**5. Conclusions**

This study presented the findings on the levels of driving stress depending on the tra ffic conditions and road types, using the EDA signals. The study confirmed that EDA is a significant indicator of the psychological stress and an e ffective tool to determine the stress levels in real driving conditions. Advantages of the developed method are that both developed models have an ability to predict stress levels under actual driving conditions with over 80% accuracy. The developed models can also be used to classify driving stress in di fferent tra ffic situations and di fferent road types. The definition of a tra ffic jam state in a driving situation is one of the important contributions of this study. One of the characteristics of the models is the ability to classify the stress level without considering the rest state of the driver. This makes the models universal for use in a variety of situations without the preliminary intervention. The proposed model and the experimental procedure are expected to be easily reconfigured by other researchers. One of the disadvantages of the proposed model is that the tra ffic congestion concept needs to be defined and improved more specifically based on tra ffic flow theory. To make a better model in the future, the following points should be considered. First, researchers can collect more road information, such as lane-keeping status, as well as bio-signals, such as ECG. In addition, various kinds of machine-learning techniques can be applied instead of logistic regression. Finally, researchers can incorporate other factors, such as seasonal factors, in-vehicle temperature, and a driver's attention into the model as key features.

Obtained results can be used for practical application. Examples of such uses are sensors for monitoring of the autonomic nervous system, smartphone/computer applications and wearable wireless devices with EDA sensors.

**Author Contributions:** J.P. proposed the main idea and conducted the experiment; O.V.B. finished the draft manuscript and drew the figures and tables; J.P. and J.K. analyzed the data; H.K.K. revised and finished the manuscript.

**Funding:** This work was supported by the Incheon National University Research Grant in 2017 (Grant No.: 20170467).

**Conflicts of Interest:** The authors declare no conflict of interest.


**1. Introduction**

The COVID-19 pandemic has certainly unprecedently affected all spheres of life including school life. The educational institutions' closure and the necessary gradual move to the online means of instructing inevitably impacted scholastic achievement. Many of these institutions all around the world still grapple with how and when to make informed decisions about when to return to in-person courses.

Studies on how and how much COVID-19 impacted students learning would serve as a backdrop against which balanced decisions about jeopardies associated with the teaching staff and students would be taken.

The COVID-19 pandemic created a state of health emergency and the kingdom has taken drastic measures to prevent its expansion. The Ministry of National Education, Vocational Training, Higher Education, and Scientific Research took decisions to maintain the continuity of school life all along the educational year and decided to stop in-present education in all sectors including higher education levels. Courageous preventive measures were immediately taken and other decisions to adopt online courses and move to distance learning and teaching were also made. Many questions were raised related to teachers' readiness, ability, and aptitude to use information and communication technologies (ICTs), students' availability and ability to be online every time situations call for it, and the abrupt impact that technology could produce while moving from face-to-face to face-to-monitor.

Amidst all this unclarity and uncertainty about how much COVID-19 impacted scholastic achievement, there is growing consensus that this novel virus lockdown closures had negative effects on students' learning. However, the elucidation of this situation comes from the fact that after a year there is ample data in hand to go beyond prophetic measures and begin the testing of this data. Schools have endured the effects and are now after a year can measure the post-COVID-19 pandemic consequences.

For more than 15 years, the Ministry of National Education has introduced the teaching of information and communication technologies (ICTs) for the benefit of university students (ICT module for students of semester 5). A large number of teachers in the school have benefited from continuing training provided within the framework of the strategy of the GENIE program (Generalization of information and communication technologies in education in Morocco) and by the Moroccan-Korean Training Center [1].

For the first time in 2005, the Moroccan government adopted a new ICTs program under the appellation, GENIE, as one way of operationally implementing the national strategy of digitizing the public education sector and enabling students in primary, secondary, and tertiary levels to benefit from four components: infrastructure, teacher training, digital resources, and development of uses. This ICTs' generalization strategy, launched in 2006 and revised in 2009, has set a basis for the equipment of schools with Internet-connected ends with rich digital resources. Encouragement of creation and innovation of these resources as well as support to users was enhanced beside training modules for all stakeholders to guarantee good use and usage. By and large, the main objectives set by the GENIE strategy in 2006 were to convince teachers to accept and to stop resist and to actively involve them in to integrate ICT in their teaching and to contribute to the improvement of the quality of teaching and learning through the use of ICT. However, it is noteworthy to mention here that many teachers did not participate as the training was optional and many of them choose to resist and not to drop in and only some, the motivated, expressed their interest. Some of these were motivated by the awards the ministry granted [1].

The primary focus of this study is to determine whether the students of the faculty of Languages, Lettres, and Arts, Ibn Tofail University suffer psychologically under the influence of the lockdown caused by the COVID-19 pandemic. Using data from hundreds of students, 297 participants, these research questions are set forth for the present study:

1.Is students' perceived scholastic achievement affected by COVID-19 pandemic conditions?

*From Face-to-Face to Face-to-Screen: A Correlational Analysis of Psychological Impacts… DOI: http://dx.doi.org/10.5772/intechopen.102547*

The hypotheses emanating from the above research questions are:

#### **2. Review of the literature**

The COVID-19 pandemic has caused unprecedentedly considerable repercussions on human life in every aspect and the education sector is no exception to this devastating contagion. Against this backdrop, governments were quick in responding to this devastating contamination as educational institutions halted every aspect of contact and were forced to shift to online classes. This novel virus that was declared a pandemic on March 11, 2020, has spread speedily ravaging peoples' lives and turning them upside down, school life was no exception. The outbreak of this pandemic pushed several if not all scientists to try their best to study its ramifications and effects at various levels, particularly at the educational level.

