# 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

H0(P)

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

Hi(P) = {0}.

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

$\dotsb {\overset {\partial _{n+1}}{\longrightarrow \,}}C_{n}{\overset {\partial _{n}}{\longrightarrow \,}}C_{n-1}{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}C_{1}{\overset {\partial _{1}}{\longrightarrow \,}}C_{0}{\overset {\partial _{0}}{\longrightarrow \,}}0$ and define the homology groups by $H_{n}(X)=\ker(\partial _{n})/\mathrm {im} (\partial _{n+1})$ .

$\dotsb {\overset {\partial _{n+1}}{\longrightarrow \,}}C_{n}{\overset {\partial _{n}}{\longrightarrow \,}}C_{n-1}{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}C_{1}{\overset {\partial _{1}}{\longrightarrow \,}}C_{0}{\overset {\epsilon }{\longrightarrow \,}}\mathbb {Z} \to 0$ where $\epsilon \left(\sum _{i}n_{i}\sigma _{i}\right)=\sum _{i}n_{i}$ . Now we define the reduced homology groups by
${\tilde {H_{n}}}(X)=\ker(\partial _{n})/\mathrm {im} (\partial _{n+1})$ for positive n and ${\tilde {H}}_{0}(X)=\ker(\epsilon )/\mathrm {im} (\partial _{1})$ .
One can show that $H_{0}(X)={\tilde {H}}_{0}(X)\oplus \mathbb {Z}$ ; evidently $H_{n}(X)={\tilde {H}}_{n}(X)$ for all positive n.