Connected Fundamental Groups and Homotopy Contacts in Fibered Topological ( C , R ) Space

: The algebraic as well as geometric topological constructions of manifold embeddings and homotopy offer interesting insights about spaces and symmetry. This paper proposes the construction of 2-quasinormed variants of locally dense p -normed 2-spheres within a non-uniformly scalable quasinormed topological ( C , R ) space. The ﬁbered space is dense and the 2-spheres are equivalent to the category of 3-dimensional manifolds or three-manifolds with simply connected boundary surfaces. However, the disjoint and proper embeddings of covering three-manifolds within the convex subspaces generates separations of p -normed 2-spheres. The 2-quasinormed variants of p -normed 2-spheres are compact and path-connected varieties within the dense space. The path-connection is further extended by introducing the concept of bi-connectedness, preserving Urysohn separation of closed subspaces. The local fundamental groups are constructed from the discrete variety of path-homotopies, which are interior to the respective 2-spheres. The simple connected boundaries of p -normed 2-spheres generate ﬁnite and countable sets of homotopy contacts of the fundamental groups. Interestingly, a compact ﬁbre can prepare a homotopy loop in the fundamental group within the ﬁbered topological ( C , R ) space. It is shown that the holomorphic condition is a requirement in the topological ( C , R ) space to preserve a convex path-component. However, the topological projections of p -normed 2-spheres on the disjoint holomorphic complex subspaces retain the path-connection property irrespective of the projective points on real subspace. The local fundamental groups of discrete-loop variety support the formation of a homotopically Hausdorff ( C , R ) space.


Introduction
In general, a path-connected topological space is considered to be locally path-connected within a path-component maintaining the equivalence relation.A first countable path-connected topological space admits countable fundamental groups if the space is a homotopically Hausdorff variety [1].Interestingly, a homotopically Hausdorff topological space containing countable fundamental groups has universal cover.However, the nature of a fundamental group is different in the lower dimensional topological spaces as compared to the higher dimensional spaces.For example, in a onedimensional topological space X the fundamental group ) ( X  becomes a free group if the space is a simply connected type [1].In this case the topological space successfully admits a suitable metric structure.A regular and separable topological space can be Citation: Bagchi uniquely generated from a given regular as well as separable topological space [2].For example, suppose X is a regular and separable topological space.If we consider that X A  and U is a neighbourhood of A then a unique topological space can be gen- erated from X if A is closed and A U \ is a countable or finite sum of disjoint open sets.Note that the uniquely generated topological space is also a regular and separable topological space.This paper proposes the topological construction and analysis of 2-quasinormed variants of  p normed 2-spheres, path-connected fundamental groups and associated homotopy contacts in a fibered as well as quasinormed topological ) , ( R C space [3].In this paper the 2-quasinormed variants of  p normed 2-spheres in X are generically denoted as

) ( X CRS
. The space is non-uniformly scalable and the fundamental groups are interior to dense subspaces of 2-quasinormed variant of  p normed 2-spheres generating a set of homotopy contacts.First, the brief descriptions about various contact structures, fundamental group varieties and associated homotopies are presented to establish introductory concepts (Sections 1.1 and 1.2).Next, the motivation for this work is illustrated in Section 1.3.In this paper, the symbols R , C , N and Z represent sets of extended real numbers, complex numbers, natural numbers and integers, respectively.Moreover, for clarity, in this paper a 3D manifold is called a three-manifold category in the proposed constructions and topological analysis.Furthermore, the surfaces of threemanifolds and 2-spheres are often alternatively named as respective boundaries for the simplicity of presentation.

