Singular homology

# Singular homology

Discussion

Encyclopedia
In algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which 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 up to homotopy equivalence.Although algebraic topology...

, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, singular homology refers to the study of a certain set of algebraic invariants of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

X, the so-called homology groups .
Intuitively spoken, singular homology counts, for each dimension n, the n-dimensional holes of a space.
Singular homology is a particular example of a homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...

, 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 standard n-simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

to a topological space, and composing them into formal sums, called singular chains. The boundary
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

operation on a simplex induces a singular chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...

. The singular homology is then the homology
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...

of the chain complex. The resulting homology groups are the same for all homotopically 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
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, where the homology group becomes a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

from the category of topological spaces
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...

to the category of graded abelian group
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...

s. These ideas are developed in greater detail below.

## Singular simplices

A singular n-simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

is a continuous mapping from the standard n-simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

to a topological space X. Notationally, one writes . This mapping need not be injective, and there can be non-equivalent singular simplices with the same image in X.

The boundary of , denoted as , 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 a free abelian group
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...

on the simplices. The basis for the group is the infinite set of all possible images of standard simplices. The group operation is "addition" and the sum of image a with image b is usually simply designated a+b, but a+a=2a and so on. Every image a has a negative −a.) Thus, if we designate the range of by its vertices

corresponding to the vertices of the standard n-simplex (which of course does not fully specify the standard simplex image produced by ), then

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 image of applied to a designation of a face of which depends on the order that its vertices are listed.) Thus, for example, the boundary of (a curve going from to ) is the formal sum (or "formal difference") .

## Singular 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
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...

, 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 on a topological space X. This set may be used as the basis of a free abelian group
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...

, so that each 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 . Elements of are called singular n-chains; they are formal sums of singular simplices with integer coefficients. In order for the theory to be placed on a firm foundation, it is commonly required that a chain be a sum of only a finite number of simplices.

The boundary
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

is readily extended to act on singular n-chains. The extension, called the boundary operator, written as
,

is a homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

of groups. The boundary operator, together with the , form a chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...

of abelian groups, called the singular complex. It is often denoted as or more simply .

The kernel of the boundary operator is , and is called the group of singular n-cycles. The image of the boundary operator is , and is called the group of singular n-boundaries.

It can also be shown that . The -th homology group of is then defined as the factor group
.

The elements of are called homology classes.

## Homotopy invariance

If X and Y are two topological spaces with the same homotopy type, then

for all n ≥ 0. This means homology groups are topological invariants.

In particular, if X is a connected contractible space
Contractible space
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point....

, then all its homology groups are 0, except .

A proof for the homotopy invariance of singular homology groups can be sketched as follows. A continuous map f: XY induces a homomorphism

It can be verified immediately that

i.e. f# is a chain map, which descends to homomorphisms on homology

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

that, geometrically speaking, takes a basis element σ: ΔnX of Cn(X) to the "prism" P(σ): Δn × IY. The boundary of P(σ) can be expressed as

So if α in Cn(X) is an n-cycle, then f#(α ) and g#(α) differ by a boundary:

i.e. they are homologous. This proves the claim.

## 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
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

. In particular, the homology group can be understood to be a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

from the category of topological spaces
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...

Top to the category of abelian groups
Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....

Ab.

Consider first that is a map from topological spaces to free abelian groups. This suggests that might be taken to be a functor, provided one can understand its action on the morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

s of Top. Now, the morphisms of Top are continuous functions, so if is a continuous map of topological spaces, it can be extended to a homomorphism of groups

by defining

where is a singular simplex, and is a singular n-chain, that is, an element of . This shows that is a functor

from the category of topological spaces
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...

to the category of abelian groups
Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....

.

The boundary operator commutes with continuous maps, so that . This allows the entire chain complex to be treated as a functor. In particular, this shows that the map is a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

from the category of topological spaces to the category of abelian groups. By the homotopy axiom, one has that is also a functor, called the homology functor, acting on hTop, the quotient homotopy category:

This distinguishes singular homology from other homology theories, wherein 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
Subcategory
In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and...

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
Cellular homology
In mathematics, cellular homology in algebraic topology is a homology theory for CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules.- Definition :...

.

More generally, the homology functor is defined axiomatically, as a functor on an abelian category
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

, or, alternately, as a functor on chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...

es, 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

which maps topological spaces as and continuous functions as . Here, then, is understood to be the singular chain functor, which maps topological spaces to the category of chain complexes
Category of chain complexes
In mathematics, chain complexes arise naturally in topology and geometry. For example, homology and cohomology theories all make use of chain complexes. To define a chain complex, fix an abelian category, say the category of modules over a commutative ring...

Comp (or Kom). The category of chain complexes has chain complexes as its objects, and chain maps as its morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

s.

The second, algebraic part is the homology functor

which maps

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
Quotient category
In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. The notion is similar to that of a quotient group or quotient space, but in the categorical setting.-Definition:Let C be a category...

hComp or K, the homotopy category of chain complexes
Homotopy category of chain complexes
In homological algebra in mathematics, the homotopy category K of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences...

.

## Coefficients in R

Given any unital ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

R, the set of singular n-simplices on a topological space can be taken to be the generators of a free R-module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...

. 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

which is now an R-module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

. Of course, it is usually not a free module. The usual homology group is regained by noting that

when one takes the ring to be the ring of integers. The notation Hn(X, R) should not be confused with the nearly identical notation Hn(X, A), which denotes the relative homology (below).

## Relative homology

For a subspace , the relative homology
Relative homology
In algebraic topology, a branch of mathematics, the homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways...

Hn(X, A) is understood to be the homology of the quotient of the chain complexes, that is,

where the quotient of chain complexes is given by the short exact sequence

## Cohomology

By dualizing the homology chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...

(i.e. applying the functor Hom(-, R), R being any ring) we obtain a cochain complex with coboundary map . The cohomology groups of X are defined as the cohomology groups of this complex; in a quip, "cohomology is the homology of the co- (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
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure.- Definition :...

as follows:
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

;
• this can be given the structure of a graded R-algebra
Algebra (ring theory)
In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....

using the cup product
Cup product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a...

;
• the Bockstein homomorphism
Bockstein homomorphism
In homological algebra, the Bockstein homomorphism, introduced by , is a connecting homomorphism associated with a short exact sequenceof abelian groups, when they are introduced as coefficients into a chain complex C, and which appears in the homology groups as a homomorphism reducing degree by...

β gives a differential.

Cohomology operation
In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from...

s, 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
Steenrod algebra
In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.For a given prime number p, the Steenrod algebra Ap is the graded Hopf algebra over the field Fp of order p, consisting of all stable cohomology operations for mod p...

structure.

## Betti homology and cohomology

Since the number of homology theories has become large (see :Category:Homology theory), the terms Betti homology and Betti cohomology are sometimes applied (particularly by authors writing on algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

) to the singular theory, as giving rise to the Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....

s of the most familiar spaces such as simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...

es and closed manifold
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....

s.

## Extraordinary homology

If one defines a homology theory axiomatically (via the Eilenberg-Steenrod axioms
Eilenberg-Steenrod axioms
In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common...

), 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
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...

and cobordism theory. In this context, singular homology is referred to as ordinary homology.