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A Singular n-simplex, singular ''n''-simplex in a topological space ''X'' is a continuous function (also called a map) $\backslash sigma$ from the standard ''n''-simplex $\backslash Delta^n$ to ''X'', written $\backslash sigma:\backslash Delta^n\backslash to\; X.$ This map need not be injective, and there can be non-equivalent singular simplices with the same image in ''X''.
The boundary of $\backslash sigma,$ denoted as $\backslash partial\_n\backslash sigma,$ is defined to be the formal sum of the singular (''n'' − 1)-simplices represented by the restriction of $\backslash sigma$ 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 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 $\backslash sigma$ by its vertices
:$[p\_0,p\_1,\backslash ldots,p\_n]=[\backslash sigma(e\_0),\backslash sigma(e\_1),\backslash ldots,\backslash sigma(e\_n)]$
corresponding to the vertices $e\_k$ of the standard ''n''-simplex $\backslash Delta^n$ (which of course does not fully specify the singular simplex produced by $\backslash sigma$), then
:$\backslash partial\_n\backslash sigma=\backslash sum\_^n(-1)^k\; \backslash sigma\; \backslash mid\; \_$
is a formal sum of the faces of the simplex image designated in a specific way. (That is, a particular face has to be the restriction of $\backslash sigma$ to a face of $\backslash Delta^n$ which depends on the order that its vertices are listed.) Thus, for example, the boundary of $\backslash sigma=[p\_0,p\_1]$ (a curve going from $p\_0$ to $p\_1$) is the formal sum (or "formal difference") $[p\_1]\; -\; [p\_0]$.

_{#} is a chain complex#Chain maps, chain map, which descends to homomorphisms on homology
:$f\_*\; :\; H\_n(X)\; \backslash rightarrow\; H\_n(Y).$
We now show that if ''f'' and ''g'' are homotopically equivalent, then ''f''_{*} = ''g''_{*}. From this follows that if ''f'' is a homotopy equivalence, then ''f''_{*} is an isomorphism.
Let ''F'' : ''X'' × [0, 1] → ''Y'' be a homotopy that takes ''f'' to ''g''. On the level of chains, define a homomorphism
:$P\; :\; C\_n(X)\; \backslash rightarrow\; C\_(Y)$
that, geometrically speaking, takes a basis element σ: Δ^{''n''} → ''X'' of ''C_{n}''(''X'') to the "prism" ''P''(σ): Δ^{''n''} × ''I'' → ''Y''. The boundary of ''P''(σ) can be expressed as
:$\backslash partial\; P(\backslash sigma)\; =\; f\_(\backslash sigma)\; -\; g\_(\backslash sigma)\; -\; P(\backslash partial\; \backslash sigma).$
So if ''α'' in ''C_{n}''(''X'') is an ''n''-cycle, then ''f''_{#}(''α'' ) and ''g''_{#}(''α'') differ by a boundary:
:$f\_\; (\backslash alpha)\; -\; g\_(\backslash alpha)\; =\; \backslash partial\; P(\backslash alpha),$
i.e. they are homologous. This proves the claim.

_{''n''}(''X'', ''R'') should not be confused with the nearly identical notation ''H''_{''n''}(''X'', ''A''), which denotes the relative homology (below).

_{''n''}(''X'', ''A'') is understood to be the homology of the quotient of the chain complexes, that is,
:$H\_n(X,A)=H\_n(C\_\backslash bullet(X)/C\_\backslash bullet(A))$
where the quotient of chain complexes is given by the short exact sequence
:$0\backslash to\; C\_\backslash bullet(A)\; \backslash to\; C\_\backslash bullet(X)\; \backslash to\; C\_\backslash bullet(X)/C\_\backslash bullet(A)\; \backslash to\; 0.$

''Algebraic topology.''

Cambridge University Press, and * J.P. May, ''A Concise Course in Algebraic Topology'', Chicago University Press * Joseph J. Rotman, ''An Introduction to Algebraic Topology'', Springer-Verlag, {{isbn, 0-387-96678-1 Homology theory

algebraic topology
250px, A torus, one of the most frequently studied objects 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 (mathem ...

, a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...

