Exploring Algebraic Topology and Homotopy Theory: Methods, Empirical Data, and Numerical Examples

Algebraic topology is a powerful branch of mathematics that bridges algebra and topology to study qualitative properties of spaces. Homotopy theory, a core component of algebraic topology, deals with the concept of continuous deformation between functions and spaces. This paper explores the fundamental concepts of algebraic topology and homotopy theory, supported by empirical methodologies and numerical examples. A key emphasis is placed on computational tools such as persistent homology and the use of simplicial complexes to analyze real-world datasets, including image datasets and sensor networks. By integrating theoretical foundations with applied examples, this study demonstrates how algebraic topology can be used not only to understand abstract mathematical spaces but also to draw insights from complex data structures.

Algebraic ,Topology ,Homotopy Theory, Empirical Data, Numerical Examples

1. Hatcher, A. (2002). Algebraic Topology. Cambridge University Press. [Google Scholar] [Crossref]

2. Munkres, J. (1984). Elements of Algebraic Topology. Addison-Wesley. [Google Scholar] [Crossref]

3. Carlsson, G. (2009). Topology and Data. Bulletin of the American Mathematical Society. [Google Scholar] [Crossref]

4. Edelsbrunner, H., & Harer, J. (2008). Persistent Homology — A Survey. Contemporary Mathematics. [Google Scholar] [Crossref]

5. Zomorodian, A., & Carlsson, G. (2005). Computing Persistent Homology. Discrete & Computational Geometry. [Google Scholar] [Crossref]

6. Oudot, S. (2015). Persistence Theory. AMS. [Google Scholar] [Crossref]

• Interplay of Students’ Emotional Intelligence and Attitude toward Mathematics on Performance in Grade 10 Algebra
• Numerical Simulation of Fitzhugh-Nagumo Dynamics Using a Finite Difference-Based Method of Lines
• Fixed Point Theorem in Controlled Metric Spaces
• Usage of Moving Average to Heart Rate, Blood Pressure and Blood Sugar
• Degree of Approximation of Function in the Generalized Zygmund Class By (E, Q) (N̅, Pn) Means of Fourier Series