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| In [[mathematics]], '''reduced homology''' is a minor modification made to [[homology theory]] in [[algebraic topology]], designed to make a point have all its [[homology group]]s zero. This change is required to make statements without some number of exceptional cases ([[Alexander duality]] being an example).
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| If ''P'' is a single-point space, then with the usual definitions the integral homology group
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| :''H''<sub>0</sub>(''P'')
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| is an [[infinite cyclic group]], while for ''i'' ≥ 1 we have | |
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| :''H''<sub>''i''</sub>(''P'') = {0}.
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| More generally if ''X'' is a [[simplicial complex]] or finite [[CW complex]], then the group ''H''<sub>0</sub>(''X'') is the [[free abelian group]] with the [[connected space|connected component]]s 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.
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| In the usual definition of [[homology (mathematics)|homology]] of a space ''X'', we consider the chain complex
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| :<math>\dotsb\overset{\partial_{n+1}}{\longrightarrow\,}C_n | |
| \overset{\partial_n}{\longrightarrow\,}C_{n-1}
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| \overset{\partial_{n-1}}{\longrightarrow\,}
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| \dotsb
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| \overset{\partial_2}{\longrightarrow\,}
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| C_1
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| \overset{\partial_1}{\longrightarrow\,}
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| C_0\overset{\partial_0}{\longrightarrow\,} 0</math>
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| and define the homology groups by <math>H_n(X) = \ker \partial_n / \mathrm{im}(\partial_{n+1})</math>.
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| To define reduced homology, we start with the ''augmented'' chain complex
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| <math>\dotsb\overset{\partial_{n+1}}{\longrightarrow\,}C_n
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| \overset{\partial_n}{\longrightarrow\,}C_{n-1}
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| \overset{\partial_{n-1}}{\longrightarrow\,}
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| \dotsb
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| \overset{\partial_2}{\longrightarrow\,}
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| C_1
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| \overset{\partial_1}{\longrightarrow\,}
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| C_0\overset{\epsilon}{\longrightarrow\,} \mathbb{Z} \to 0
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| </math>
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| where <math>\epsilon \left( \sum_i n_i \sigma_i \right) = \sum_i n_i </math>. Now we define the ''reduced'' homology groups by
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| :<math> \tilde{H_n}(X) = \ker(\partial_n) / \mathrm{im}(\partial_{n+1})</math> for positive ''n'' and <math>\tilde{H}_0(X) = \ker(\epsilon) / \mathrm{im}(\partial_1)</math>.
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| One can show that <math> H_0(X) = \tilde{H}_0(X) \oplus \mathbb{Z}</math>; evidently <math>H_n(X) = \tilde{H}_n(X)</math> for all positive ''n''.
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| Armed with this modified complex, the standard ways to obtain homology with coefficients by applying the [[tensor product]], or ''reduced'' [[cohomology group]]s from the [[cochain complex]] made by using a [[Hom functor]], can be applied.
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| ==References==
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| * Hatcher, A., (2002) ''[http://www.math.cornell.edu/~hatcher/AT/ATchapters.html Algebraic Topology]'' Cambridge University Press, ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
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| {{DEFAULTSORT:Reduced Homology}}
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| [[Category:Homology theory]]
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