In this regard, in the educational context, the devastating impacts were also felt [2]. Several concurring factors have converged to make the COVID-19 pandemic a worldwide catastrophe without precedent on school life and hence produced great psychological sufferings [3]. Therefore, there are different types of the risk factors related to COVID-19 psychological perceived impact. Prabhakar, Kapoor, and Mahajan [4]. Stress the fact that the crisis created by the pandemic is cruel to school children as it can lead to stress, mental illnesses, and various psychological problems such as lack of coping skills and even enhanced self-harm behaviors. In the same vein, Handayani & Sri Sumartiningsih [5] claim that like any other pandemic, COVID-19 had diverse effects on different groups with different degrees of impact. University students are a vulnerable group that suffered directly from stress, depression, and anxiety effects due to disruption or cancelation of usual life events like final in-present final exams and/or graduation ceremonies.

Empirically and in a study on stressors, coping strategies, and mental health impacts of COVID-19 in international students studying abroad led by [6], the team discovered that more than 80% of the students had moderate-to-high perceived stress. Exhaustively, in the sample studied, they found that stress related to academics, health, and lack of social support were predictive of higher perceived stress levels and more severe anxiety and depression symptoms. In another study conducted by [7, 8], these scholars have found that the risk impact factors differ according to social groups to which students pertain.  They also found themselves disadvantaged insofar as the availability of online teaching and learning means and hence cause

unwillingness and feeling of bitterness not to pursue their studies online like others where internet connectivity is better.

Extra variables that impacted students' academic achievement, as well as life, can be explained through the learning theories that marked the history of humans as to their learning and environment with particular focus on the bioecological systems theory. The behaviorists with their revolutionizing of our understanding by advancing the principle of operant conditioning trial and error and reinforcement or punishment. Later came the social learning theory by Albert Bandura who elaborated on the behaviorist theory but considered humans as thinking beings who anticipate, punish themselves, and are capable of storing information for later use. The cognitive-developmental theory of the Swiss Linguist Jean Piaget followed strongly later. The theory emphasized the importance of 'constructing' an advanced mode of thinking through a mechanism of combining maturation and experience. Other views of cognitive development along with that of Piaget's were very influential too, the reference here is to Vygotsky's sociocultural and the informative-processing perspectives. The theory adopted here insofar as the academic achievement of our respondents is the Bioecological systems theory by [9].

This persuasive model lays the ground for another deeper understanding of human development in relation to the changing environment and its effects on academic achievement. The system theories postulate that this development is impacted by an interplay between the changing context and the changing environment where one affects the other and where they are joined at the hip units [10]. The complex mutual impacts are seen as a co-acting process between the two parties which are entangled in a dynamic system. According to Bronfenbrenner's perspective, there exist five systems that interact with the individual and many bidirectional or reciprocal impacts are at play as is shown in **Figure 1**.

The first system is the microsystem which is described as the "immediate setting containing that person" [6, p. 514], that is, the direct context in which the person gets in first touch with other people like family, neighbors, and peers. It is the proximal level of context to the individual. The mesosystem is the second layer of the model and is referred to as 'a system of microsystems' through which youth's life is influenced by one or more environments without direct contact with the youth [11]. The macrosystem comprises broad social environments or social beliefs and values that impact the youth incorporating features such as political, religious, and educational values as well as appropriate standards of behavior. The exosystem is described as the system that involves social settings where the individuals

#### **Figure 1.**

*Bronfenbrenner's ecological model of individual development. (Adopted from Sigelman & Rider [10], p. 114).*

*From Face-to-Face to Face-to-Screen: A Correlational Analysis of Psychological Impacts… DOI: http://dx.doi.org/10.5772/intechopen.102547*

do not interact with but can still influence their development [10]. It shows how distal this system is from the individual, however, decisions taken significantly affect him/her. In addition to the nested series of systems, the chronosystem is the last added system to the model and stipulates that the interplay between the environment and the individual change over time and have different ramifications and schemes [12].

The evolutionary, psychoanalytic, learning, cognitive-developmental, and bioecological systems perspectives capture the complexity of human life with its interplay with the environment be it immediate or distant. However, the bioecological systems theory is more probing as to its complete depiction of the whole human development especially when it comes to academic achievement.


**Physiological and Pathological Role of ROS**

**Benefits and Limitations of Antioxidant Treatment**

Special Issue Editors

**Sergio Di Meo Paola Venditti Gaetana Napolitano**

MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin

*Special Issue Editors* Sergio Di Meo University of Naples Federico II Italy

Gaetana Napolitano University of Naples Parthenope Italy

*Editorial Office* MDPI St. Alban-Anlage 66 4052 Basel, Switzerland Paola Venditti University Federico II of Naples Italy

This is a reprint of articles from the Special Issue published online in the open access journal *International Journal of Molecular Sciences* (ISSN 1422-0067) (available at: https://www.mdpi.com/ journal/ijms/special issues/ROS).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. *Journal Name* **Year**, *Article Number*, Page Range.

**ISBN 978-3-03936-282-0 (Hbk) ISBN 978-3-03936-283-7 (PDF)**

c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications.

The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND.


**8.1 Successes achieved**

After the ICHF/KHF mission in our center in 2003, pediatric cardiology and surgery activities in our center dwindled owing to dilapidated equipment, poor workers' remuneration, and brain drain as well as poor leadership, coupled with government directive

## *Establishing Sustainable Pediatric Cardiac Surgery Program in Nigeria: Challenges and Prospects DOI: http://dx.doi.org/10.5772/intechopen.102737*

to move UNTH to the permanent site without any building for pediatric cardiac surgery activities. In 2007, UNTH moved to the permanent site and then came a change in the leadership of NCTCE. Through continuous appropriate dialog and advocacy, NCTCE benefitted from TETFUND program. With advice from Prof. Tom Pezzella of ICHF, USA, a training center was found in India (ICHH, Hyderabad India), with institutional collaboration. With this development, pediatric cardiac anesthetists and intensivists, technicians, ICU and perioperative nurses including pediatric cardiologists got training within periods ranging from 6 months to 2 years.