, singular homology refers to the study of a certain set of algebraic invariants of a topological space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

''X'', the so-called homology groups $H\_n(X).$ Intuitively, singular homology counts, for each dimension ''n'', the ''n''-dimensional holes of a space. Singular homology is a particular example of a homology theory, 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.
In brief, singular homology is constructed by taking maps of the 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 – induces the singular chain complex. The singular homology is then the homology (mathematics), homology of the chain complex. The resulting homology groups are the same for all Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalent spaces, which is the reason for their study. These constructions can be applied to all topological spaces, and so singular homology can be expressed in terms of category theory, where homology is expressible as a functor from the category of topological spaces to the category of graded abelian groups.
Singular simplices

Image:2D-simplex.svg, 150px, The standard 2-simplex ΔSingular chain complex

The usual construction of singular homology proceeds by defining formal sums of simplices, which may be understood to be elements of a free abelian group, and then showing that we can define a certain group, the homology group of the topological space, involving the boundary operator. Consider first the set of all possible singular ''n''-simplices $\backslash sigma\_n(X)$ on a topological space ''X''. This set may be used as the basis of a free abelian group, so that each singular ''n''-simplex is a generator of the group. This set of generators is of course usually infinite, frequently uncountable, as there are many ways of mapping a simplex into a typical topological space. The free abelian group generated by this basis is commonly denoted as $C\_n(X)$. Elements of $C\_n(X)$ are called singular ''n''-chains; they are formal sums of singular simplices with integer coefficients. The boundary $\backslash partial$ is readily extended to act on singular ''n''-chains. The extension, called the boundary operator, written as :$\backslash partial\_n:C\_n\backslash to\; C\_,$ is a homomorphism of groups. The boundary operator, together with the $C\_n$, form a chain complex of abelian groups, called the singular complex. It is often denoted as $(C\_\backslash bullet(X),\backslash partial\_\backslash bullet)$ or more simply $C\_\backslash bullet(X)$. The kernel of the boundary operator is $Z\_n(X)=\backslash ker\; (\backslash partial\_)$, and is called the group of singular ''n''-cycles. The image of the boundary operator is $B\_n(X)=\backslash operatorname\; (\backslash partial\_)$, and is called the group of singular ''n''-boundaries. It can also be shown that $\backslash partial\_n\backslash circ\; \backslash partial\_=0$. The $n$-th homology group of $X$ is then defined as the factor group :$H\_(X)\; =\; Z\_n(X)\; /\; B\_n(X).$ The elements of $H\_n(X)$ are called homology classes.Homotopy invariance

If ''X'' and ''Y'' are two topological spaces with the same homotopy type (i.e. are homotopy equivalent), then :$H\_n(X)\; \backslash cong\; H\_n(Y)\backslash ,$ for all ''n'' ≥ 0. This means homology groups are topological invariants. In particular, if ''X'' is a connected contractible space, then all its homology groups are 0, except $H\_0(X)\; \backslash cong\; \backslash mathbb$. A proof for the homotopy invariance of singular homology groups can be sketched as follows. A continuous map ''f'': ''X'' → ''Y'' induces a homomorphism :$f\_\; :\; C\_n(X)\; \backslash rightarrow\; C\_n(Y).$ It can be verified immediately that :$\backslash partial\; f\_\; =\; f\_\; \backslash partial,$ i.e. ''f''Functoriality

