In Homology and Cohomology, we showed how to study the topology of spaces using homology and cohomology groups, which are obtained via the construction of chain complexes out of abelian groups of subspaces of the topological space. However, we have not elaborated how this is achieved. In this post we introduce the notion of a simplex, which gives us a method of “triangulating” the space so that we can have construct the chains that make up our chain complex.
An -simplex can be thought of as the -dimensional analogue of a triangle. More technically, it is the smallest convex set in some Euclidean space containing points , called its vertices, such that the “difference vectors” defined by are linearly independent. We will use the notation to denote a simplex. We will keep track of the ordering of the vertices of the simplex, and we will always make use of the convention that the subscripts indexing the vertices are to be written in increasing order.
To make things more concrete, we discuss one of the most basic examples of an -simplex, the standard -simplex. It is defined to be the subset of -dimensional Euclidean space given by
The standard -simplex is a point (actually the point on the real line ), the standard -simplex is a line segment connecting the points and in the – plane, the standard -simplex is a triangle (including its interior) whose vertices are located at , , and in -dimensional Euclidean space, and the standard -simplex is a tetrahedron (again including its interior) whose vertices are located at , , , and in -dimensional Euclidean space. The standard higher-dimensional simplices have analogous descriptions. Here is an image depicting the standard -simplex, courtesy of user Tosha of Wikipedia:
Consider now a -simplex . This is of course a triangle. To this -simplex there are three related -simplices, namely , , and . They can be thought of as the “edges” of the -simplex , and together they form the boundary of the -simplex, written .
We want to use the concept of simplices in order to construct chains, which in turn form chain complexes, so that we can make use of the techniques of homology and cohomology. Crucial to the notion of chains is the abelian group structure, so that we can “add” and “subtract” -chains, for a given , to form new -chains. This abelian group structure will also help us in making the idea of a boundary of a simplex more concrete, and at the same time provide us with an explicit expression for the boundary operator (also known as the boundary map, or boundary function) also crucial to the idea of a chain complex and its homology. This boundary operator (written ) is given by
where means that the vertex is to be omitted. Therefore, for the -simplex , our boundary is given by
Simplices, with the boundary operators, can therefore be used to form chain complexes. The chains in this chain complex consist of “linear combinations” of simplices. We can then apply the notions of cycles and boundaries, and the principle that a space that is the boundary of another space has itself no boundary (but not all spaces that have no boundaries are the boundaries of other spaces – this is what the homology groups express), to study topology.
Of course, not all spaces look like simplices. But for many spaces, we can always map simplices to them via a homoeomorphism. Intuitively, this corresponds to “triangulating” the space. For example, we may map the boundary of a tetrahedron (made up of four triangles – -simplices – with certain edges and vertices in common) homeomorphically onto the sphere. What this means is that we can essentially then take the techniques that we have developed for the tetrahedron and apply them to the sphere.
Similarly, we can take a square with a diagonal (made up of two -simplices, again with certain edges and vertices in common), identify opposite edges of the boundary, and map it homeomorphically to the torus. This allows us to calculate the homology groups of the torus.
The use of simplices to construct chain complexes for taking the homology groups of a topological space is called simplicial homology. A generalization that involves maps that may not be homeomorphic is called singular homology. There are also other ways to construct chain complexes, for instance, we also have cellular cohomology, which instead makes use of of “cells” instead of simplices. Just as simplices are generalizations of triangles and tetrahedrons, cells are generalizations of discs and balls. A space made up of simplices is called a simplicial complex, while a space made up of cells is called a CW-complex.
Aside from algebraic topology in the usual sense, the notion of simplices are also useful in higher category theory. We recall from Category Theory that a category is made up of objects and morphisms, sometimes also called arrows, between these objects. In higher category theory, we also consider “morphisms between morphisms”, “morphisms between morphisms between morphisms”, and so on. This is reminiscent of simplices, in which we have vertices, edges, faces, and higher-dimensional analogues. Hence, the idea of simplices can be abstracted so that they can be used for the constructions of higher category theory. This leads to the theory of simplicial categories.