Barycentric subdivision
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.
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).
Moreover, barycentric subdivision induces maps on homology groups and is helpful for computational concerns, see Excision and the Mayer–Vietoris sequence.
equal as sets and as topological spaces, only the simplicial structure changes.
The barycentric subdivision of a simplex can be defined inductively by its dimension.
The barycentric subdivision is then defined to be the geometric simplicial complex whose maximal simplices of dimension
One can generalize the subdivision for simplicial complexes whose simplices are not all contained in a single simplex of maximal dimension, i.e. simplicial complexes that do not correspond geometrically to one simplex.
This can be done by effectuating the steps described above simultaneously for every simplex of maximal dimension.
[3] 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.
[4] The vertices of the barycentric subdivision correspond to the faces of all dimensions of the original polytope.
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.
Therefore, by applying barycentric subdivision sufficiently often, the largest edge can be made as small as desired.
[6] For some statements in homology-theory one wishes to replace simplicial complexes by a subdivision.
Moreover, the induced map is an isomorphism: Subdivision does not change the homology of the complex.
In an analogous way as described for simplicial homology groups, barycentric subdivision can be interpreted as an endomorphism of singular chain complexes.
Therefore it is crucial for statements in singular homology theory, see Mayer–Vietoris sequence and excision.
Each point in a geometric complex lies in the inner of exactly one simplex, its support.
The simplicial approximation theorem guarantees for every continuous function
[8] 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.
The Lefschetz number is a useful tool to find out whether a continuous function admits fixed-points.
In the proof this is first shown only for simplicial maps and then generalized for any continuous functions via the approximation theorem.
Now, Brouwer's fixpoint theorem is a special case of this statement.
[9] The Mayer–Vietoris sequence is often used to compute singular homology groups and gives rise to inductive arguments in topology.
There is an exact sequence where we consider singular homology groups,
For the construction of singular homology groups one considers continuous maps defined on the standard simplex
It allows in certain cases to forget about subsets of topological spaces for their homology groups and therefore simplifies their computation: Let
Analogously those can be understood as a sum of images of smaller simplices obtained by the barycentric subdivision.
