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에일렌베르크-스틴로드 (Eilenberg-Steenrod) 공리

  • '호몰로지 이론은 에일렌베르크-스틴로드 공리를 만족하는 functor이다'


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

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