barycentric subdivision

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.

Let $\backslash mathcal\{S\}\backslash subset\; \backslash mathbb\{R\}^n$ be a geometric simplicial complex. A complex $\backslash mathcal\{S\text{'}\}$ is said to be a subdivision of $\backslash mathcal\{S\}$ if

• each simplex of $\backslash mathcal\{S\text{'}\}$ is contained in a simplex of $\backslash mathcal\{S\}$
• each simplex of $\backslash mathcal\{S\}$ is a finite union of simplices of $\backslash mathcal\{S\text{'}\}$

These conditions imply that $\backslash mathcal\{S\}$ and $\backslash mathcal\{S\text{'}\}$ equal as sets and as topological spaces, only the simplicial structure changes.{{citation|surname1=James R. Munkres|title=Elements of algebraic topology|publication-place=Menlo Park, Calif.|at=p. 96|isbn=0-201-04586-9|language=German

}}

File:Barycentric subdivision of 2simplex.png

For a simplex $\backslash Delta$ spanned by points $p\_0,...,p\_n$, the barycenter is defined to be the point $b\_\{\backslash Delta\}=\; \backslash frac\{1\}\{n+1\}(p\_0+p\_1+...+$

p_n) . 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.

Suppose then for a simplex $\backslash Delta$ of dimension $n$ that its faces $\backslash Delta\; \_i$ of dimension $n-1$ are already divided. Therefore, there exist simplices $\backslash Delta\; \_\{i,1\},\; \backslash ;\; \backslash Delta\; \_\{i,2\}...,\; \backslash Delta\; \_\{i,\; n!\}$ covering $\backslash Delta\_i$. The barycentric subdivision is then defined to be the geometric simplicial complex whose maximal simplices of dimension $n$ are each a convex hulls of $\backslash Delta\_\{i,j\}\; \backslash cup\; b\_\{\backslash Delta\}$ for one pair $i,j$ for some $i\; \backslash in\; \{0,...,n\},\; \backslash ;\; j\backslash in\; \{1,...,n!\}$, so there will be $(n\; +\; 1)!$ simplices covering $\backslash Delta$.

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. The induction will then be based on the $n$-th skeleton of the simplicial complex. It allows effectuating the subdivision more than once.{{citation|surname1=James R. Munkres|title=Elements of algebraic topology|publication-place=Menlo Park, Calif.|at=pp. 85 f|isbn=0-201-04586-9|language=German

}}

{{see also|Schläfli orthoscheme#Characteristic simplex of the general regular polytope}}

File:Modell, Kristallform Hexakisoktaeders -Krantz 421-.jpg, the barycentric subdivision of a cube]]

The operation of barycentric subdivision can be applied to any convex polytope of any dimension, producing another convex polytope of the same dimension.{{citation

| last1 = Ewald | first1 = G.

| last2 = Shephard | first2 = G. C.

| doi = 10.1007/BF01344542

| journal = Mathematische Annalen

| mr = 350623

| pages = 7–16

| title = Stellar subdivisions of boundary complexes of convex polytopes

| volume = 210

| year = 1974}} 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.{{citation|title=Convex Polytopes and Tilings with Few Flag Orbits|type=Doctoral dissertation|last=Matteo|first=Nicholas|publisher=Northeastern University|year=2015|id={{ProQuest|1680014879}}}} See p. 22, where the omnitruncation is described as a "flag graph". 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.{{citation

| last1 = Langer | first1 = Joel C.

| last2 = Singer | first2 = David A.

| doi = 10.1007/s00032-010-0124-5

| issue = 2

| journal = Milan Journal of Mathematics

| mr = 2781856

| pages = 643–682

| title = Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem

| volume = 78

| year = 2010}} 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.

