Reduced Cohomology

In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptional cases (as in the homology groups of spheres).

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

:''H''₀(''P'')

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

:''H''_''i''(''P'') = .

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
:$\dotsb\oversetC_n \oversetC_ \overset \dotsb \overset C_1 \overset C_0\overset 0$

and define the homology groups by $H_n(X) = \ker(\partial_n) / \mathrm(\partial_)$.

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

$\dotsb\oversetC_n \oversetC_ \overset \dotsb \overset C_1 \overset C_0\overset \mathbb \to 0$

where $\epsilon \left( \sum_i n_i \sigma_i \right) = \sum_i n_i$. Now we define the ''reduced'' homology groups by

:$\tilde_n(X) = \ker(\partial_n) / \mathrm(\partial_)$ for positive ''n'' and $\tilde_0(X) = \ker(\epsilon) / \mathrm(\partial_1)$.

One can show that $H_0(X) = \tilde_0(X) \oplus \mathbb$; evidently $H_n(X) = \tilde_n(X)$ 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, . Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.

{{DEFAULTSORT:Reduced Homology
Homology theory

de:Singuläre_Homologie#Reduzierte_Homologie