Definitions

Singular homology

In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of topological invariants of a topological space X, the so-called homology groups $H_n\left(X\right)$. 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 standard n-simplex to a topological space, and composing them into formal sums, called singular chains. The boundary operation on a simplex induces a singular chain complex. The singular homology is then the homology 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, where the homology group becomes a functor from the category of topological spaces to the category of graded abelian groups. These ideas are developed in greater detail below.

Singular simplices

A singular n-simplex is a continuous mapping $sigma$ from the standard n-simplex $Delta^n$ to a topological space X. Notationally, one writes $sigma_n:Delta^nto X$. This mapping need not be injective, and there can be non-equivalent singular simplices with the same image in X.

The boundary of $sigma$, denoted as $partial_nsigma_n$, is defined to be the formal sum of the singular (n−1)-simplices represented by the restriction of $sigma$ to the faces of the standard n-simplex, with an alternating sign to take orientation into account. That is, if

$sigma_n = \left[p_0,p_1,cdots,p_n\right]=sigma_n\left(\left[e_0,e_1,cdots,e_n\right]\right)$

are the corners of the n-simplex corresponding to the vertices $e_k$ of the standard n-simplex $Delta^n$, then

$partial_nsigma_n=sum_\left\{k=0\right\}^n\left(-1\right)^k \left[p_0,cdots,p_\left\{k-1\right\},p_\left\{k+1\right\},cdots p_n\right]$

is the formal sum of the (oriented) faces of the simplex. Thus, for example, the boundary of a 1-simplex $sigma=\left[p_0,p_1\right]$ is the formal difference $sigma_1 - sigma_0=\left[p_1\right] - \left[p_0\right]$.

Singular chain complex

The usual construction of singular homology proceeds by defining a chain of simplices, which may be understood to be elements of a free abelian group, and then showing that the boundary operator leads to a certain group, the homology group of the topological space.

Consider first the set $sigma_n\left(X\right)$ of all possible singular n-simplices on a topological space X. This set may be used as the basis of a free abelian group, so that each $sigma_n$ is a generator of the group. This group is, of course, very large, usually infinite, frequently uncountable, as there are many ways of mapping a simplex into a typical topological space. This group is commonly denoted as $C_n\left(X\right)$. Elements of $C_n\left(X\right)$ 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 $partial$ is readily extended to act on singular n-chains. The extension, called the boundary operator, written as

$partial_n:C_nto C_\left\{n-1\right\}$,

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 $\left(C_bullet\left(X\right),partial_bullet\right)$ or more simply $C_bullet\left(X\right)$.

The kernel of the boundary operator is $Z_n\left(X\right)=ker \left(partial_\left\{n\right\}\right)$, and is called the group of singular n-cycles. The image of the boundary operator is $B_n\left(X\right)=operatorname\left\{im\right\} \left(partial_\left\{n+1\right\}\right)$, and is called the group of singular n-boundaries.

Clearly, one has $partial_ncirc partial_\left\{n+1\right\}=0$. The $n$-th homology group of $X$ is then defined as the factor group

$H_\left\{n\right\}\left(X\right) = Z_n\left(X\right) / B_n\left(X\right)$.

The elements of $H_n\left(X\right)$ are called homology classes.

Homotopy invariance

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

$H_n\left(X\right)=H_n\left(Y\right),$

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

In particular, if X is a contractible space, then all its homology groups are 0, except $H_0\left(X\right) = mathbb\left\{Z\right\}$.

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

$f_\left\{sharp\right\} : C_n\left(X\right) rightarrow C_n\left(Y\right).$

It can be verified immediately that

$partial f_\left\{sharp\right\} = f_\left\{sharp\right\} partial,$

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

$f_* : H_n\left(X\right) rightarrow H_n\left(Y\right).$

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

$P : C_n\left(X\right) rightarrow C_\left\{n+1\right\}\left(Y\right)$

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

$partial P\left(sigma\right) = f_\left\{sharp\right\}\left(sigma\right) - g_\left\{sharp\right\}\left(sigma\right) + P\left(partial sigma\right).$

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

$f_\left\{sharp\right\} \left(alpha\right) - g_\left\{sharp\right\}\left(alpha\right) = partial P\left(alpha\right),$

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. 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 $Xmapsto C_n\left(X\right)$ is a map from topological spaces to free abelian groups. This suggests that $C_n\left(X\right)$ 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 $f:Xto Y$ is a continuous map of topological spaces, it can be extended to a homomorphism of groups

$f_*:C_n\left(X\right)to C_n\left(Y\right),$

by defining

$f_*left\left(sum_i a_isigma_iright\right)=sum_i a_i \left(fcirc sigma_i\right)$

where $sigma_i:Delta^nto X$ is a singular simplex, and $sum_i a_isigma_i,$ is a singular n-chain, that is, an element of $C_n\left(X\right)$. This shows that $C_n$ is a functor

$C_n:bold\left\{Top\right\} to bold\left\{Ab\right\}$

from the category of topological spaces to the category of abelian groups.

The boundary operator commutes with continuous maps, so that $partial_n f_*=f_*partial_n$. This allows the entire chain complex to be treated as a functor. In particular, this shows that the map $Xmapsto H_n \left(X\right)$ is a functor

$H_n:bold\left\{Top\right\}tobold\left\{Ab\right\}$

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:

$H_n:bold\left\{hTop\right\}tobold\left\{Ab\right\}.$

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

$C_bullet:bold\left\{Top\right\}tobold\left\{Comp\right\}$

which maps topological spaces as $Xmapsto \left(C_bullet\left(X\right),partial_bullet\right)$ and continuous functions as $fmapsto f_*$. Here, then, $C_bullet$ 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 objects, and chain maps as its morphisms.

The second, algebraic part is the homology functor

$H_n:bold\left\{Comp\right\}tobold\left\{Ab\right\}$

which maps

$C_bulletmapsto H_n\left(C_bullet\right)=Z_n\left(C_bullet\right)/B_n\left(C_bullet\right)$

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 R, the set of singular n-simplices on a topological space can be taken to be the generators of a 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

Hn(X, R)

which is now an R-module. Of course, it is usually not a free module. The usual homology group is regained by noting that

$H_n\left(X,mathbb\left\{Z\right\}\right)=H_n\left(X\right)$

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

$H_n\left(X,A\right)=H_n\left(C_bullet\left(X\right)/C_bullet\left(A\right)\right)$

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

$0to C_bullet\left(A\right) to C_bullet\left(X\right) to C_bullet\left(X\right)/C_bullet\left(A\right) to 0.$

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 $delta$. The cohomology groups of X are defined as the cohomology groups of this complex. They form a graded R-module, which can be given the structure of a graded R-algebra using the cup product.

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.