, after replacing homotopic vortex
$\mbox{hVor}E$ with homotopic nerve $\mbox{hVorNrv}E$ and replacing the
homotopic vortex basis with basis $\mathcal{B}$, the set of generator vertexes
either on the end points of the homotopic bridges between the vortex cycles in
$\mbox{hVorNrv}E$ or in the intersection of the $\mbox{hVor}E$ homotopic
cycles in $\mbox{hVorNrv}E$. ∎

######  Definition 18.

Edelsbrunner-Harer Nerve [4, §III.2,p. 59].
Let $F$ be a finite collection of sets. A nerve consists of all nonempty sets
$X$ in $F$ that have nonvoid intersection, i.e.,

$\mbox{Nrv}F=\left\\{X\subseteq F:\bigcap
X\neq\emptyset\right\\}.\mbox{\qquad{\color[rgb]{0,0,1}\SquareSteel}}$

######  Remark 2.

Def. 18 is a formal definition of the original view of an Alexandrov nerve
complex [2, 1]. ◼

######  Theorem 13.

[4, §III.2,p. 59].
Let $F$ be a finite collection of closed, convex sets in Euclidean space. Then
the nerve of $F$ and union of the sets in $F$ have the same homotopy type.

Let $i_{X},i_{Y}$ be identity maps, i.e.,

$\displaystyle i_{X}(x)$ $\displaystyle=x,\ \mbox{for all}\ x\in X.$
$\displaystyle i_{Y}(y)$ $\displaystyle=y,\ \mbox{for all}\ y\in Y.$

Recall that two spaces $X$ and $Y$ have the same homotopy type [10, §19,p.
108] (i.e., spaces $X$ and $Y$ are homotopy equivalent denoted by $X\equiv Y$
using Hilton’s notation [7]), provided the maps

$f:X\to Y\ \mbox{and}\ g:Y\to X$

are both homotopic such that

$f\circ g=i_{X}\ \mbox{and}\ g\circ f=i_{Y}.$

######  Theorem 14.

Let $F$ be a vortex nerve that is a finite collection of closed, convex sets
in Euclidean space. Then the nerve of $F$ and union of the sets in $F$ have
the same homotopy type.

###### Proof.

There are a number of cases to consider, namely,

  1. Case.1o

1-cycle nerve: Let $\mbox{Nrv}F$ be a finite collection of closed, convex
1-cycles in Euclidean space, i.e.,

$\mbox{Nrv}CycF=\left\\{\mbox{cyc}E\subseteq
F:\bigcap\mbox{cyc}E\neq\emptyset\right\\}.$

From Theorem 13, $\mbox{Nrv}CycF$ and the union of the 1-cycles in $F$ have
the homotopy type.

  2. Case.2o

homotopic 1-cycle nerve: After replacing $\mbox{cyc}E$ with $\mbox{hCyc}E$,
the proof for this case is symmetric with Case.1o.

  3. Case.3o

vortex nerve: Let $\mbox{Nrv}F$ be a finite collection of closed, convex,
nested, intersecting vortexes in Euclidean space. After replacing
$\mbox{cyc}E$ with $\mbox{vor}E$, the proof for this case is symmetric with
Case.1o, i.e.,

$\mbox{Nrv}VorF=\left\\{\mbox{vor}E\subseteq
F:\bigcap\mbox{vor}E\neq\emptyset\right\\}.$

From Theorem 13, $\mbox{Nrv}VorF$ and the union of the vortexes $\mbox{vor}E$
in $F$ have the homotopy type.

  4. Case.4o

homotopic vortex nerve: After replacing $\mbox{vor}E$ with $\mbox{hVor}E$ in
$\mbox{Nrv}VorF$ in Case.3o, the proof for this case is symmetric with
Case.3o.

∎

##  5\. Application

An application of homotopic vortex nerves is given in term of the persistence
of descriptively proximal video frame shapes. The motivation for considering
free group presentations of such nerves is that we can then describe frame
shapes in terms of the Betti numbers of the vortex nerves that lie within the
interior of the frame shapes. A Betti number is a count of the number
generators in a free group [10, §4,p. 24]. Frame shapes are, for example,
descriptively close, provided the difference between the Betti numbers of the
free group presentations of the corresponding homotopic vortex nerves, is
close. Determining the persistence of frame shapes then reduces to tracking
the appearance, disappearance and possible reappearance of the shapes in terms
of their recurring Betti numbers. For more about this, see [14]. ◼

##  Appendix A Cell Complexes

A planar Whitehead cell complex $K$ [13] (usually called a CW complex) is a
collection of n-dimensional minimal cells $e^{n}_{\alpha},n\in\\{0,1,2\\}$,
i.e.,

$K=\left\\{e^{n}_{\alpha}\subset\mathbb{R}^{2}:n\in\\{0,1,2\\}\right\\}.$

in the Euclidean plane $\pi$.