The construction above can be defined for any topological space, and is preserved by the action of continuous maps. This generality implies that singular homology theory can be recast in the language of category theory. In particular, the homology group can be understood to be a functor from the category of topological spaces Top to the category of abelian groups Ab. Consider first that $X\backslash mapsto\; C\_n(X)$ is a map from topological spaces to free abelian groups. This suggests that $C\_n(X)$ might be taken to be a functor, provided one can understand its action on the morphisms of Top. Now, the morphisms of Top are continuous functions, so if $f:X\backslash to\; Y$ is a continuous map of topological spaces, it can be extended to a homomorphism of groups :$f\_*:C\_n(X)\backslash to\; C\_n(Y)\backslash ,$ by defining :$f\_*\backslash left(\backslash sum\_i\; a\_i\backslash sigma\_i\backslash right)=\backslash sum\_i\; a\_i\; (f\backslash circ\; \backslash sigma\_i)$ where $\backslash sigma\_i:\backslash Delta^n\backslash to\; X$ is a singular simplex, and $\backslash sum\_i\; a\_i\backslash sigma\_i\backslash ,$ is a singular ''n''-chain, that is, an element of $C\_n(X)$. This shows that $C\_n$ is a functor :$C\_n:\backslash mathbf\; \backslash to\; \backslash mathbf$ from the category of topological spaces to the category of abelian groups. The boundary operator commutes with continuous maps, so that $\backslash partial\_n\; f\_*=f\_*\backslash partial\_n$. This allows the entire chain complex to be treated as a functor. In particular, this shows that the map $X\backslash mapsto\; H\_n\; (X)$ is a functor :$H\_n:\backslash mathbf\backslash to\backslash mathbf$ from the category of topological spaces to the category of abelian groups. By the homotopy axiom, one has that $H\_n$ is also a functor, called the homology functor, acting on hTop, the quotient homotopy category: :$H\_n:\backslash mathbf\backslash to\backslash mathbf.$ This distinguishes singular homology from other homology theories, wherein $H\_n$ is still a functor, but is not necessarily defined on all of Top. In some sense, singular homology is the "largest" homology theory, in that every homology theory on a subcategory of Top agrees with singular homology on that subcategory. On the other hand, the singular homology does not have the cleanest categorical properties; such a cleanup motivates the development of other homology theories such as cellular homology. More generally, the homology functor is defined axiomatically, as a functor on an abelian category, or, alternately, as a functor on chain complexes, satisfying axioms that require a boundary morphism that turns short exact sequences into long exact sequences. In the case of singular homology, the homology functor may be factored into two pieces, a topological piece and an algebraic piece. The topological piece is given by :$C\_\backslash bullet:\backslash mathbf\backslash to\backslash mathbf$ which maps topological spaces as $X\backslash mapsto\; (C\_\backslash bullet(X),\backslash partial\_\backslash bullet)$ and continuous functions as $f\backslash mapsto\; f\_*$. Here, then, $C\_\backslash bullet$ is understood to be the singular chain functor, which maps topological spaces to the category of chain complexes Comp (or Kom). The category of chain complexes has chain complexes as its object (category theory), objects, and chain maps as its morphisms. The second, algebraic part is the homology functor :$H\_n:\backslash mathbf\backslash to\backslash mathbf$ which maps :$C\_\backslash bullet\backslash mapsto\; H\_n(C\_\backslash bullet)=Z\_n(C\_\backslash bullet)/B\_n(C\_\backslash bullet)$ and takes chain maps to maps of abelian groups. It is this homology functor that may be defined axiomatically, so that it stands on its own as a functor on the category of chain complexes. Homotopy maps re-enter the picture by defining homotopically equivalent chain maps. Thus, one may define the quotient category hComp or K, the homotopy category of chain complexes.Coefficients in ''R''

Given any unital ring (mathematics), ring ''R'', the set of singular ''n''-simplices on a topological space can be taken to be the generators of a free module, free ''R''-module. That is, rather than performing the above constructions from the starting point of free abelian groups, one instead uses free ''R''-modules in their place. All of the constructions go through with little or no change. The result of this is :$H\_n(X,\; R)\backslash $ which is now an module (mathematics), ''R''-module. Of course, it is usually ''not'' a free module. The usual homology group is regained by noting that :$H\_n(X,\backslash mathbb)=H\_n(X)$ when one takes the ring to be the ring of integers. The notation ''H''Relative homology

For a subspace $A\backslash subset\; X$, the relative homology ''H''Cohomology

By dualizing the homology chain complex (i.e. applying the functor Hom(-, ''R''), ''R'' being any ring) we obtain a cochain complex with coboundary map $\backslash delta$. The cohomology groups of ''X'' are defined as the homology groups of this complex; in a quip, "cohomology is the homology of the co [the dual complex]". The cohomology groups have a richer, or at least more familiar, algebraic structure than the homology groups. Firstly, they form a differential graded algebra as follows: * the graded set of groups form a graded ''R''-Module (mathematics), module; * this can be given the structure of a graded ''R''-Algebra (ring theory), algebra using the cup product; * the Bockstein homomorphism ''β'' gives a differential. There are additional cohomology operations, and the cohomology algebra has addition structure mod ''p'' (as before, the mod ''p'' cohomology is the cohomology of the mod ''p'' cochain complex, not the mod ''p'' reduction of the cohomology), notably the Steenrod algebra structure.Betti homology and cohomology

Since the number of homology theories has become large (see :Homology theory), the terms ''Betti homology'' and ''Betti cohomology'' are sometimes applied (particularly by authors writing on algebraic geometry) to the singular theory, as giving rise to the Betti numbers of the most familiar spaces such as simplicial complexes and closed manifolds.Extraordinary homology

If one defines a homology theory axiomatically (via the Eilenberg–Steenrod axioms), and then relaxes one of the axioms (the ''dimension axiom''), one obtains a generalized theory, called an extraordinary homology theory. These originally arose in the form of extraordinary cohomology theories, namely K-theory and cobordism theory. In this context, singular homology is referred to as ordinary homology.See also

* Derived category * Excision theorem * Hurewicz theorem * Simplicial homology * Cellular homologyReferences

* Allen Hatcher''Algebraic topology.''

Cambridge University Press, and * J.P. May, ''A Concise Course in Algebraic Topology'', Chicago University Press * Joseph J. Rotman, ''An Introduction to Algebraic Topology'', Springer-Verlag, {{isbn, 0-387-96678-1 Homology theory