There was the arrangement for pediatric cardiac surgeons to do 1 year training at the Indian center with the Indian team coming to our center to see a smooth takeoff of sustainable pediatric cardiac surgery program. However, that arrangement did not materialize because cardiac mission model was adopted. This cardiac mission model started in February 2013 and ended in October 2019. During the period, William Global Cardiac Alliance, Vincent Ohaju Memorial Foundation, Save-A-Heart-Nigeria, Cardiostart International, O'Pobic, Santarina all visited our center and performed operations in 113 in pediatric patients over 7 year period [23, 24].

Currently, our cardiologists can adequately handle all echocardiographic investigations. In addition, our adult cardiologists are able to handle all forms of cardiac pacemaker insertions, coronary angiography, and some coronary artery stenting and angioplasties. Our cardiothoracic surgeons with anesthesiologists and other theater staff can handle all non-open heart surgical procedures and some open-heart surgical procedures such as repair of atrial septal defects (ASD), repair of some ventricular septal defects (VSD), excision of some types of intracardiac tumors, and replacement of the mitral valve. We are still not able to do intracardiac repair for the blue babies in addition to some other types of congenital cardiac anomalies as well as some types of valve repair/replacement and coronary artery bypass procedures.

The COVID-19 pandemic has slowed the engagement of a resident foreign pediatric cardiac surgeon (GhanianModerl–ref) at NCTCE by the present UNTH/NCTCE management to equip the local surgeon with adequate skills that will make the program very sustainable.

*Congenital Heart Defects - Recent Advances*


**Abstract**

With the rapid development of high-performance perovskite solar cell, its instability has become an urgent problem to be solved. 2D perovskite is considered as a potential light absorbing material for perovskite solar cells due to its excellent stability. However, the preparation of high quality 2D perovskite films suitable for photovoltaic devices remains a challenge. In this chapter, based on the structural and photophysical properties of 2D perovskite thin film materials, the latest progress in 2D perovskite cells in recent years and the strategy of controlling the film quality of 2D perovskite are summarized, which is of great significance for the further development of 2D perovskite photovoltaic devices.

**Keywords:** perovskite solar cell, stability, 2D perovskite, material property, film quality


*3.2. Composition of Fungal Microbiomes in MOR and AOF Samples*

In the MOR group, the dominant phylum was Ascomycota with a relative abundance of 49.53–94.32%, whereas the other phyla were low in abundance (Figure 2a). Eurotiomycetes, Dothideomycetes, and Sordariomycetes were dominant at the class level, accounting for 2.09–78.91%, 1.88–57.02%, and 0.60–20.37% of the fungal reads, respectively (Figure 2b). At the order level, Capnodiales was predominant in MCK1, GDM3, GXM1, GXM2, and GXM3, whereas Eurotiales was predominant in MCK2, MCK3, GDM1, and GDM2 (Figure 2c). Further taxonomical classification demonstrated that Aspergillaceae (1.90–76.73%) was the most dominant at the family level, followed by Cladosporiaceae (1.83–56.78%) and Trichocomaceae (0.12–20.45%, Figure 2d). At the genus level, *Penicillium* (1.86–76.14%), *Cladosporium* (1.82–56.65%), *Trichoderma* (0.12–19.71%), *Monascus* (0.00–19.35%), and *Talaromyces* (0.18–19.71%) were the top 5 genera with the highest relative abundance (Figure 3a). The top 20 abundant genera were visualized using a heatmap, which showed the relative abundance of fungal genera in different samples (Figure 3b).

In the AOF group, Ascomycota was the most abundant phylum representing 14.81–81.85%, followed by Basidiomycota (0.36–52.57%, Figure 4a). At the class level, Eurotiomycetes (0.57–56.84%), Tremellomycetes (0.11–51.81%), and Dothideomycetes (3.13– 46.37%) were prevalent (Figure 4b). Among the 39 orders detected, Eurotiales, Tremellales, and Capnodiales were dominant with the relative abundances of 0.41–56.29%, 0.09–51.78%, and 3.11–46.30%, respectively (Figure 4c). At the family level, the dominant fungi in samples ACK1 and HNA3 were Aspergillaceae (56.17%) and Rhynchogastremataceae (51.71%), respectively. Cladosporiaceae was dominant in the rest of the samples representing 5.71– 46.30% (Figure 4d). *Penicillium* (0.27–56.06%), *Papiliotrema* (0.04–51.71%), and *Cladosporium* (3.08–44.41%) were dominant at the genus level (Figure 5a). *Candida* was the subdominant genus in the ACK1 with the relative abundance of 11.95%, while it accounted for low levels in other samples (0.00–4.51%). A heatmap of the 20 most abundant genera is presented in Figure 5b.

**Figure 2.** Fungal composition of the MOR samples at the phylum (**a**), class (**b**), order (**c**), and family (**d**) levels.

**Figure 3.** Composition analyses of the fungal microbiomes in the MOR samples. (**a**) Fungal composition in MOR samples at genus level; (**b**) Heatmap of the top 20 genera in MOR samples.

**Figure 4.** Fungal composition of the AOF samples at the phylum (**a**), class (**b**), order (**c**), and family (**d**) levels.

