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- In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptional cases (as in the homology groups of spheres). If P is a single-point space, then with the usual definitions the integral homology group H0(P) is isomorphic to (an infinite cyclic group), while for i ≥ 1 we have Hi(P) = {0}. More generally if X is a simplicial complex or finite CW complex, then the group H0(X) is the free abelian group with the connected components of X as generators. The reduced homology should replace this group, of rank r say, by one of rank r − 1. Otherwise the homology groups should remain unchanged. An ad hoc way to do this is to think of a 0-th homology class not as a formal sum of connected components, but as such a formal sum where the coefficients add up to zero. In the usual definition of homology of a space X, we consider the chain complex and define the homology groups by . To define reduced homology, we start with the augmented chain complex where . Now we define the reduced homology groups by for positive n and . One can show that ; evidently for all positive n. Armed with this modified complex, the standard ways to obtain homology with coefficients by applying the tensor product, or reduced cohomology groups from the cochain complex made by using a Hom functor, can be applied. (en)
- 축소 호몰로지(reduced homology)와 축소 코호몰로지(reduced cohomology)는 호몰로지 군에 약간 수정을 가한 것이다. (ko)
- Приведённые гомологии — незначительная модификация теории гомологий, позволяющая формулировать некоторые утверждения алгебраической топологии, как например двойственность Александера, без исключительных случаев. Приведённые гомологии и когомологии обычно обозначающийся волной.При этом отличие от обычных гомологий проявляется только в нулевой размерности;то есть и для всех положительных n. (ru)
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- 축소 호몰로지(reduced homology)와 축소 코호몰로지(reduced cohomology)는 호몰로지 군에 약간 수정을 가한 것이다. (ko)
- Приведённые гомологии — незначительная модификация теории гомологий, позволяющая формулировать некоторые утверждения алгебраической топологии, как например двойственность Александера, без исключительных случаев. Приведённые гомологии и когомологии обычно обозначающийся волной.При этом отличие от обычных гомологий проявляется только в нулевой размерности;то есть и для всех положительных n. (ru)
- In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptional cases (as in the homology groups of spheres). If P is a single-point space, then with the usual definitions the integral homology group H0(P) is isomorphic to (an infinite cyclic group), while for i ≥ 1 we have Hi(P) = {0}. and define the homology groups by . for positive n and . (en)
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- 축소 호몰로지 (ko)
- Reduced homology (en)
- Приведённые гомологии (ru)
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