# 호몰로지

둘러보기로 가기 검색하러 가기

## 에일렌베르크-스틴로드 (Eilenberg-Steenrod) 공리

• '호몰로지 이론은 에일렌베르크-스틴로드 공리를 만족하는 functor이다'

## 노트

### 말뭉치

1. Homology theory was introduced towards the end of the 19th century by H. Poincaré (cf.
2. 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.
3. There exists a cohomology theory dual to a homology theory (cf.
4. The axioms are formulated in the same manner as for homology, with the obvious reversal of the direction of the homomorphisms.
5. 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.
6. Conceptually, however, it can be useful to understand homology as a special kind of homotopy.
7. This is maybe most vivid in the dual picture: cohomology derives its name from that fact that chain homology and cohomology are dual concepts.
8. One good way of understanding homology of CW complexes is with cellular homology.
9. Homology groups were originally defined in algebraic topology .
10. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes.
11. Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold .
12. A particular type of mathematical object, such as a topological space or a group , may have one or more associated homology theories.
13. Historically, the term "homology" was first used in a topological sense by Poincaré.
14. 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.
15. To simplify the definition of homology, Poincaré simplified the spaces he dealt with.
16. Eventually, Poincaré's version of homology was dispensed with and replaced by the more general singular homology.
17. The starting point will be simplicial complexes and simplicial homology.
18. The simplicial homology depends on the way these simplices fit together to form the given space.
19. 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.
20. One of the main results in the module will be the proof that simplicial homology and singular homology agree for simplicial complexes.
21. We can think of homology as a perspective that allows one to distinguish shapes and forms disregarding changes made by stretching and bending.
22. 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.
23. This is the reason why according to homology those objects are not the same.
24. Homology is a mathematical way of counting different types of loops and holes in topological spaces.
25. Through analysis of viral and simulated genomic datasets, we show how persistent homology captures fundamental evolutionary aspects not directly inferred from phylogeny.
26. 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.
27. C. We then perform persistent homology, which computes the homology groups of dimension k at all scales ε.
28. Our aim, then, is to apply persistent homology to the study of evolution.
29. Exercise Show that E(V) has zero homology.
30. For each cover we obtain a homology group.
31. This leads to a homomorphism of homology groups.
32. Let us look at .Cech homology again.
33. In this section, we categorize the persistent homology enabled applications as single graph and multiple graph analysis.
34. 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.
35. (2018) use persistent homology to detect clique communities and their evolution in weighted networks.
36. Persistent homology is also used to analyze the brain networks by computing distributions of cliques (brain regions) and cycles (strongly connected regions) in them.
37. 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.
38. At first, calculations of group homology were restricted to those groups π which were fundamental groups of familiar topological spaces, using the bar complex.
39. By measuring leaves from throughout the seed plants using persistent homology, a defined morphospace comparing all leaves is demarcated.
40. Landmark analysis excels in its interpretability, because each landmark is an identifiable feature with biological meaning imparted by the shared homology between samples.
41. 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.
42. Using persistent homology, we then use a linear discriminant analysis (LDA) to classify leaves by plant family.
43. 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.
44. Historically, the term homology was first used in a topological sense by Poincaré .
45. 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.
46. Eventually, Poincaré's version of homology was dispensed with and replaced by the more general Singular Homology.
47. In modern usage, however, the word homology is used to mean Homology Group.
48. This workshop will explore topics of current interest in the theory of Floer homology for 3-manifolds.
49. Floer homology is a powerful tool for studying the topology of 3- and 4-dimensional manifolds, and the relations between them.
50. There are a wide variety of ways to define the Floer homology of a 3-manifold.
51. Many of these are now known to be equivalent, but their relation to the original instanton homology remains mysterious.
52. 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).
53. Rather, our main goal is to develop a systematic filtration scheme to study persistent homology in unweighted and undirected networks.
54. 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.
55. The last observation is a reflection of the differences in the persistent homology of the clique complexes corresponding to spherical and hyperbolic networks.
56. 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.
57. The -th homology group of a simplicial complex , denoted , is the quotient vector space .
58. Therefore, computing homology generalizes the graph-theoretic methods of computing connected components.
59. The quotient construction imposes that two vertices (two elements of the homology group) are considered equivalent if their difference is a boundary.
60. 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 .