Reduced homology

In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, designed to make a point have all its homology groups zero. This change is required to make statements without some number of exceptional cases (Alexander duality being an example).

If P is a single-point space, then with the usual definitions the integral homology group

is isomorphic to ${\displaystyle \mathbb {Z} }$ (an infinite cyclic group), while for i ≥ 1 we have

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 ${\displaystyle H_{n}(X)=\ker(\partial _{n})/\mathrm {im} (\partial _{n+1})}$.

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