**Figure 5.** Composition analyses of the fungal microbiomes in the AOF samples. (**a**) Fungal composition in AOF samples at genus level; (**b**) Heatmap of the top 20 genera in AOF samples.

It is worth noting that a potential toxigenic fungus, namely, *Trichoderma atroviride*, was detected in all samples. Furthermore, another potential toxigenic fungus (*Fusarium equiseti*) was detected in four non-moldy samples (MCK1, MCK2, ACK1, and ACK3). The species and relative abundance of harmful fungi in each sample are shown in Supplementary Table S2.


**2. Materials and Methods**

#### *2.1. Automatic Apple Grader Design*

The structure of the automatic apple grader designed in this paper is shown in Figure 1. It consists of Feeding and material handling lifting mechanism, turnover detection conveyor, visual inspection and automatic grading control system, and graded actuators. The design is based on a two-level layout to reduce the space required.

**Figure 1.** Structure of the automatic apple grader.

(1) Feeding and material handling lifting mechanism. The Feeding and material handling lifting mechanism is a scraper elevator, as shown in Figure 2. The scraper elevator consists of a funnel-shaped storage tank and a vertical conveyor belt, where the funnel-shaped storage tank includes the back plate of the hopper and the support plate, the three-dimensional conveyor belt includes the guide plate and the curved scraper, and the whole mechanism is placed at an inclination of 45◦. The scraper elevator moves the conveyor belt by means of an AC motor driven by a frequency converter, which organizes the disordered apples into an orderly quadruple queue, transporting them from the bottom upwards and conveying them into the Turnover detection conveyor.

**Figure 2.** Scraper elevator. (1—hopper, 2—support plate, 3—guide plate, 4—curved scraper).

(2) Turnover detection conveyor. The Turnover detection conveyor is shown in Figure 3 and consists of sprockets, chains, sponge rollers, and motors. The apples are lifted by the scraper elevator into the turnover detection conveyor. The turnover detection conveyor uses pairs of double-tapered rollers to turn the apples axially, and a CCD industrial camera mounted on top of the lampshade collects images of the tumbling apples several times to obtain complete surface information about the apples in a moving position.

**Figure 3.** Turnover detection conveyor. (1—sprockets, 2—chains, 3—sponge rollers, 4—motor).

(3) Visual inspection and automatic grading control system. The visual inspection and automatic grading control system are shown in Figure 4 and consist of a CCD industrial camera and automatic grading control system. The visual inspection and automatic grading control system determines the grading of apples according to the information collected by the CCD industrial camera on the whole surface of the apples and finally sends the grading results to the graded actuators.

**Figure 4.** Visual inspection and automatic grading control system structure diagram.

(4) Graded actuators. The graded actuator is shown in Figure 5 and consists of a Trigger grading mechanism, sprocket chain drive, grading fruit cup, grading channel, and a motor. The grading fruit cup is shown in Figure 5b and consists of a cup body, a drop door, and rollers. The grading actuator receives the grading results from the image detection and automatic grading system and allows the apples to reach the corresponding grade position and then open the cups and fall into the corresponding grade storage bin.

**Figure 5.** Graded actuators. (**a**) Grading actuator; (**b**) Detail of grading fruit cup. (1-Trigger grading mechanism; 2-Sprocket chain drive; 3-Grading fruit cup; 4-Grading channel; 5-Motor).


**2. Foliations, WZW** *σ***-models and exotic R**4

An exotic **R**4 is a topological space with **R**4−topology but with a smooth structure different (i.e. non-diffeomorphic) from the standard **R**4 *std* obtaining its differential structure from the product **R** × **R** × **R** × **R**. The exotic **R**4 is the only Euclidean space **R***n* with an exotic smoothness structure. The exotic **R**4 can be constructed in two ways: by the failure to split arbitrarily a smooth 4-manifold into pieces (large exotic **R**4) and by the failure of the so-called smooth h-cobordism theorem (small exotic **R**4). Here, we deal with the later kind of exotics. We refer the reader to Asselmeyer-Maluga & Brans (2007) for general presentation of various topological and geometrical constructions and their physical perspective. Another useful mathematical books are Gompf & Stipsicz (1999); Scorpan (2005). The reader can find further results in original scientific papers.

Even though there are known, and by now rather widely discussed (see the Introduction), difficulties with making use of different differential structures on **R**4 (and on other open 4-manifolds) in explicit coordinate-like way (see e.g. Asselmeyer-Maluga & Brans (2007)), it was, however, established, in a series of recent papers, the way how to relate these 4-exotics with some structures on *S*3 (see e.g. Asselmeyer-Maluga & Król (2009a;b; 2011b)). This *S*3 is supposed to fulfil specific topological conditions: it has to lie in ambient **R**4 such that it is a part of the boundary of some compact 4-submanifold with boundary, i.e. Akbulut cork. If so, one can prove that exotic smoothness of the **R**4 is tightly related to codimension-one foliations of this *S*3, hence, with the 3-rd real cohomology classes of *S*3. Reformulating Theorem 1 we have [Asselmeyer-Maluga & Król (2009a)]:

*The exotic* **R**4*'s, from the radial family of exotic* **R**4*'s embedded in standard* **R**4*, are determined by the codimension-1 foliations,* F*'s, with non-vanishing Godbillon-Vey (GV) class in H*3(*S*3, **R**) *of a 3-sphere lying at the boundary of the Akbulut corks of* **R**4*'s. The radius in the family, ρ, and value of GV are related by* GV = *ρ*2*. We maintain: the exoticness is localized at a 3-sphere inside the small exotic* **R**4 *(seen as a submanifold of* **R**4*).*

Let us explain briefly, following Asselmeyer-Maluga & Król (2011d), how the codimension-1 foliations of *S*3 emerges from the structure of exotic **R**4. The complete construction and proof can be found in Asselmeyer-Maluga & Król (2009a).

