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위키데이터

• ID : Q1095056

말뭉치

1. However, the importance of the singular homology groups is not limited to this.
2. now we come to another kind of homology groups, called singular homology groups of a topological space.
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.
4. Singular theory is closely related to homotopy theory; there is a natural homomorphism from homotopy groups to singular homology groups.
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 ) .
6. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories.
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.
8. A proof for the homotopy invariance of singular homology groups can be sketched as follows.
9. The main purpose of this book is to give a systematic treatment of singular homology and cohomology theory.
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.
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.
12. They describe the singular homology groups and prove that they satisfy the Eilenberg-Steenrod axioms on the class of all topological spaces.
13. The zero'th singular homology group of a non-empty path-connected space is isomorphic to the group of integers.
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.
15. Corollary: homotopy equivalent spaces have isomorphic singular homology groups (and in particular, contractible spaces have all reduced homology groups equal to 0).
16. I am having some difficulties understanding the difference between simplicial and singular homology.
17. My question is why this homology group equals the singular homology group?

메타데이터

위키데이터

• ID : Q1095056

Spacy 패턴 목록

• [{'LOWER': 'singular'}, {'LEMMA': 'homology'}]