Barycentric Subdivision - Barycentric Subdivision in Topology

Barycentric Subdivision in Topology

Barycentric subdivision is an important tool in simplicial homology theory, where it is used as a means of obtaining finer simplicial complexes (containing the original ones, i.e. with more simplices). This in turn is crucial to the simplicial approximation theorem, which roughly states that one can approximate any continuous function between polyhedra by a (finite) simplicial map, given a sufficient amount of subdivision of the respective simplicial complexes whom they realize. Ultimately, this approximation technique is a standard ingredient in the proof that simplicial homology groups are topological invariants.

A generalization of barycentric subdivision can also be defined for a cell complex. Informally, such an object can be thought of as an assemblage of one or more chunks of rubber (cells), each shaped like a convex polytope, which are glued to each other by their facets — possibly with much stretching and twisting.

The topological version of BCS replaces each cell by an assemblage of rubber simplices, likewise glued together by their facets and possibly deformed. The procedure is (1) select for each cell a deformation map that converts it into a geometric convex polytope, preserving its incidence and topological connections; (2) perform the geometric BCS on this polytope; and, then (3) map the resulting subdivision back to the original cells.

The result of barycentric subdivision, when viewed as an abstract simplicial complex, is an example of a flag complex. It has one vertex for every cell of the original cell complex and one maximal-dimensional cell for every flag (a collection of cells of different dimensions, all related to each other by inclusion) of the original cell complex.