# 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

*H*_{0}(*P*)

is isomorphic to (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 *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 .

To define reduced homology, we start with the *augmented* chain complex

where . Now we define the *reduced* homology groups by

- for positive
*n*and .

One can show that ; evidently 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.

## References

- Hatcher, A., (2002)
*Algebraic Topology*Cambridge University Press, ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.