# 호몰로지

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## 개요

- 단체 호몰로지 (simplicial homology)
- 특이 호몰로지 (singular homology)
- \(\langle S,d\omega\rangle=\langle \partial S,\omega \rangle\)

## 에일렌베르크-스틴로드 (Eilenberg-Steenrod) 공리

- '호몰로지 이론은 에일렌베르크-스틴로드 공리를 만족하는 functor이다'

## 역사

- 1752 다면체에 대한 오일러의 정리 V-E+F=2
- 1827 가우스, 1848 보네 가우스-보네 정리
- 1851 리만 connectivity = maximum number of non separating curves
- 1863 뫼비우스, 곡면의 분류
- 1871 베티 넘버
- 푸앵카레
- 브라우어
- 1920년대 Veblen, Alexander, Lefschetz
- 수학사 연표

## 메모

## 관련된 항목들

## 계산 리소스

## 사전 형태의 자료

## 관련논문

- Weibel, Charles A. 1999. History of Homological Algebra History of Topology: 797–836.
- Hilton, Peter A Brief, Subjective History of Homology and Homotopy Theory in This Century (1988), Mathematics Magazine (Mathematical Association of America) 60 (5): 282-291

## 노트

### 위키데이터

- ID : Q1144780

### 말뭉치

- Homology theory was introduced towards the end of the 19th century by H. Poincaré (cf.
^{[1]} - Axiom 6, which requires the invariance under excision and which has a number of different formulations, displays the property of homology theory by which it differs from homotopy theory.
^{[1]} - There exists a cohomology theory dual to a homology theory (cf.
^{[1]} - The axioms are formulated in the same manner as for homology, with the obvious reversal of the direction of the homomorphisms.
^{[1]} - By analysis of the lifting problem it introduces the funda mental group and explores its properties, including Van Kampen's Theorem and the relationship with the first homology group.
^{[2]} - Conceptually, however, it can be useful to understand homology as a special kind of homotopy.
^{[3]} - This is maybe most vivid in the dual picture: cohomology derives its name from that fact that chain homology and cohomology are dual concepts.
^{[3]} - One good way of understanding homology of CW complexes is with cellular homology.
^{[4]} - Homology groups were originally defined in algebraic topology .
^{[5]} - The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes.
^{[5]} - Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold .
^{[5]} - A particular type of mathematical object, such as a topological space or a group , may have one or more associated homology theories.
^{[5]} - Historically, the term "homology" was first used in a topological sense by Poincaré.
^{[6]} - To him, it meant pretty much what is now called a bordism, meaning that a homology was thought of as a relation between manifolds mapped into a manifold.
^{[6]} - To simplify the definition of homology, Poincaré simplified the spaces he dealt with.
^{[6]} - Eventually, Poincaré's version of homology was dispensed with and replaced by the more general singular homology.
^{[6]} - The starting point will be simplicial complexes and simplicial homology.
^{[7]} - The simplicial homology depends on the way these simplices fit together to form the given space.
^{[7]} - It is not that hard to prove that singular homology is a homotopy invariant but very hard to compute singular homology directly from the definition.
^{[7]} - One of the main results in the module will be the proof that simplicial homology and singular homology agree for simplicial complexes.
^{[7]} - We can think of homology as a perspective that allows one to distinguish shapes and forms disregarding changes made by stretching and bending.
^{[8]} - For example, homology sees no difference between the beach ball and the jumping ball, which can be constructed from the ball shape by pulling out two fingers.
^{[8]} - This is the reason why according to homology those objects are not the same.
^{[8]} - Homology is a mathematical way of counting different types of loops and holes in topological spaces.
^{[8]} - Through analysis of viral and simulated genomic datasets, we show how persistent homology captures fundamental evolutionary aspects not directly inferred from phylogeny.
^{[9]} - We can define a topological invariant called the “homology group” H k as an algebraic structure that encompasses all holes in dimension k, and the “Betti number” b k is the count of these holes.
^{[9]} - C. We then perform persistent homology, which computes the homology groups of dimension k at all scales ε.
^{[9]} - Our aim, then, is to apply persistent homology to the study of evolution.
^{[9]} - Exercise Show that E(V) has zero homology.
^{[10]} - For each cover we obtain a homology group.
^{[10]} - This leads to a homomorphism of homology groups.
^{[10]} - Let us look at .Cech homology again.
^{[10]} - In this section, we categorize the persistent homology enabled applications as single graph and multiple graph analysis.
^{[11]} - In some applications, persistent homology is used to detect global structural features of a single network such as complexity and distributions of strongly connected regions.
^{[11]} - (2018) use persistent homology to detect clique communities and their evolution in weighted networks.
^{[11]} - Persistent homology is also used to analyze the brain networks by computing distributions of cliques (brain regions) and cycles (strongly connected regions) in them.
^{[11]} - If I is an ideal of R, he considers the homology of the kernel of F * → F * /I and shows that it is independent of the choice of resolution.
^{[12]} - At first, calculations of group homology were restricted to those groups π which were fundamental groups of familiar topological spaces, using the bar complex.
^{[12]} - By measuring leaves from throughout the seed plants using persistent homology, a defined morphospace comparing all leaves is demarcated.
^{[13]} - Landmark analysis excels in its interpretability, because each landmark is an identifiable feature with biological meaning imparted by the shared homology between samples.
^{[13]} - Here, we present a morphometric technique based on topology, using a persistent homology framework, to measure the outlines of leaves and classify them by plant family.
^{[13]} - Using persistent homology, we then use a linear discriminant analysis (LDA) to classify leaves by plant family.
^{[13]} - The theory has applications in many branches of mathematics, including spectral theory, the theory of de Rham homology in differential geometry, automatic continuity theory and K-theory.
^{[14]} - Historically, the term ``homology
*was first used in a topological sense by Poincaré .*^{[15]} - To him, it meant pretty much what is now called a Cobordism, meaning that a homology was thought of as a relation between Manifolds mapped into a Manifold.
^{[15]} - Eventually, Poincaré's version of homology was dispensed with and replaced by the more general Singular Homology.
^{[15]} - In modern usage, however, the word homology is used to mean Homology Group.
^{[15]} - This workshop will explore topics of current interest in the theory of Floer homology for 3-manifolds.
^{[16]} - Floer homology is a powerful tool for studying the topology of 3- and 4-dimensional manifolds, and the relations between them.
^{[16]} - There are a wide variety of ways to define the Floer homology of a 3-manifold.
^{[16]} - Many of these are now known to be equivalent, but their relation to the original instanton homology remains mysterious.
^{[16]} - In the Theory section, we have shown that the number of critical simplices determines the effective number of filtration weights to study the persistent homology of a clique complex (See Eq. 10).
^{[17]} - Rather, our main goal is to develop a systematic filtration scheme to study persistent homology in unweighted and undirected networks.
^{[17]} - A visual inspection of the barcode diagrams for the five model networks (Figs 3 and 5 and SI Figs S1–S4) suggests that the different models can be distinguished based on their persistent homology.
^{[17]} - The last observation is a reflection of the differences in the persistent homology of the clique complexes corresponding to spherical and hyperbolic networks.
^{[17]} - In this paper, we propose to use the persistent homology to systematically study the osmolytes’ molecular aggregation and their hydrogen-bonding network from a global topological perspective.
^{[18]} - The -th homology group of a simplicial complex , denoted , is the quotient vector space .
^{[19]} - Therefore, computing homology generalizes the graph-theoretic methods of computing connected components.
^{[19]} - The quotient construction imposes that two vertices (two elements of the homology group) are considered equivalent if their difference is a boundary.
^{[19]} - It is easy to see that (indeed, just by the first four generators of the image) all vertices are equivalent to 0, so there is a unique generator of homology, and the vector space is isomorphic to .
^{[19]}

