Singular Simplices
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. (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 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") .
Read more about this topic: Singular Homology
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