######  Definition 19.

A cell subcomplex
$\mbox{sh}E\mathrel{\mathop{:}}=\left\\{e^{n}_{\alpha}\right\\}\in 2^{K}$
(shape complex) is a closed subcomplex, provided the subcomplex includes both
a nonempty interior (denoted by $\mbox{int}(e^{n}_{\alpha})$) and its boundary
(denoted by $\mbox{bdy}(e^{n}_{\alpha})$). In effect, $\mbox{sh}E$ is closed,
provided

$\mbox{sh}E=\mbox{int}(\mbox{sh}E)\cup\mbox{bdy}(\mbox{sh}E)\ \mbox{({\bf
Closed subcomplex})}.$

Let $2^{\pi}$ be the collection of all subsets in the Euclidean plane $\pi$.
In the plane, a Whitehead Closure-finite Weak (CW) cell complex $K\in 2^{\pi}$
has two properties, namely,

C:

A cell complex $K$ is closure-finite, provided each cell $e^{n}_{\alpha}\in K$
is contained in a finite subcomplex of $K$. In addition, each cell
$e^{n}_{\alpha}\in K$ has a finite number of immediate faces. One cell
$e^{n}_{\alpha}$ is an immediate face of another cell $e^{m}_{\alpha}$,
provided $e^{n}_{\alpha}\cap e^{m}_{\alpha}\neq\emptyset$ [19] (also called a
common face).

W:

The plane $\pi$ has a weak topology induced by cell complex $K$, i.e., a
subset $S\in 2^{\pi}$ is closed, if and only if $S\cap e^{n}_{\alpha}$ is also
closed in $e^{n}_{\alpha}$ for each $n,\alpha$ [19, §5.3, p. 65].

A collection $K\in 2^{\pi}$ is called a CW complex, provided it has the
closure-finite property and $\pi$ has the weak topology property induced by
$K$.

Minimal cell planar complexes are given in Table 1.

Table 1. Minimal Planar Cell Complexes Minimal Complex |  Cell $e^{n}:n\in\\{0,1,2\\}$ | Planar Geometry | Interior
---|---|---|---
| $e^{0}$ | Vertex | nonempty
| $e^{1}$ | Edge | line segment w/o end points
| $e^{2}$ | Filled triangle | nonempty triangle interior w/o edges

######  Remark 3.

Closure finite cell complexes with weak topology (briefly, CW complexes) were
introduced by J.C.H. Whitehead [22], later formalized in [23]. In this
work333Here, we use $cl(e^{n})$ (closure of a cell) and $\mbox{bdy}(e^{n})$
(contour of a cell), instead of Whitehead’s $\bar{e^{n}}$ and
$\partial(e^{n})$., a cell complex $K$ (or complex) [23, §4, p. 221] is a
Hausdorff space (union of disjoint open cells $e,e^{n},e_{i}^{n}$) such that
the closure of an $n$ cell $e^{n}\in K$ (denoted by $\mbox{c$\ell$}(e^{n})$ is
the image of a map $f:\sigma^{n}\to\mbox{c$\ell$}(e^{n})$, where $\sigma^{n}$
is a fixed $n$-simplex and where the boundary $\mbox{bdy}(e^{n})$ (otherwise
known as the contour of a complex) is defined by

$\mbox{bdy}(e^{n})=\overbrace{f(\mbox{bdy}(e^{n}))=\mbox{c$\ell$}(e^{n})-\mbox{int}(e^{n}).}^{\mbox{{\color[rgb]{0,0,1}\bf
Complex contour $\to$ closure $\mbox{c$\ell$}(e^{n})$ minus $Int(e^{n})$
interior}}}$

Notice that a subcomplex $X\subset K$ has the weak topology, since $X$ is the
union of a finite number intersections $X\cap\mbox{c$\ell$}(e)$ for single
cells $e\in K$ [23, §5, p. 223]. From a geometric perspective, a cell complex
is a triangulation of the CW space $K$ [22, p. 246]. ◼

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