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 formula_1 (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 "H"0("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 formula_3.
To define reduced homology, we start with the "augmented" chain complex
formula_4
where formula_5. Now we define the "reduced" homology groups by 
One can show that formula_8; evidently formula_9 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.