### 소스

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Encyclopedia of Mathematics - ↑ Homology Theory - An Introduction to Algebraic Topology
- ↑
^{3.0}^{3.1}homology in nLab - ↑ Intuition of the meaning of homology groups
- ↑
^{5.0}^{5.1}^{5.2}^{5.3}Homology (mathematics) - ↑
^{6.0}^{6.1}^{6.2}^{6.3}Homology -- from Wolfram MathWorld - ↑
^{7.0}^{7.1}^{7.2}^{7.3}MA3H6 Algebraic Topology - ↑
^{8.0}^{8.1}^{8.2}^{8.3}Mathematical Institute - ↑
^{9.0}^{9.1}^{9.2}^{9.3}Topology of viral evolution - ↑
^{10.0}^{10.1}^{10.2}^{10.3}Algebraic Topology: Homology - ↑
^{11.0}^{11.1}^{11.2}^{11.3}Persistence homology of networks: methods and applications - ↑
^{12.0}^{12.1}Homology Group - an overview - ↑
^{13.0}^{13.1}^{13.2}^{13.3}Topological Data Analysis as a Morphometric Method: Using Persistent Homology to Demarcate a Leaf Morphospace - ↑ Mathematics, Statistics and Physics, School of
- ↑
^{15.0}^{15.1}^{15.2}^{15.3}Homology (Topology) - ↑
^{16.0}^{16.1}^{16.2}^{16.3}Floer homology in low-dimensional topology: January 11-15, 2021 - ↑
^{17.0}^{17.1}^{17.2}^{17.3}Persistent homology of unweighted complex networks via discrete Morse theory - ↑ Persistent homology analysis of osmolyte molecular aggregation and their hydrogen-bonding networks
- ↑
^{19.0}^{19.1}^{19.2}^{19.3}Homology Theory — A Primer

## 메타데이터

### 위키데이터

- ID : Q1144780