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.
A singular n-simplex is a continuous mapping from the standard n-simplex 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. That is, if
are the corners of the n-simplex corresponding to the vertices of the standard n-simplex , then
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, so that each 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 . 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 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.
Clearly, one has . The -th homology group of is then defined as the factor group
The elements of are called homology classes.
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 contractible space, 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: X → Y 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: Δn → X of Cn(X) to the "prism" P(σ): Δn × I → Y. 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.
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 morphisms 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
where is a singular simplex, and is a singular n-chain, that is, an element of . This shows that is a functor
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
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 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
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 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
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.
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
which is now an R-module. 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).
where the quotient of chain complexes is given by the short exact sequence
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 . 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.
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.
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