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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 and eliminates many exceptional cases.
If P is a single-point space, then with the usual definitions the integral homology group
is isomorphic to Z {\displaystyle \mathbb {Z} } , while for i ≥ 1 we have
More generally if X is a simplicial complex or finite CW complex, then the group H0 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.