Star Neighborhoods in Double Barycentric Subdivision
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This Demonstration illustrates skeleta of stars and the dual of the double barycentric subdivision of a 3-simplex.
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Contributed by: Aleksandr Berdnikov (August 2018)
Open content licensed under CC BY-NC-SA
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A significant number of topological spaces are (or can be modeled by) simplicial complexes—spaces that are "glued" out of simplexes, attaching them face to face. Given a subcomplex of a complex
, one would often like to consider a neighborhood
of
in
that approximates
fairly well. It is not always possible to build such a
out of simplexes of
; For example, if
, the only simplicial neighborhood of
is the whole segment
, which fails to capture the disconnectedness of
.
However, there are such neighborhoods that are simplicial in a double barycentric subdivision of
(i.e. each simplex is subdivided barycentrically twice). Given any subcomplex
of
, such a neighborhood is given by
in
, where
is defined as the union of all open simplexes with a vertex in
.
The Demonstration illustrates this concept with the 3-simplex as an example of . Each controller toggles the display of stars (in the double subdivision) of centers of corresponding elements. For example, selecting "vertices" and "edges" gives a neighborhood of a 1-skeleton of
(i.e. of the collection of its 1-simplexes).
Barycentric subdivision is defined inductively. Subdivision of a point is the point itself. To define a subdivision of a simplex
, take the subdivision
of its boundary
and take
as the collection of simplexes with base the simplex in
and vertex the barycenter of
.
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