4 Will-be-set-by-IN-TECH

effective matter states (without gravity) are described by dual theories with gravity. Hence

In the next section we describe the relation of small exotic **R**4 with foliations of *S*3 and WZW models on *SU*(2). Then we show the connections between string theory and exotic **R**4. In particular 4-smoothness underlying spacetime emerges from superstring calculations and it modifies the spectra of charged particles in such spacetime. In Sec. 4 we discuss the Kondo state and show that it generates the same exotic 4-smoothness. Moreover, the Kondo state, when survive the high energy and relativistic limit, would couple to the gravity backgrounds of superstring theory. The backgrounds are precisely those related with exotic smooth **R**4 as in Sec. 3. We conjecture that one could encounter the experimental trace of existence of exotic

*p* in the *k*-channel, *k* > 2, Kondo effect, where the usual fusion rules of the *SU*(2)*k* WZW

Next in Sec. 5 we present the connections of branes configurations in superstring theory with

An exotic **R**4 is a topological space with **R**4−topology but with a smooth structure different

product **R** × **R** × **R** × **R**. The exotic **R**4 is the only Euclidean space **R***n* with an exotic smoothness structure. The exotic **R**4 can be constructed in two ways: by the failure to split arbitrarily a smooth 4-manifold into pieces (large exotic **R**4) and by the failure of the so-called smooth h-cobordism theorem (small exotic **R**4). Here, we deal with the later kind of exotics. We refer the reader to Asselmeyer-Maluga & Brans (2007) for general presentation of various topological and geometrical constructions and their physical perspective. Another useful mathematical books are Gompf & Stipsicz (1999); Scorpan (2005). The reader can find further

Even though there are known, and by now rather widely discussed (see the Introduction), difficulties with making use of different differential structures on **R**4 (and on other open 4-manifolds) in explicit coordinate-like way (see e.g. Asselmeyer-Maluga & Brans (2007)), it was, however, established, in a series of recent papers, the way how to relate these 4-exotics with some structures on *S*3 (see e.g. Asselmeyer-Maluga & Król (2009a;b; 2011b)). This *S*3 is supposed to fulfil specific topological conditions: it has to lie in ambient **R**4 such that it is a part of the boundary of some compact 4-submanifold with boundary, i.e. Akbulut cork. If so, one can prove that exotic smoothness of the **R**4 is tightly related to codimension-one foliations of this *S*3, hence, with the 3-rd real cohomology classes of *S*3. Reformulating Theorem 1 we

*The exotic* **R**4*'s, from the radial family of exotic* **R**4*'s embedded in standard* **R**4*, are determined by the codimension-1 foliations,* F*'s, with non-vanishing Godbillon-Vey (GV) class in H*3(*S*3, **R**) *of a 3-sphere lying at the boundary of the Akbulut corks of* **R**4*'s. The radius in the family, ρ, and value of GV are related by* GV = *ρ*2*. We maintain: the exoticness is localized at a 3-sphere inside the small*

Let us explain briefly, following Asselmeyer-Maluga & Król (2011d), how the codimension-1 foliations of *S*3 emerges from the structure of exotic **R**4. The complete construction and proof

*std* obtaining its differential structure from the

non-standard 4-smoothness of **R**4. Discussion and conclusions close the Chapter.

gravity is inherently present in description of such condensed matter states.

model would be modified to these of *SU*(2)*p* WZW in high energies.

**2. Foliations, WZW** *σ***-models and exotic R**4

(i.e. non-diffeomorphic) from the standard **R**4

results in original scientific papers.

have [Asselmeyer-Maluga & Król (2009a)]:

*exotic* **R**4 *(seen as a submanifold of* **R**4*).*

can be found in Asselmeyer-Maluga & Król (2009a).

**R**4

Small exotic **R**4 is determined by the compact 4-manifold *A* with boundary *∂A* which is homology 3-sphere, and attached several Casson handles CH's. *A* is the Akbulut cork and CH is built from many stages towers of immersed 2-disks. These 2-disks cannot be embedded and the intersection points can be placed in general position in 4D in separated double points. Every CH has infinite many stages of intersecting disks. However, as Freedman proved, CH is topologically the same as (homeomorphic to) open 2-handle, i.e. *D*2 × **R**2. Now if one replaces CH's, from the above description of small exotic **R**4, by ordinary open 2-handles (with suitable linking numbers in the attaching regions) the resulting object is standard **R**4. The reason is the existence of infinite (continuum) many diffeomorphism classes of CH, even though all are topologically the same.

Consider the following situation: one has two topologically equivalent (i.e. homeomorphic), simple-connected, smooth 4-manifolds *M*, *M*� , which are not diffeomorphic. There are two ways to compare them. First, one calculates differential-topological invariants like Donaldson polynomials Donaldson & Kronheimer (1990) or Seiberg-Witten invariants Akbulut (1996). But there is yet another possibility – one can change a manifold *M* to *M*� by using a series of operations called surgeries. This procedure can be visualized by a 5-manifold *W*, the cobordism. The cobordism *W* is a 5-manifold having the boundary *∂W* = *M* � *M*� . If the embedding of both manifolds *M*, *M*� into *W* induces a homotopy-equivalence then *W* is called an h-cobordism. Moreover, we assume that both manifolds *M*, *M*� are compact, closed (without boundary) and simply-connected. Freedman (1982) showed that every *h*-cobordism implies a homeomorphism, hence *h*-cobordisms and homeomorphisms are equivalent in that case. Furthermore, the following structure theorem for such h-cobordisms holds true [Curtis & Stong (1997)]:

