Singular homology
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 resulting homology groups are the same for all homotopy equivalent 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 functor from the category of topological spaces to the category of graded abelian groups.
This map need not be injective, and there can be non-equivalent singular simplices with the same image in
(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
), then is a formal sum of the faces of the simplex image designated in a specific way.
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.
This set may be used as the basis of a free abelian group, so that each singular
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
The extension, called the boundary operator, written as is a homomorphism of groups.
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*.
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.
[3] The table below shows the k-th homology groups
of n-dimensional real projective spaces RPn, complex projective spaces, CPn, a point, spheres Sn(
The construction above can be defined for any topological space, and is preserved by the action of continuous maps.
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.
is a map from topological spaces to free abelian groups.
might be taken to be a functor, provided one can understand its action on the morphisms of Top.
is a continuous map of topological spaces, it can be extended to a homomorphism of groups by defining where
This allows the entire chain complex to be treated as a functor.
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.
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.
For the usual homology defined on a chain complex: To define the reduced homology, we augment the chain complex with an additional
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 homology groups of this complex; in a quip, "cohomology is the homology of the co [the dual complex]".
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) to the singular theory, as giving rise to the Betti numbers of the most familiar spaces such as simplicial complexes and closed manifolds.
