This paper explores the foundational concepts of simplicial structures that form the basis of simplicial homology theory. It also introduces singular homology as a means to establish the equivalence of homology groups for homeomorphic topological spaces. The paper concludes by providing a proof of the equivalence between simplicial and singular homology groups.

We follow the structure and explanations provided by Nadathur [7] and Hatcher [3]. In particular, the definitions are taken from the introductory book of Boissonnat et al. [1], as well as from Jonsson's introduction [4] and the paper on computational topology by Melodia et al. [6]. Some individual lemmas and proof ideas are drawn from Khoury [5] or from the textbook by Edelsbrunner et al. [2], but they have been adapted, expanded, and implemented independently. To enhance readability, we have omitted source citations within the text.

1. Simplicial Complexes
2. Homology Groups
3. Singular Homology
4. Chain Complexes
5. Exact and Short Exact Sequences
6. Relative Homology Groups
7. The Equivalence of Simplicial Homology Group $H_d^\Delta$ and Singular Homology Group $H_d$

1. Boissonnat, J. D., Chazal, F., Yvinec, M. (2018). Geometric and Topological Inference (Vol. 57). Cambridge University Press.
2. Edelsbrunner, H., Harer, J. L. (2022). Computational Topology: An Introduction. American Mathematical Society.
3. Hatcher, A. (2005). Algebraic Topology. Cambridge University Press.
4. Jonsson, J. (2011). Introduction to Simplicial Homology. Königliche Technische Hochschule. URL: https://people.kth.se/~jakobj/doc/homology/homology.pdf.
5. Khoury, M. (2022). Lecture 6: Introduction to Simplicial Homology. Topics in Computational Topology: An Algorithmic View. Ohio State University. URL: http:https://web.cse.ohio-state.edu/~wang.1016/courses/788/Lecs/lec6-marc.pdf.
6. Melodia, L., Lenz, R. (2021). Estimate of the Neural Network Dimension Using Algebraic Topology and Lie Theory. In Pattern Recognition. ICPR International Workshops and Challenges.
7. Nadathur, P. (2007). An Introduction to Homology. University of Chicago. URL: https://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Nadathur.pdf.
8. Pontryagin L. S. (1952): Foundations of Combinatorial Topology. Graylock Press.