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 ...
, singular homology refers to the study of a certain set of
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
s of a
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
''X'', the so-called homology groups
Intuitively, singular homology counts, for each dimension ''n'', the ''n''-dimensional holes of a space. Singular homology is a particular example of a
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 topol ...
, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions (see also the related theory
simplicial homology In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case ...
In brief, singular homology is constructed by taking maps of the standard ''n''-simplex
to a topological space, and composing them into formal sums
, called singular chains. The boundary operation – mapping each ''n''-dimensional simplex to its (''n''−1)-dimensional
Boundary or Boundaries may refer to:
* Border, in political geography
* ''Boundaries'' (2016 film), a 2016 Canadian film
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*Boundary (cricket), the edge of the pla ...
– induces the singular
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 ...
. The singular homology is then the homology
of the chain complex. The resulting homology groups are the same for all
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...
spaces, which is the reason for their study. These constructions can be applied to all topological spaces, and so singular homology is expressible as a
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
from the category of topological spaces
to the category of graded
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
A singular ''n''-simplex
in a topological space ''X'' is a continuous function
(also called a map)
from the standard ''n''-
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimensio ...
to ''X'', written
This map need not be injective
, and there can be non-equivalent singular simplices with the same image in ''X''.
The boundary of
is defined to be the formal sum
of the singular (''n'' − 1)-simplices represented by the restriction of
to the faces of the standard ''n''-simplex, with an alternating sign to take orientation into account. (A formal sum is an element of 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 ...
on the simplices. The basis for the group is the infinite set of all possible singular simplices. The group operation is "addition" and the sum of simplex ''a'' with simplex ''b'' is usually simply designated ''a'' + ''b'', but ''a'' + ''a'' = 2''a'' and so on. Every simplex ''a'' has a negative −''a''.) Thus, if we designate
by its vertices