Barycentric subdivision

In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension on simplicial complexes is a canonical method to refine them. Therefore, the barycentric subdivision is an important tool in algebraic topology. The barycentric subdivision is an operation on simplicial complexes. In algebraic topology it is sometimes useful to replace the original spaces with simplicial complexes via triangulations: The substitution allows to assign combinatorial invariants as the Euler characteristic to the spaces. One can ask if there is an analogous way to replace the continuous functions defined on the topological spaces by functions that are linear on the simplices and which are homotopic to the original maps (see also simplicial approximation). In general, such an assignment requires a refinement of the given complex, meaning, one replaces bigger simplices by a union of smaller simplices. A standard way to effectuate such a refinement is the barycentric subdivision. Moreover, barycentric subdivision induces maps on homology groups and is helpful for computational concerns, see Excision and Mayer-Vietoris-sequence. Let be a geometric simplicial complex. A complex is said to be a subdivision of if each simplex of is contained in a simplex of each simplex of is a finite union of simplices of These conditions imply that and equal as sets and as topological spaces, only the simplicial structure changes. For a simplex spanned by points , the barycenter is defined to be the point . To define the subdivision, we will consider a simplex as a simplicial complex that contains only one simplex of maximal dimension, namely the simplex itself. The barycentric subdivision of a simplex can be defined inductively by its dimension. For points, i.e. simplices of dimension 0, the barycentric subdivision is defined as the point itself. Suppose then for a simplex of dimension that its faces of dimension are already divided. Therefore, there exist simplices covering .