In
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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, 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
(an
infinite cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary ...
), 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 set ...
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 cla ...
, 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 In mathematics, a formal sum, formal series, or formal linear combination may be:
*In group theory, an element of a free abelian group, a sum of finitely many elements from a given basis set multiplied by integer coefficients.
*In linear algebra, an ...
of connected components, but as such a formal sum where the coefficients add up to zero.
In the usual definition of
homology
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
* Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chrom ...
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
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 \otimes W ...
, or ''reduced''
cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
s from the
cochain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
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