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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, reduced homology is a minor modification made to
homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, 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 In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
or finite
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
, then the group ''H''0(''X'') is the
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
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 In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
, or ''reduced'' cohomology groups from the cochain complex made by using a
Hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...
, 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