In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X {\displaystyle X} , the so-called homology groups H n ( X ) . {\displaystyle H_{n}(X).} Intuitively, singular homology counts, for each dimension n {\displaystyle n} , the n {\displaystyle 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 (see also the related theory simplicial 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 boundary operation – mapping each n {\displaystyle n} -dimensional simplex to its ( n − 1 ) {\displaystyle (n-1)} -dimensional boundary – induces the singular chain complex. The singular homology is then the homology of the chain complex. 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.
Singular simplices
A singular n-simplex in a topological space X {\displaystyle X} is a continuous function (also called a map) σ {\displaystyle \sigma } from the standard n {\displaystyle n} -simplex Δ n {\displaystyle \Delta ^{n}} to X {\displaystyle X} , written σ : Δ n → X . {\displaystyle \sigma :\Delta ^{n}\to X.} This map need not be injective, and there can be non-equivalent singular simplices with the same image in X {\displaystyle X} . The boundary of σ , {\displaystyle \sigma ,} denoted as ∂ n σ , {\displaystyle \partial _{n}\sigma ,} is defined to be the formal sum of the singular ( n − 1 ) {\displaystyle (n-1)} -simplices represented by the restriction of σ {\displaystyle \sigma } to the faces of the standard n {\displaystyle 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 {\displaystyle a} with simplex b {\displaystyle b} is usually simply designated a + b {\displaystyle a+b} , but a + a = 2 a {\displaystyle a+a=2a} and so on. Every simplex a {\displaystyle a} has a negative − a {\displaystyle -a} .) Thus, if we designate σ {\displaystyle \sigma } by its vertices
[ p 0 , p 1 , … , p n ] = [ σ ( e 0 ) , σ ( e 1 ) , … , σ ( e n ) ] {\displaystyle [p_{0},p_{1},\ldots ,p_{n}]=[\sigma (e_{0}),\sigma (e_{1}),\ldots ,\sigma (e_{n})]}
corresponding to the vertices e k {\displaystyle e_{k}} of the standard n {\displaystyle n} -simplex Δ n {\displaystyle \Delta ^{n}} (which of course does not fully specify the singular simplex produced by σ {\displaystyle \sigma } ), then
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