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

## Abstract

**:**

## 1. Introduction

#### 1.1. Contact Structures and Fundamental Groups

#### 1.2. Homotopy and Twisting

#### 1.3. Motivation and Contributions

## 2. Preliminary Concepts

_{i}is an essential (i.e., not nullhomotopic) closed curve in $X$ then there always exists an open neighbourhood ${V}_{i}$ of $X$ in $\mathsf{\Phi}$ such that ${p}_{i}$ is also essential in ${V}_{i}$.

## 3. Fundamental Groups and Homotopy Contacts

#### 3.1. Topological $CRS(X)$

#### 3.2. Topologically Bi-Connected Subspaces

**Remark 1.**

_{a}) = 0 and$v({x}_{b})=1$. Note that the boundaries$\partial {S}_{c}^{2}$and$\partial {S}_{d}^{2}$of two respective$CRS(X)$are homotopically simply connected Hausdorff and can preserve Urysohn separation of every points on them.

#### 3.3. Discrete-Loop Homotopy Class

**Remark 2.**

#### 3.4. Local Fundamental Group

#### 3.5. Homotopy Contacts

**Remark 3.**

## 4. Main Results

**Theorem 1.**

**Proof.**

**Theorem 2.**

**Proof.**

**Corollary 1.**

**Theorem 3.**

**Proof.**

**Lemma 1.**

**Proof.**

**Theorem 4.**

**Proof.**

_{X}) generating three-manifold embeddings by considering two open sets. If we consider an open disk ${U}_{a=1}=D({z}_{m},\epsilon >0)\subset RS$ centred at ${z}_{m}\in C$ then ${f}_{a=1}({U}_{a=1})\subset (A\subset X)$ where ${f}_{a=1}({z}_{k}\in {U}_{a=1})=({f}_{a=1}({z}_{k})={z}_{n},{r}_{n}\in R)$ and $A\subset X$ is an open set. Moreover, the inverse preserves the condition given as $\forall {x}_{n}\in X,{f}_{a=1}^{-1}(({z}_{n},{r}_{n}))={f}_{a=1}^{-1}({z}_{n})={z}_{k}$. It directly follows that $\forall {N}_{xn}\subset X,{x}_{n}\in {N}_{xn}$ open neighbourhood $\exists D({z}_{m},\epsilon >0)\subset RS$ such that ${f}_{a=1}^{-1}({x}_{u}\in {N}_{xn})\in D({z}_{m},\epsilon >0)$. Furthermore, there is a coordinate identification map given as:

**Corollary 2.**

#### Homotopy Contacts and Manifold Embeddings

**Theorem 5.**

**Proof.**

**Lemma 2.**

**Remark 4.**

**Theorem 6.**

**Proof.**

_{a}and $g(1)={x}_{b}$ then a set of continuous functions given by $F=\{{g}_{e}:[0,1]\to Y,e\in {Z}^{+},{g}_{e}{|}_{g}=g\}$ can be constructed in the topological subspace. Hence, we conclude that if ${X}_{\u22b3F}$ is a path-component under $F$ then $Y\equiv {X}_{\u22b3F}$ and as a result ${\pi}_{1}(X,{x}_{a})$ and ${\pi}_{1}(X,{x}_{b})$ are path-connected in compact $Y\subset X$. □

**Remark 5.**

**Lemma 3.**

**Proof.**

_{⊳q}. □

**Corollary 3.**

**Proof.**

**Theorem 7.**

**Proof.**

**Remark 6.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Bagchi, S.
Connected Fundamental Groups and Homotopy Contacts in Fibered Topological (*C*, *R*) Space. *Symmetry* **2021**, *13*, 500.
https://doi.org/10.3390/sym13030500

**AMA Style**

Bagchi S.
Connected Fundamental Groups and Homotopy Contacts in Fibered Topological (*C*, *R*) Space. *Symmetry*. 2021; 13(3):500.
https://doi.org/10.3390/sym13030500

**Chicago/Turabian Style**

Bagchi, Susmit.
2021. "Connected Fundamental Groups and Homotopy Contacts in Fibered Topological (*C*, *R*) Space" *Symmetry* 13, no. 3: 500.
https://doi.org/10.3390/sym13030500