In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension to simplicial complexes is a canonical method to refining them. Therefore, the barycentric subdivision is an important tool in algebraic topology.

Motivation

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: This substitution allows one to assign combinatorial invariants such as the Euler characteristic to the spaces. One can ask whether there is an analogous way to replace the continuous functions defined on the topological spaces with functions that are linear on the simplices and homotopic to the original maps (see also simplicial approximation). In general, such an assignment requires a refinement of the given complex, meaning that one replaces larger simplices with a union of smaller simplices. A standard way to carry out such a refinement is the barycentric subdivision. Moreover, barycentric subdivision induces maps on homology groups and is helpful for computational concerns, see Excision and the Mayer–Vietoris sequence.

Definition

Subdivision of simplicial complexes

if

equal as sets and as topological spaces, only the simplicial structure changes.

Barycentric subdivision of a simplex

. 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.

.

-th skeleton of the simplicial complex. It allows effectuating the subdivision more than once.

Barycentric subdivision of a convex polytope

The operation of barycentric subdivision can be applied to any convex polytope of any dimension, producing another convex polytope of the same dimension. In this version of barycentric subdivision, it is not necessary for the polytope to form a simplicial complex: it can have faces that are not simplices. This is the dual operation to omnitruncation. The vertices of the barycentric subdivision correspond to the faces of all dimensions of the original polytope. Two vertices are adjacent in the barycentric subdivision when they correspond to two faces of different dimensions with the lower-dimensional face included in the higher-dimensional face. The facets of the barycentric subdivision are simplices, corresponding to the flags of the original polytope.

For instance, the barycentric subdivision of a cube, or of a regular octahedron, is the disdyakis dodecahedron. The degree-6, degree-4, and degree-8 vertices of the disdyakis dodecahedron correspond to the vertices, edges, and square facets of the cube, respectively.

Properties

Mesh

obtained by the barycentric subdivision. Then, the following estimation holds:

. Therefore, by applying barycentric subdivision sufficiently often, the largest edge can be made as small as desired.

Homology

For some statements in homology-theory one wishes to replace simplicial complexes by a subdivision. On the level of simplicial homology and such that the maps induces endomorphisms of chain complexes. Moreover, the induced map is an isomorphism: Subdivision does not change the homology of the complex.

To compute the singular homology groups of a topological space to a linear combination . This map also induces an endomorphism of chain complexes.

Applications

The barycentric subdivision can be applied on whole simplicial complexes as in the simplicial approximation theorem or it can be used to subdivide geometric simplices. Therefore it is crucial for statements in singular homology theory, see Mayer–Vietoris sequence and excision.

Simplicial approximation

induces a map between the geometric realizations of the complexes. Each point in a geometric complex lies in the inner of exactly one simplex, its support. Consider now a continuous. is said to be a simplicial approximation if and only if by defining it on each simplex; there, it always exists, because simplices are contractible.

The simplicial approximation theorem guarantees for every continuous function by its iterated barycentric subdivision. The theorem plays an important role for certain statements in algebraic topology in order to reduce the behavior of continuous maps on those of simplicial maps, as for instance in Lefschetz's fixed-point theorem.

Lefschetz's fixed-point theorem

The Lefschetz number induces homomorphisms can be determined and their alternating sum

is called the Lefschetz number has a fixed-point. In the proof this is first shown only for simplicial maps and then generalized for any continuous functions via the approximation theorem.

has a fixpoint.

Mayer–Vietoris sequence

The Mayer–Vietoris sequence is often used to compute singular homology groups and gives rise to inductive arguments in topology. The related statement can be formulated as follows:

.

There is an exact sequence

denotes the direct sum of abelian groups.

induces an isomorphism on homology which is needed to compare the homology groups.

Excision

Excision can be used to determine relative homology groups. It allows in certain cases to forget about subsets of topological spaces for their homology groups and therefore simplifies their computation:

Again, in singular homology, maps may appear such that their image is not part of the subsets mentioned in the theorem. Analogously those can be understood as a sum of images of smaller simplices obtained by the barycentric subdivision.

References