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**Table of Content:**
1. Introduction
1.1 Background of the Study
1.2 Research Questions
1.3 Significance of the Study
1.4 Definition of Terms
2. Literature Review
2.1 Overview of Algebraic Topology
2.2 Homotopy Groups
2.3 Homology Groups
2.4 Previous Research on Homotopy Groups and Homology Groups
3. Research Methodology
3.1 Data Collection Methods
3.2 Sample Selection
3.3 Data Analysis Techniques
4. Discussion of Findings
4.1 Analysis of Homotopy Groups
4.2 Analysis of Homology Groups
4.3 Comparison of Homotopy and Homology Groups
5. Conclusion and Summary
5.1 Summary of Findings
5.2 Conclusion
5.3 Recommendations for Future Research

**Brief Overview:**

Algebraic Topology is a branch of mathematics that studies spaces through the concept of algebraic invariants. In this thesis, the focus is on two key algebraic invariants – Homotopy Groups and Homology Groups.

Homotopy Groups are fundamental invariants that classify continuous maps between topological spaces up to deformation. These groups provide information about the shape of a space and are used to distinguish different types of spaces.

Homology Groups, on the other hand, are derived from singular homology theory and provide a way to associate algebraic structures to topological spaces. These groups are used to study the properties of spaces by assigning algebraic objects to them.

The literature review in this thesis provides an overview of Algebraic Topology, Homotopy Groups, and Homology Groups, along with a discussion of previous research in this field.

The research methodology section outlines the data collection methods, sample selection procedures, and data analysis techniques used in this study.

The discussion of findings section presents the analysis of both Homotopy and Homology Groups, comparing and contrasting their properties.

In conclusion, this thesis provides valuable insights into the world of Algebraic Topology, specifically focusing on Homotopy and Homology Groups, and offers recommendations for future research in this area.

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