*Let W be a h-cobordism between M*, *M*� *. Then there are contractable submanifolds A* ⊂ *M*, *A*� ⊂ *M*� *together with a sub-cobordism V* ⊂ *W with ∂V* = *A* � *A*� *(the disjoint oriented sum), so that the h-cobordism W* \ *V induces a diffeomorphism between M* \ *A and M*� \ *A*� *.*

Thus, the smoothness of *M* is completely determined (see also Akbulut & Yasui (2008; 2009)) by the contractible submanifold *A* (Akbulut cork) and its embedding *A* → *M* determined by a map *τ* : *∂A* → *∂A* with *τ* ◦ *τ* = *id∂A* and *τ* �= ±*id∂A*(*τ* is an involution). Again, according to Freedman (1982), the boundary of every contractible 4-manifold is a homology 3-sphere. This *h*-cobordism theorem is employed to construct an exotic **R**4. First, one considers a neighborhood (tubular) of the sub-cobordism *V* between *A* and *A*� . The interior of *V*, *int*(*V*), (as open manifold) is homeomorphic to **R**4. However, if (and only if) *M* and *M*� are not diffeomorphic (still being homeomorphic), then *int*(*V*) ∩ *M* is an exotic **R**4.

Next, Bizaca (1994) and Bi ˘ zaca & Gompf (1996) showed how to construct an explicit handle ˘ decomposition of the exotic **R**4 by using *int*(*V*). The details of the construction can be found in their papers or in the book Gompf & Stipsicz (1999). The idea is simply to use the cork *A* and add some Casson handle to it. The interior of this resulting structure is an exotic **R**4. The key feature here is the appearance of *CH*. Briefly, a Casson handle *CH* is the result of attempts to embed a disk *D*2 into a 4-manifold. In most cases this attempt fails and Casson (1986) searched for a possible substitute, which is just what we now call a Casson handle. Freedman (1982) showed that every Casson handle *CH* is homeomorphic to the open 2-handle *D*2 × **R**2 but in nearly all cases it is not diffeomorphic to the standard handle, Gompf (1984; 1989). The Casson handle is built by iteration, starting from an immersed disk in some 4-manifold *M*, i.e. a map *D*2 → *M* which has injective differential. Every immersion *D*2 → *M* is an embedding except on a countable set of points, the double points. One can "kill" one double point by immersing

det *g* = −1. The left invariant 1-form *g*−1*dg* generates locally the cotangent space connected to the unit. The forms *ωk* = *Tr*((*g*−1*dg*)*k*) are complex *k*−forms generating the deRham cohomology of the Lie group. The cohomology classes of the forms *ω*1, *ω*2 vanish and *ω*3 ∈ *H*3(*SU*(2), **R**) generates the cohomology group. Then, we obtain as the value for the

Quantum Gravity Insights from Smooth 4-Geometries on Trivial 4 59

*ω*3 = 1 .

*Tr*((*G*−1*dG*)3) ∈ **Z**

*k 's, k* ∈ **Z** *from the radial family determines the WZ term of the k WZW*

*k* , then the WZ term

*k* with

*l , k*, *l* ∈ **Z** *both from the radial family, corresponds*

*k* i.e. *S*3 × **R**. This end cannot be standard smooth

*k* . The appearance of the *SU*(2)*k* × **R** is

*S*3=*SU*(2)

Ω3 = 1 8*π*2

This integral can be interpreted as winding number of *g*. Now, we consider a smooth map

*S*3

is the winding number of *G*. Every Godbillon-Vey class with integer value like (1) is generated by a 3-form Ω3. Therefore, the Godbillon-Vey class is the WZ term of the *SU*(2)*k*. Thus, we

This WZ term enables one for the cancellation of the quantum anomaly due to the conformal invariance of the classical *σ*-model on *SU*(2). Thus, we have a method of including this cancellation term from smooth 4-geometry: when a smoothness of the ambient 4-space, in

of the classical *σ*-model with target *S*3 = *SU*(2), i.e. *SU*(2)*k* WZW, is precisely generated by

1 8*π*2

*G* : *S*3 → *SU*(2) with 3-form Ω3 = *Tr*((*G*−1*dG*)3) so that the integral

which *S*3 is placed as a part of the boundary of the cork, is this of exotic **R**4

this 4-smoothness. As the conclusion, we have the important correlation:

*to the change of the level k of the WZW model on SU*(2)*, i.e. k* WZW →*l* WZW*.*

*k to exotic* **R**4

and it is in fact fake smooth *S*3 ×Θ*k* **R**, Freedman (1979). Given the connection of **R**4

the WZ term as above, we have determined the "quantized" geometry of *SU*(2)*k* × **R** as

a source for various further constructions. In particular, we will see that gravitational effects

Let us, following Asselmeyer-Maluga & Król (2011a) (see also Król (2011a;b)), begin with a charged quantum particle, say *e*, moving through non-flat gravitational background, i.e. smooth 4-spacetime manifold. The amount of gravity due to the curvature of this background affects the particle trajectory as predicted by GR. There should exist, however, a high energy limit where gravity contained in this geometrical background becomes quantum rather than classical and the particle may not be described by perturbative field theory any longer. This rather natural, from the point of view of physics, scenario requires, however, quantum gravity calculations which is not in reach in dimension 4. Moreover, mathematics underlying classical

*k* on the quantum level are determined via string theory where one replaces consistently

*S*3=*SU*(2)

1 8*π*2

integral of the generator

obtain the relation:

*model on SU*(2)*.*

of **R**4

*The structure of exotic* **R**4

*The change of smoothnes of exotic* **R**4

Let us consider now the end of the exotic **R**4

**3. 10d string theory and 4d-smoothness**

corresponding to the exotic geometry of the end of **R**4

flat **R**4 part of the background by curved 4D *SU*(2)*k* × **R**.