Contact Structures and Fundamental Groups
The constructions of geometric contact structures and the analysis of their topological properties on manifolds are required to understand the characteristics of associated group algebraic varieties.The contact structure on a manifold M is a hyperplane field in the corresponding tangent subbundle.A 1 2  n dimensional contact manifold structure is essentially a Hausdorff topological space, which is in the  C class [4].In general, the topological analysis considers that a contact manifold is in the compact category and the contact form  is regular.As a consequence, the integral curves on such contact manifolds are homeomorphic to 1 S .It is shown that if a contact structure A is constructed on a three-manifold 3 T then the fundamental group ) , ( 3 1 A T  includes an infinite cyclic group [5].However, a similar variety of results can also be extended on The topological contact structures on three-manifolds can be further generalized towards higher dimensions.However, in case of n -manifolds ( 3  n ) the theory of contact homology plays an important role.Note that if we consider  as a contact structure and n M as a n -manifold then the contact homology ) , ( *  n M HC is invariant of the corresponding contact structure [6].In this case the contact homology is defined as a chain complex.Interestingly, the higher dimensional manifolds and contact homology can be useful to prove some topological results in the lower dimensional contact structures.For example, the formulations of fundamental group for the n -manifold ( 3  n ) and the associated higher order homotopy groups are successfully realized by employing the higher dimensional contact homology [7].The analytic and geometric properties of the higher dimensional contact structures in n -manifold show some very interesting observations.A 2-torus can be generated by attaching a projection of J -hol- omorphic cylinder to a n -manifold n M along with homotopy pairs, which results in the preparation of a ) , ( 2Z M H n homology class [7].

Homotopy and Twisting
The contact structures can be twisted and can also be classified.According to the Eliashberg definition, a contact structure  on a three-manifold 3 M is called overtwisted if it can successfully allow embedding of an overtwisted disc [8].There is a relationship between the homotopy theory of algebraic topology and the corresponding twisted contact structures.It is shown by Eliashberg that all oriented 2-plane fields on a 3 M structure are essentially homotopic to a contact structure in the overtwisted category.
The Haefliger classifications of foliations in the contact manifolds are in a generalized form considering the open manifold variety [9].The Haefliger categories are further extended by constructing homotopy classifications of foliations on the open contact manifolds [10].However, in this case the leaves are the open contact submanifolds in the topological space.The contact structures, twisting and manifolds are often viewed in geometric perspectives.The construction and analysis of holomorphic curves on the symplectic manifolds are proposed by Gromov [11].Note that the contact geometry is an odd dimensional variety of symplectic geometry.

Motivation and Contributions
The anomalous behaviour in homotopy theory is observed when the uniform limit of a map from a nullhomotopic loop is the essential homotopy loop, which is not nullhomotopic in nature [12].Moreover, the Baire categorizations of a topological subspace influence the properties of structural embedding within the space.Suppose we consider a path-connected subset A of 2 S , where the topological space c A (complement of A ) is a dense subspace.It is shown that the fundamental group of A successfully embeds the fundamental group of Sierpinski curve [1].In this case, the nullhomotopic loop in the topological space X given by X S f 

:
factors through a surjective map on the planar topological subspace.Interestingly, in view of algebraic topology one can construct a fundamental group ) , ( 1x M  from a set of equivalence classes of paths on a manifold M [13].As a result, the covering map given by Y X p  : between the two topological spaces induces another map given by )) ( , ( ) , ( : [13].This paper proposes the topological construction and analysis of multiple path-connected fundamental groups of discrete variety within the non-uniformly scaled as well as quasinormed topological ) , ( R C space.The topological ) , ( R C space supports fibrations in two varieties, such as compact fibres and non-compact fibres.It is considered that the fundamental groups generating homotopy contacts are interior to the 2-quasinormed variants of  p normed 2-spheres within the topological ) , ( R C space.This paper addresses two broad questions in the relevant topological contexts such as, (1) what the topological properties of the resulting structures are if the space is dense and, (2) how the homotopy contacts, covering manifold embeddings and path-connections interplay within the topological ) , ( R C space.Moreover, the question is: how the concept of homotopically Hausdorff fundamental groups influences the proposed structures.The presented construction and analysis employ the combined standpoints of general topology as well as algebraic topology as required.The elements of geometric topology are often used whenever necessary.
The main contributions made in this paper can be summarized as follows.The construction of multiple locally dense  p normed 2-spheres within the dense and fibered non-uniformly scalable topological ) , ( R C space is proposed in this paper.The threemanifold embeddings and the corresponding formation of covering separation of ) ( X CRS are analysed.The generation of path-connected components in a holomorphic convex subspace is formulated and the concept of bi-connectedness is introduced.This paper illustrates that the local and discrete variety of fundamental groups interior to the ) ( X CRS generate the finite and countable sets of homotopy contacts with the simply connected boundaries of

