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barycentric subdivision (Definition)

Recall that an abstract $ n$-simplex is an abstract simplicial complex $ K$ such that $ \bigcup K\in K$ and the cardinality of $ \bigcup K$ is $ n+1$. It can be identified as the powerset (minus the empty set element) of a set $ V_K$ of elements $ v_1,\ldots, v_n$, called the vertices of $ K$.

The barycentric subdivision of an abstract simplex $ K$ is the construction of a certain abstract simplicial complex $ K'$ from $ K$. $ K'$ itself is called the barycentric subdivision of $ K$. Before giving the general construction, let us describe some simple cases, specifically, when $ n=1,2,$ and $ 3$ and when $ V_K$ is embedded in some ambient Euclidean space $ \mathbb{R}^m$ where $ m\ge n$:

  • When $ n=1$, $ V_K$ is just a one-point space. We define the barycentric subdivision of $ K$ to be $ K$ itself: $ K'=K$.
  • When $ n=2$, $ V_K$ consists of two distinct points $ v_1$ and $ v_2$. Take the midpoint $ w$ of $ v_1$ and $ v_2$. Then the barycentric subdivision $ K'$ is defined to be the union of powersets of $ \lbrace v_1, w\rbrace$ and $ \lbrace v_2, w\rbrace$. Abstracting this process, $ K'$ can be thought of as the union of all the $ 2$-simplices having $ w$ as a vertex.
    bary.eps
  • When $ n=3$, $ V_K$ consists of three distinct non-collinear points, $ v_1,v_2,v_3$. We may picture them as the vertices of a triangle. From each of these vertices, draw a line connecting the vertex to the midpoint of the opposite side. This construction creates six smaller triangles, such that each pair either intersect at a common vertex or a common side. The barycentric subdivision $ K'$ is defined as the union of each of these smaller triangles considered as a simplex. In other words, $ K'$ is the union of the powerset of the three-point sets that correspond to these smaller triangles.
    bary2.eps
    Abstracting this process, we may take $ w$ as the barycenter of $ v_1,v_2,$ and $ v_3$. Label $ w=w(123)$, so
    $\displaystyle w(123)=\frac{1}{3}(v_1+v_2+v_3).$
    Next, take $ w(12),w(23),$ and $ w(13)$ as the midpoints of $ (v_1,v_2)$, $ (v_2,v_3)$ and $ (v_1,v_3)$. Finally, relabel $ v_1,v_2,v_3$ as $ w(1),w(2),w(3)$. We form six three-point sets:
    $\displaystyle W(32):=\lbrace w(123), w(12), w(1) \rbrace,\quad W(31):=\lbrace w(123), w(12), w(2) \rbrace,$
    $\displaystyle W(23):=\lbrace w(123), w(13), w(1) \rbrace,\quad W(21):=\lbrace w(123), w(13), w(3) \rbrace,$
    $\displaystyle W(13):=\lbrace w(123), w(23), w(2) \rbrace,\quad W(12):=\lbrace w(123), w(23), w(3) \rbrace.$
    The formation of any one of these sets goes as follows:
    1. it must contain the barycenter $ w(123)$;
    2. from $ w(123)$, pick one of the midpoints $ w(12),w(23),w(13)$ to be the next point in the set;
    3. once the midpoint is chosen, pick one of the initial vertices so that the label occurs in the labeling of the midpoint. For example, if $ w(23)$ were picked, only one of vertices $ w(2)$ or $ w(3)$ is allowed to be picked next.
    The logic behind the labeling $ ab$ of $ W(ab)$ comes from the fact that the labeling of the midpoint in $ W(ab)$ does not contain $ a$ and the labeling of the initial vertex in $ W(ab)$ does not contain $ b$.

    Finally, we form $ K'$ as the union of the powersets of $ W(ab)$'s.

In the last example, one can abstract the construction one step further. Since each labelled point is the barycenter of at least one of the initial vertices $ v_i$, we can uniquely identify any non-empty subset $ V$ of $ V_K$ with the labelled point that is the barycenter of the point(s) in $ V$. Then each $ W(ab)$ above can be identified as a maximal chain (ordered by inclusion) in $ K$ with $ \varnothing$ deleted.

This suggests the general construction of the barycentric subdivision of an abstract $ n$-simplex.

Definition. Let $ K$ be an abstract $ n$-simplex. Order $ K$ by inclusion $ \subseteq$. Let

$\displaystyle \mathcal{C}_K:=\big\lbrace P(C) \mid C$ is a maximal chain in $\displaystyle K\big\rbrace.$
The barycentric subdivision $ K'$ of $ K$ is:
$\displaystyle K'=\bigcup \mathcal{C}_K - \lbrace \varnothing\rbrace.$

It is easy to see that every maximal chain in $ K$ is an $ (n-1)$-simplex whose powerset is an $ n$-simplex (so isomorphic to $ K$). In addition, the barycentric subdivision $ K'$ of $ K$ is a simplicial complex with $ n!$ maximal simplices, each of which is isomorphic to $ K$.

Remark. This definition can be generalized to include the barycentric subdivision of an abstract simplicial complex. If $ K$ is an abstract simplicial complex, then the barycentric subdivision $ K'$ of $ K$ is the union of the barycentric subdivisions of the individual maximal simplicies in $ K$. Below are two examples:

  • In this example (pictured above), the maximal simplices of $ K$ consist of a triangle, and two line segments.

    \begin{pspicture} % latex2html id marker 142 (0,-1)(8,3) \pspolygon[fillstyle=so... ...psdots(8,1.5) \psdots(6.5,3) \psdots(6,1) \rput[b](6.5,-1){$K'$} \end{pspicture}
  • Here (pictured below), the maximal simplices are two triangles meeting at a common edge.

    \begin{pspicture} % latex2html id marker 150 (0,-1)(8,3) \pspolygon[fillstyle=so... ...) \psline(5,3)(8,1.5) \psline(8,0)(6.5,3) \rput[b](6.5,-1){$K'$} \end{pspicture}
In both examples, the vertex sets of the original simplicial complexes are the same.

It can be shown that the barycentric subdivision $ K'$ of an abstract simplicial complex $ K$ can be constructed as follows: $ V_{K'}:=\lbrace S\mid S\in K\rbrace$ is the set of vertices of $ K'$, and $ T\in K'$ iff $ T$ is a chain of simplexes in $ K$: $ T=\lbrace S_1,\ldots,S_{n(T)}\rbrace$, $ S_i\subset S_j$ for $ i<j$.



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See Also: centre of mass of polygon
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Cross-references: iff, edge, line segments, simplicial complex, addition, isomorphic, easy to see, order, inclusion, chain, subset, logic, labeling, occur ins, contain, label, barycenter, side, intersect, opposite side, line, triangle, union, midpoint, points, Euclidean space, simple, vertices, empty set, powerset, cardinality, abstract simplicial complex
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This is version 17 of barycentric subdivision, born on 2007-03-15, modified 2007-05-29.
Object id is 9080, canonical name is BarycentricSubdivision.
Accessed 1162 times total.

Classification:
AMS MSC: 55U10 (Algebraic topology :: Applied homological algebra and category theory :: Simplicial sets and complexes)
 55U05 (Algebraic topology :: Applied homological algebra and category theory :: Abstract complexes)
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