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* ID :  [https://www.wikidata.org/wiki/Q1095056 Q1095056]

2020년 12월 26일 (토) 05:08 판

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말뭉치

  1. However, the importance of the singular homology groups is not limited to this.[1]
  2. now we come to another kind of homology groups, called singular homology groups of a topological space.[2]
  3. Singular homology groups were first defined by S. Lefschetz in 1933 and were perfected in their present form by S. Eilenberg (1913–1998) in the beginning of the 1940’s.[2]
  4. Singular theory is closely related to homotopy theory; there is a natural homomorphism from homotopy groups to singular homology groups.[3]
  5. In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H n ( X ) .[4]
  6. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories.[4]
  7. In brief, singular homology is constructed by taking maps of the standard n-simplex to a topological space, and composing them into formal sums, called singular chains.[4]
  8. A proof for the homotopy invariance of singular homology groups can be sketched as follows.[4]
  9. The main purpose of this book is to give a systematic treatment of singular homology and cohomology theory.[5]
  10. Singular homology and cohomology theory has been the subject of a number of textbooks in the last couple of decades, so the basic outline of the theory is fairly well established.[5]
  11. This corresponds to Eilenberg's innovation in the definition of singular homology and cohomology: one uses ordered p-simplexes rather than oriented p-simplexes.[6]
  12. They describe the singular homology groups and prove that they satisfy the Eilenberg-Steenrod axioms on the class of all topological spaces.[7]
  13. The zero'th singular homology group of a non-empty path-connected space is isomorphic to the group of integers.[8]
  14. For a map between topological spaces, definition of the induced chain map between the respective chain complexes, and the induced homomorphisms between singular homology groups.[8]
  15. Corollary: homotopy equivalent spaces have isomorphic singular homology groups (and in particular, contractible spaces have all reduced homology groups equal to 0).[8]
  16. I am having some difficulties understanding the difference between simplicial and singular homology.[9]
  17. My question is why this homology group equals the singular homology group?[10]

소스

  1. ↑ Encyclopedia of Mathematics
  2. ↑ 2.0 2.1 Singular Homology
  3. ↑ Singular Homology Theory (Chapter 8)
  4. ↑ 4.0 4.1 4.2 4.3 Singular homology
  5. ↑ 5.0 5.1 Singular Homology Theory
  6. ↑ Singular Homology - an overview
  7. ↑ Singular Homology Theory
  8. ↑ 8.0 8.1 8.2 Ver Post · Topologia Algébrica
  9. ↑ Difference between simplicial and singular homology?
  10. ↑ singular homology of a differential manifold

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