) ( X CRS
. Interestingly, a compact fibre in the topological ) , ( R C space may prepare a homotopy loop.It is shown that the holomorphic condition is required to be maintained in the convex subspace topological ) , ( R C space to support the respective convex path-component.However, it is observed that the path-connected homotopy loops are not always guaranteed to be bi-connected as an implication.
The rest of the paper is organized as follows.The preliminary concepts are presented Section 2 in brief.The definitions and descriptions of

) ( X CRS
, homotopy contacts and fundamental groups are presented in Section 3. The analyses of topological properties are presented in Section 4 in details.Finally, Section 5 concludes the paper.

Preliminary Concepts
In this section, the introduction to topological represents a path-homotopy and additionally it supports the condition that: such that the restriction to the boundary components are the given loops.A topological space ) , (

Fundamental Groups and Homotopy Contacts
In this section, the construction of 2-quasinormed variants of  p normed 2-spheres and the associated definitions of connected fundamental groups as well as homotopy contacts are presented.The constructions consider that the underlying space is a quasinormed as well as non-uniformly scalable topological if they are equivalent (i.e., identified by following the equivalence relation or quotient).Furthermore, the homotopic path equivalence between A and B is denoted as B A H  , whereas the homotopic path joining them is algebraically denoted by B A  maintaining the respective sequence.In the remainder of this paper, the category of 3D manifold is termed as a three-manifold whereas the surfaces of a threemanifold category x X remains a 2-quasinormed topological space.
Note that, in general a ) ( X CRS is a closed and locally dense subspace in the 3- S is equivalent to a compact three-manifold

Topologically Bi-Connected Subspaces
Let X A  and X B  be two locally dense (i.e., locally dense in respective convex subspaces) as well as disjoint such that Remark 1.If A and B are bi-connected then they are also path-connected subspaces in a dense topological space.Moreover, it is possible to formulate an Urysohn separation of A and B un- The main reason for such construction is to generate a set of homotopy contacts as defined in Section 3.5.First we define the discrete variety of path-homotopy loops and associated homotopy class within the topological ) , ( R C space.

Discrete-Loop Homotopy Class
Note that effectively the path-homotopy loops as defined above give rise to the formation of a discrete variety of fundamental group ) , ( 1  c x X  within the topological space at the base point, which is the centre of corresponding

) ( X CRS
. In other words, a set of discrete homotopy loops can be constructed from the path-homotopy loops at a base point centred within

Local Fundamental Group
Note that the discrete variety of a local fundamental group preserves the concept of homotopically Hausdroff property.This is because Once a local fundamental group is prepared within the dense subspace of a topological space ) , ( X X  , the set of homotopy contacts generated by the local fundamental group can be formulated.Recall that a topological space X is defined as simply con- nected if every continuous function is homotopic to a constant function.It is important to note that the homotopically simple connectedness of facilitates the existence of finite as well as countable homotopy contacts.

Let
) , ( 1  c x X  be a local fundamental group in the corresponding subspace

Main Results
This section presents the analysis and a set of topological properties related to the constructed homotopy contacts and the associated fundamental groups of connected variety.The holomorphic condition on the topological space is not imposed as a precondition to maintain generality and it is later established that holomorphic condition should be maintained within a convex path-connected component.It is shown that the bi-connected functions between subspaces and their extensions preserve holomorphic condition.Moreover, the homotopy contacts maintain simple connectedness of the boundary of a

) ( X CRS
, which are essentially dense three-manifolds.First we show that a continuous bi-connection between two

) ( X CRS
is two-points compact in the respective sets of homotopy contacts.
within the topological space if and only if . According to the definition of topologically bi-connected subspaces, then it is a two-points compactification of (.) g

) , ( R C
space is essentially a two-point compactification of a path-connection involving the sets of respective homotopy contacts.Interestingly, the two-point compactification can be performed by employing axiom of choice if the fundamental group is not a trivial variety.In any case, a two-point compact bi-connection between two

) ( X CRS
and its extension are holomorphic in ) , ( X X  .The following theorem pre- sents this observation.