another disk into that point. These disks form the first stage of the Casson handle. By iteration one can produce the other stages. Finally, we consider a tubular neighborhood *D*2 × *D*2 of this immersed disk, called a kinky handle, on each stage. The union of all neighborhoods of all stages is the Casson handle. So, there are two input data involved with the construction of a *CH*: the number of double points in each stage and their orientation ±. Thus, we can visualize the Casson handle *CH* by a tree: the root is the immersion *D*2 → *M* with *k* double points, the first stage forms the next level of the tree with *k* vertices connected with the root by edges etc. The edges are evaluated using the orientation ±. Every Casson handle can be represented by such an infinite tree. The structure of CH as immersed many-layers 2-disks will be important in Sec. 4 where we will assign fermion fields to CH's.

Next, we turn again to the radial family of small exotic **R**4, i.e. a continuous family of exotic {**R**4 *ρ*}*ρ*∈[0,+∞] with parameter *ρ* so that **R**4 *ρ* and **R**4 *ρ*� are non-diffeomorphic for *ρ* �= *ρ*� . The point is that this radial family has a natural foliation (see Theorem 3.2 in DeMichelis & Freedman (1992)) which can be induced by a polygon *P* in the two-dimensional hyperbolic space **H**2. The area of *P* is a well-known invariant, the Godbillon-Vey class as the element in *H*3(*S*3, **R**). Every GV class determines a codimension-one foliation on the 3-sphere (firstly constructed by Thurston (1972); see also the book Tamura (1992) chapter VIII for the details). This 3-sphere is a part of the boundary *∂A* of the Akbulut cork *A* (there is an embedding *S*3 → *∂A*). Furthermore, one can show that the codimension-one foliation of the 3-sphere induces a codimension-one foliation of *∂A* so that the area of the corresponding polygons (and therefore the foliation invariants) agree. The Godbillon-Vey invariant [*GV*] ∈ *H*3(*S*3, **R**) of the foliation is related to the parameter of the radial family by *GV*, [*S*3] = *ρ*2 using the pairing between cohomology and homology (the fundamental homology class [*S*3] ∈ *H*3(*S*3)).

Thus, the relation between an exotic **R**4 (of Bizaca as constructed from the failure of the smooth h-cobordism theorem) and codimension-one foliation of the *S*3 emerges. Two non-diffeomorphic exotic **R**4 imply non-cobordant codimension-one foliations of the 3-sphere described by the Godbillon-Vey class in *H*3(*S*3, **R**) (proportional to the surface of the polygon). This relation is very strict, i.e. if we change the Casson handle, then we must change the polygon. But that changes the foliation and vice verse. Finally, we obtain the result:

*The exotic* **R**4 *(of Bizaca) is determined by the codimension-1 foliations with non-vanishing Godbillon-Vey class in H*3(*S*3, **R**3) *of a 3-sphere seen as submanifold S*3 ⊂ **R**4*. We say: the exoticness is localized at a 3-sphere inside the small exotic* **R**4*.*

In the particular case of integral *H*3(*S*3, **Z**) one yields the relation of exotic **R**4 *k* , *k*[ ] ∈ *H*3(*S*3, **Z**), *k* ∈ **Z** with the WZ term of the *k* WZW model on *SU*(2). This is because the integer classes in *H*3(*S*3, **Z**) are of special character. Topologically, this case refers to flat *PSL*(2, **R**)−bundles over the space (*S*2 \ {k punctures}) × *S*1 where the gluing of *k* solid tori produces a 3-sphere (so-called Heegard decomposition). Then, one obtains the relation [Asselmeyer-Maluga & Król (2009a)]:

$$\frac{1}{(4\pi)^2} \langle GV(\mathcal{F})\_\prime [S^3] \rangle = \frac{1}{(4\pi)^2} \int\_{S^3} GV(\mathcal{F}) = \pm (2-k) \tag{1}$$

in dependence on the orientation of the fundamental class [*S*3]. We can interpret the Godbillon-Vey invariant as WZ term. For that purpose, we use the group structure *SU*(2) = *S*3 of the 3-sphere *S*3 and identify *SU*(2) = *S*3. Let *g* ∈ *SU*(2) be a unitary matrix with 6 Will-be-set-by-IN-TECH

another disk into that point. These disks form the first stage of the Casson handle. By iteration one can produce the other stages. Finally, we consider a tubular neighborhood *D*2 × *D*2 of this immersed disk, called a kinky handle, on each stage. The union of all neighborhoods of all stages is the Casson handle. So, there are two input data involved with the construction of a *CH*: the number of double points in each stage and their orientation ±. Thus, we can visualize the Casson handle *CH* by a tree: the root is the immersion *D*2 → *M* with *k* double points, the first stage forms the next level of the tree with *k* vertices connected with the root by edges etc. The edges are evaluated using the orientation ±. Every Casson handle can be represented by such an infinite tree. The structure of CH as immersed many-layers 2-disks will be important

Next, we turn again to the radial family of small exotic **R**4, i.e. a continuous family of

