Reduced homology

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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 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

H₀(P)

is an infinite cyclic group, while for i ≥ 1 we have

H_i(P) = {0}.

More generally if X is a simplicial complex or finite CW complex, then the group H₀(X) is the free abelian group on generators the connected components of X. 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.

A more fundamental way to do the same thing is to go back to the chain complex defining homology, and tweak the C₀ term in it. Namely, define the augmentation ε from C₀ to the integers, which expresses the sum of coefficients. Replace C₀ by the subgroup with sum 0, or in other words the kernel of ε. Then calculate homology groups as usual, with the modified chain complex. 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.