Theorem 2. If a function
and the extended function maintains the following two conditions: . Hence, the extended bi-connection

) ( X CRS
components within the space and it can be transformed into a bi-connected form by a bounded continuous function.
space and the topological projections in one-dimension are given as where Re(.) and Im(.) represent the real and imaginary components of a complex projective subspace.Suppose the entire 1D topological projective spaces are given by within the topological space.Thus there exist a set of continuous functions are at least path-connected in W . How- ever, if we consider that  Interestingly, the bi-connectedness of two homotopy loops cannot always be guaranteed by the path-connected fundamental groups within multiple

) ( X CRS
. The locality of existence of

) ( X CRS
within the topological space is an important parameter in determining the bi-connectedness implication derived from the path-connectedness.This observation is presented in the next lemma.

Lemma 1. If
within the topological space.
Recall that a ) ( X CRS is a dense subspace which supports continuity of . Thus the fundamental groups The topological separation within a space is an important phenomenon to analyse the connectedness of a space as well as the properties of embedded algebraic and geometric structures.It is important to note that two compact

) ( X CRS
Let us consider two three-manifold category chart-maps and X A  is an open set.Moreover, the in- verse preserves the condition given as . It di- . Furthermore, there is a coordinate identification map given as: Note that it maintains the condition that because the projections on real subspace do not directly predetermine the locality of embeddings.Let us consider two such embedded subspaces given as As a result we can con- clude that the embedded three-manifolds maintain . Recall that the topological space ) , ( X X  is dense every- where.Hence, it can be concluded that and as a result the compact

) ( X CRS
are also separable compact subspaces if they can be covered by disjoint compact three-manifolds.This observation is presented in the next corollary. The separable embeddings of Schoenflies variety in a connected as well as dense topological space invite a set of interesting topological properties in view of the Jordan Curve Theorem (JCT).For example, the interrelationship between connected fundamental groups within the multiple compact

) ( X CRS
and the corresponding homotopy contacts are affected by the connectedness of the topological space.The topological properties related to the interplay between connected fundamental groups, homotopy contacts and manifold embeddings within a dense topological ) , ( R C space are presented in the following subsection.

Homotopy Contacts and Manifold Embeddings
The embeddings of three-manifolds within the dense topological space ensure that multiple ) ( X CRS are separable, which affects the bi-connectedness property involving respective homotopy contacts.The following theorem illustrates that if the embedded three-manifolds are dense then the different projections of multiple ) ( X CRS into the complex subspaces retain path-connectedness.

Theorem . If
Suppose the corresponding two

) ( X CRS
and there is a ) , ( X X  is path-connected and dense then there exists a continuous function within the topological space and the complex subspace This indicates that the corresponding projection under composition It is important to note that the holomorphic condition is a requirement to maintain the path-connectedness under respective complex projections fixed at different points on the real subspace.Interestingly, if the homotopy contacts are present then the complex projections retain bi-connectedness of disjoint complex holomorphic subspaces.This observation is presented in the following lemma.

Lemma 2. If there exist the contacts of homotopy classes
)

Remark 4. Interestingly, if we relax the condition of interior embedding further such
. Note that in this case we are considering that the sets of contacts of homotopy classes are not empty.
Proof.The proof is relatively straightforward.First consider two local fundamental groups ) , ( 1  c x X  and ) , ( The homeomorphisms between two discrete varieties of local fundamental groups can be established once the homotopy equivalences are established.Note that it is considered that the local fundamental groups are path-connected in nature.The condition for formation of a homeomorphism between the two path-connected discrete fundamental groups is presented in the following corollary.

Corollary 3. If
) , ( ) , ( ) , ( : then the function maintains the condition given by, . If we restrict that ) , ( ) , ( : Hence, it can be concluded that the bijective function ) , ( ) , ( : Interestingly there is an interrelationship between the path-connection between the base points of two fundamental groups within the respective two dense .Interestingly, the retraction is independent of the influence of real subspace and it can be fixed at any arbitrary point in the real subspace.