The point is that this radial family has a natural foliation (see Theorem 3.2 in DeMichelis & Freedman (1992)) which can be induced by a polygon *P* in the two-dimensional hyperbolic space **H**2. The area of *P* is a well-known invariant, the Godbillon-Vey class as the element in *H*3(*S*3, **R**). Every GV class determines a codimension-one foliation on the 3-sphere (firstly constructed by Thurston (1972); see also the book Tamura (1992) chapter VIII for the details). This 3-sphere is a part of the boundary *∂A* of the Akbulut cork *A* (there is an embedding *S*3 → *∂A*). Furthermore, one can show that the codimension-one foliation of the 3-sphere induces a codimension-one foliation of *∂A* so that the area of the corresponding polygons (and therefore the foliation invariants) agree. The Godbillon-Vey invariant [*GV*] ∈ *H*3(*S*3, **R**)

pairing between cohomology and homology (the fundamental homology class [*S*3] ∈ *H*3(*S*3)). Thus, the relation between an exotic **R**4 (of Bizaca as constructed from the failure of the smooth h-cobordism theorem) and codimension-one foliation of the *S*3 emerges. Two non-diffeomorphic exotic **R**4 imply non-cobordant codimension-one foliations of the 3-sphere described by the Godbillon-Vey class in *H*3(*S*3, **R**) (proportional to the surface of the polygon). This relation is very strict, i.e. if we change the Casson handle, then we must change the

*The exotic* **R**4 *(of Bizaca) is determined by the codimension-1 foliations with non-vanishing Godbillon-Vey class in H*3(*S*3, **R**3) *of a 3-sphere seen as submanifold S*3 ⊂ **R**4*. We say: the exoticness*

*H*3(*S*3, **Z**), *k* ∈ **Z** with the WZ term of the *k* WZW model on *SU*(2). This is because the integer classes in *H*3(*S*3, **Z**) are of special character. Topologically, this case refers to flat *PSL*(2, **R**)−bundles over the space (*S*2 \ {k punctures}) × *S*1 where the gluing of *k* solid tori produces a 3-sphere (so-called Heegard decomposition). Then, one obtains the relation

(4*π*)2

in dependence on the orientation of the fundamental class [*S*3]. We can interpret the Godbillon-Vey invariant as WZ term. For that purpose, we use the group structure *SU*(2) = *S*3 of the 3-sphere *S*3 and identify *SU*(2) = *S*3. Let *g* ∈ *SU*(2) be a unitary matrix with

*S*3

polygon. But that changes the foliation and vice verse. Finally, we obtain the result:

In the particular case of integral *H*3(*S*3, **Z**) one yields the relation of exotic **R**4

*ρ* and **R**4

*ρ*� are non-diffeomorphic for *ρ* �= *ρ*�

*GV*, [*S*3]

*GV*(F) = ±(2 − *k*) (1)

= *ρ*2 using the

*k* , *k*[ ] ∈

.

in Sec. 4 where we will assign fermion fields to CH's.

*is localized at a 3-sphere inside the small exotic* **R**4*.*

1

(4*π*)2 �*GV*(F), [*S*3]� = 1

[Asselmeyer-Maluga & Król (2009a)]:

*ρ*}*ρ*∈[0,+∞] with parameter *ρ* so that **R**4

of the foliation is related to the parameter of the radial family by

exotic {**R**4

det *g* = −1. The left invariant 1-form *g*−1*dg* generates locally the cotangent space connected to the unit. The forms *ωk* = *Tr*((*g*−1*dg*)*k*) are complex *k*−forms generating the deRham cohomology of the Lie group. The cohomology classes of the forms *ω*1, *ω*2 vanish and *ω*3 ∈ *H*3(*SU*(2), **R**) generates the cohomology group. Then, we obtain as the value for the integral of the generator

$$\frac{1}{8\pi^2} \int\_{S^3 = SU(2)} \omega\_3 = 1\,\text{J}$$

This integral can be interpreted as winding number of *g*. Now, we consider a smooth map *G* : *S*3 → *SU*(2) with 3-form Ω3 = *Tr*((*G*−1*dG*)3) so that the integral

$$\frac{1}{8\pi^2} \int\_{S^3 = SU(2)} \Omega\_3 = \frac{1}{8\pi^2} \int\_{S^3} \text{Tr}((G^{-1}dG)^3) \in \mathbb{Z}^3$$

is the winding number of *G*. Every Godbillon-Vey class with integer value like (1) is generated by a 3-form Ω3. Therefore, the Godbillon-Vey class is the WZ term of the *SU*(2)*k*. Thus, we obtain the relation:

*The structure of exotic* **R**4 *k 's, k* ∈ **Z** *from the radial family determines the WZ term of the k WZW model on SU*(2)*.*

This WZ term enables one for the cancellation of the quantum anomaly due to the conformal invariance of the classical *σ*-model on *SU*(2). Thus, we have a method of including this cancellation term from smooth 4-geometry: when a smoothness of the ambient 4-space, in which *S*3 is placed as a part of the boundary of the cork, is this of exotic **R**4 *k* , then the WZ term of the classical *σ*-model with target *S*3 = *SU*(2), i.e. *SU*(2)*k* WZW, is precisely generated by this 4-smoothness. As the conclusion, we have the important correlation:

*The change of smoothnes of exotic* **R**4 *k to exotic* **R**4 *l , k*, *l* ∈ **Z** *both from the radial family, corresponds to the change of the level k of the WZW model on SU*(2)*, i.e. k* WZW →*l* WZW*.*

Let us consider now the end of the exotic **R**4 *k* i.e. *S*3 × **R**. This end cannot be standard smooth and it is in fact fake smooth *S*3 ×Θ*k* **R**, Freedman (1979). Given the connection of **R**4 *k* with the WZ term as above, we have determined the "quantized" geometry of *SU*(2)*k* × **R** as corresponding to the exotic geometry of the end of **R**4 *k* . The appearance of the *SU*(2)*k* × **R** is a source for various further constructions. In particular, we will see that gravitational effects of **R**4 *k* on the quantum level are determined via string theory where one replaces consistently flat **R**4 part of the background by curved 4D *SU*(2)*k* × **R**.