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Homology: A Basic Notion of Algebraic Topology

Updated: Aug 13, 2023

Written by Adanur Nas


Topology, known as one of the most interesting areas of the mathematical world recently, is a sub-branch of science in which spaces and their properties that do not change under continuous deformations are studied. [1] Some historians suggest that the origin of topology began with Euler's work, while others suggest that the starting point of topology was the 1895 book Analysis Situs by the French Mathematician Henri Poincaré. [2]


Although applying the topology requires advanced mathematics and geometry knowledge, it is illustrated below in its simplest form and together with the topological equivalence. There is a topological equivalence between the square and the circle, since the square does not break when it is deformed to become a circle. However, there is no such equivalence between these two shapes because it is broken when it is tried to be made into 8-shape.

Reference: University of Waterloo


Homology, known as a basic algebraic concept of topology, is the general way of relating a set of algebraic sequences, such as Abelian Groups or modules, to other mathematical objects, such as topological spaces. [3] The history of homology, which is used by mathematicians especially to better understand topology, began with the work of the German mathematician Bernhard Riemann, who also found the Riemann Sum model, which is often used in calculus) and the Italian mathematician Enrico Betti. [4] Below is an illustration of one of the areas where homology is often used:

Reference: Wikipedia



Areas of Application of Homology


Science and Engineering

Topological data analysis is one of the most commonly used areas of homology, which is widely used especially in the fields of science and engineering. Topological data analysis is a new and rapidly growing field used to find and interpret more complex structures using topological and geometric tools. [6] In this analysis, data sets are considered as a point cloud sample of a mathematical set or algebraic changes in Euclidean space and enter into a data trilogy by connecting with the data closest to them. Thanks to this trilogy, the simplified homology of the data is calculated. [6] Below, the point clouds where these data can be found are shown in the first image and Euclidean space is shown in the second image:

Reference: Wikipedia

Reference: Wikipedia


Along with topological data analysis, homology is also used in: wireless sensor networks, explaining the dynamical systems theory of physics, the finite element method, often used in engineering. [7] Even if its use in these areas is more simulation-based, it is also gaining vital importance, especially in areas that directly affect human life, such as wireless sensor networks (wireless sensor systems are actively used in areas such as air quality control). [8] Below, the theory of dynamical systems from the areas where homology is used as a simulation is shown in the first image and the finite element method in the second image:

Reference: Wikipedia

Reference: Wikipedia



Software Programs

The influence of homology on software programs is so great that mathematicians and computer experts study software programs by taking a separate subheading. Many different software packages have been developed solely for the purpose of developing homology groups of finite cell complexes. One of the software programs that has the most diverse catalog related to homology is the C++ language. LinBox is a C++ language that performs fast matrix calculations using homology and linear algebra. Moreover, this language also has the ability to connect with software programs known as Gap and Maple.


In addition to these languages, the works such as Chomp, CAPD: Redhom, and Perseus were also written in C++. All three preprocessor algorithms are based on simple homology equations and Discrete Morse Theory. The common goal of all three is to perform reductions of the input cell complexes to preserve homology before resorting to matrix algebra. [9] The graphic reconstruction of Discrete Morse Theory is illustrated below:

Reference: Graph Reconstruction by Discrete Morse Theory



Abstract Mathematics

There is still an ongoing debate in the mathematical world about which branch of mathematics topology and homology belong to. Although the vast majority of mathematicians say that topology and indirect homology should be called strictly abstract mathematics, especially because of the scientific and technological developments brought along by topology and homology in the 21st century are being used more and more in applied mathematics every day. [10]


Some of the theorems that mathematicians have been working on for many years have been solved and proved thanks to homology. Some of these theorems are:


1. The Brouwer Fixed Point Theorem

Reference: Wikipedia


2. Invariance of Domain Theorem

Reference: Wikipedia


3. The Hairy Ball Theorem

Reference: Wikipedia


4. The Borsuk–Ulam Theorem

Reference: Stock Exchange


5. Invariance of Dimension Theorem

Reference: A Topological Proof of the Invariance of Dimension Theorem


The Future of Homology and Final Comments

The fact that homology is still a young and dynamic field and helps to protect the safety of human life makes it one of the most discussed and researched topics of recent times. In particular, there are two topics that scientists are focusing on: rapid and structured drug discovery and a new modeling system for predicting the structures of GPCRs.


Thanks to Homology Modeling, which is the subject of many research articles even now, the 3D structures of proteins from amino acid sequences can be determined computationally. [11] This modeling will help to propose hypotheses about drug design, ligand binding site, substrate specificity and function explanations, and make serious advances in molecular biology.¹²



References:
  1. Chavez, C., Cruz-Becerra, G., Fei, J., Kassavetis, G. A., & Kadonaga, J. T. (2019, October 1). The tardigrade damage suppressor protein binds to nucleosomes and protects DNA from hydroxyl radicals. eLife. Retrieved November 19, 2022, from https://elifesciences.org/articles/47682

  2. Mínguez-Toral, M., Cuevas-Zuviría, B., Garrido-Arandia, M., & Pacios, L. F. (2020, August 7). A computational structural study on the DNA-protecting role of the tardigrade-unique dsup protein. Nature News. Retrieved November 19, 2022, from https://www.nature.com/articles/s41598-020-70431-1

  3. NBCUniversal News Group. (2019, October 14). What is a tardigrade? NBCNews.com. Retrieved November 19, 2022, from https://www.nbcnews.com/mach/science/what-tardigrade-ncna1065771

  4. Tardigrade. Animals. (n.d.). Retrieved November 19, 2022, from https://www.nationalgeographic.com/animals/invertebrates/facts/tardigrades-water-bears

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