Conclusions
A  q quasinormed topological space can equally admit a corresponding topology generated by the respective  p norm function.The resulting structures provide a set of interesting topological properties in view of homotopy theory and fundamental groups.The proposed constructions of 2-quasinormed variety of locally dense  p normed 2- spheres within a non-uniformly scalable quasinormed topological ) , ( R C space enable the formulation of path-connected fundamental groups interior to it.The space is fibered and, in view of Baire category the topological space is dense, which supports path-connection as well as the concept of bi-connection between multiple  p normed 2-spheres as long as the continuous functions in the respective convex subspace are holomorphic in nature.The 2-quasinormed varieties of  p normed 2-spheres are equivalent to the cate- gory of connected three-manifolds with simply connected boundaries in terms of nullhomotopy.The  p normed 2-spheres admit Urysohn separation of the closed sub- spaces.Moreover, the separations can also be formed by proper embeddings of respective covering three-manifolds within the topological ) , ( R C space.The homotopically simple connected boundaries of 2-quasinormed varieties of  p normed 2-spheres support a finite and countable set of homotopy contacts generated by a set of discrete-loop local fundamental groups.Interestingly, a compact fibre in the space can prepare a homotopy loop in the local fundamental group within the fibered topological ) , ( R C space.It is shown that the path-connected homotopy loops are not guaranteed to be bi-connected as an implication.Moreover, the topological projections of 2-quasinormed varieties of  p normed 2-spheres on the disjoint holomorphic complex subspaces successfully retain path-connection irrespective of the projective points on real subspace.The algebraic topological properties, the properties of compactness of holomorphic convex path-components and the homeomorphism between local fundamental groups are analysed in detail.
A topological space X is termed as homotopically Hausdorff if there is an open neighbourhood at a base point X x  0 such that any element of a non-trivial homotopy class of the fundato the corresponding open neighbourhood [1].
this paper a 2-quasinormed variant of  p normed 2-sphere centred at point c x in the topological ) , ( R C space is algebraically represented as 2 c S and it is generically termed as ) ( X CRS without specifying any prefixed centre as indicated earlier.Note that an arbitrary point p Gauss origin.In this paper o A and A represent interior and closure of an arbitrary set

3
(and alternatively called as boundaries).If the interior of a three-manifold category 3

3 M
in view of category.It indicates that the closed subspace2 cS is locally dense in a convex subspace within the topo- connected variety enabling the existence of a finite number of homotopy contacts on 2 c S  .

Remark 2 . 5 )
Interestingly, there is a relationship between a compact fibre and a homotopy loop in the fibered topologicalIt is relatively straightforward to observe that in this case the fibration maintains

Corollary 1 .
The above theorem indicates that the within the topological space often facilitates the generation of connected components and the determination of separation of multiple) ( X CRSwithin the topological space.It is illustrated in the following theorem that the placement of centres of multiple) ( X CRSin one-dimensional projective subspaces prepares path-connected space.Suppose we consider the compact (i.e., bounded and finite) and continuous (i.e., holomorphic) function in the topological form (i.e., without any specific restrictions imposed on codomain) such that

2 STheorem 4 . 3 forming the separations of 2  S and 2 S
are not necessarily separable even if we simply consider that a relatively stronger condition is required involving Riemannian covering manifolds and the corresponding embeddings as presented in the following theorem.If RS is a smooth and compact Riemann complex-sphere with) if and only if o

X
 where  is an index set.Note that the open sets three-manifold embed- dings by considering two open sets.If we consider an open disk above-mentioned separation property enforces a stronger condition in the multidimensional topological ) , ( R C space; however it is in line with the Urysohn separation concept.The embeddings of separable three-manifolds within a topological ) , ( R C space invite the possibility of generation of multiple components.The main reasons are that the topological space ) , ( X X  is dense and the multiple

3 M and 3 N 3 M and 3 N.
are two disjoint covering three-manifolds in path-connected dense ) be two three-manifolds in path-connected dense) Recall that we are considering compact three-manifolds such that

3 M and 3 N
are the two disjoint covering three-manifolds of2

Remark 5 .
The above theorem reveals a property in view of geometric topology.If the base point of a fundamental group Moreover, the simple connectedness property allows inward retraction of boundary of that the boundaries (