Let $\backslash Delta\; \backslash subset\; \backslash mathbb\{R\}^n$ a simplex and define $\backslash operatorname\{diam\}(\backslash Delta)\; =\; \backslash operatorname\{max\}\; \backslash Bigl\backslash \{\; \backslash |a-b\backslash |\_\{\backslash mathbb\{R\}^n\}\; \backslash ;\; \backslash Big|\backslash ;\; a,\; b\; \backslash in\; \backslash Delta\; \backslash Bigr\backslash \}$. One way to measure the mesh of a geometric, simplicial complex is to take the maximal diameter of the simplices contained in the complex. Let $\backslash Delta\text{'}$ be an $n$- dimensional simplex that comes from the covering of $\backslash Delta$ obtained by the barycentric subdivision. Then, the following estimation holds:

$\backslash operatorname\{diam\}(\backslash Delta\text{'})\backslash leq\; \backslash left(\backslash frac\{n\}\{n+1\}\backslash right)\backslash ;\; \backslash operatorname\{diam\}(\backslash Delta)$. Therefore, by applying barycentric subdivision sufficiently often, the largest edge can be made as small as desired.{{citation|title=Algebraic Topology|year=2001|url=https://pi.math.cornell.edu/~hatcher/AT/AT.pdf|last=Hatcher|first= Allen|page=120}}

For some statements in homology-theory one wishes to replace simplicial complexes by a subdivision. On the level of simplicial homology groups one requires a map from the homology-group of the original simplicial complex to the groups of the subdivided complex. Indeed it can be shown that for any subdivision $\backslash mathcal\{K\text{'}\}$ of a finite simplicial complex $\backslash mathcal\{K\}$ there is a unique sequence of maps between the homology groups $\backslash lambda\_n:\; C\_n(\backslash mathcal\{K\})\backslash rightarrow\; C\_n(\backslash mathcal\{K\text{'}\})$ such that for each $\backslash Delta$ in $\backslash mathcal\{K\}$ the maps fulfills $\backslash lambda(\backslash Delta)\backslash subset\; \backslash Delta$ 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 $X$ one considers continuous functions $\backslash sigma:\; \backslash Delta^n\; \backslash rightarrow\; X$ where $\backslash Delta^n$ denotes the $n$-dimensional-standard-simplex. In an analogous way as described for simplicial homology groups, barycentric subdivision can be interpreted as an endomorphism of singular chain complexes. Here again, there exists a subdivision operator $\backslash lambda\_n:\; C\_n(X)\backslash rightarrow\; C\_n(X)$ sending a chain $\backslash sigma:\; \backslash Delta\; \backslash rightarrow\; X$ to a linear combination $\backslash sum\; \backslash varepsilon\_\{B\_\{\backslash Delta\}\}\; \backslash sigma\backslash vert\_\{B\_\{\backslash Delta\}\}$ where the sum runs over all simplices $B\_\{\backslash Delta\}$ that appear in the covering of $\backslash Delta$ by barycentric subdivision, and $\backslash varepsilon\_\{B\_\{\backslash Delta\}\}\backslash in\; \backslash \{1,\; -1\backslash \}$ for all of such $B\_\{\backslash Delta\}$. This map also induces an endomorphism of chain complexes.{{sfnp|Hatcher|2001|pages=122 f}}

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.

Let $\backslash mathcal\{K\}$, $\backslash mathcal\{L\}$ be abstract simplicial complexes above sets $V\_K$, $V\_L$. A simplicial map is a function $f:V\_K\; \backslash rightarrow\; V\_L$ which maps each simplex in $\backslash mathcal\{K\}$ onto a simplex in $\backslash mathcal\{L\}$. By affin-linear extension on the simplices, $f$ 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 map $f:\backslash mathcal\{K\}\backslash rightarrow\; \backslash mathcal\{L\}$. A simplicial map $g:\backslash mathcal\{K\}\backslash rightarrow\; \backslash mathcal\{L\}$ is said to be a simplicial approximation of $f$ if and only if each $x\; \backslash in\; \backslash mathcal\{K\}$ is mapped by $g$ onto the support of $f(x)$ in $\backslash mathcal\{L\}$. If such an approximation exists, one can construct a homotopy $H$ transforming $f$ into $g$ by defining it on each simplex; there, it always exists, because simplices are contractible.

The simplicial approximation theorem guarantees for every continuous function $f:V\_K\; \backslash rightarrow\; V\_L$ the existence of a simplicial approximation at least after refinement of $\backslash mathcal\{K\}$, for instance by replacing $\backslash mathcal\{K\}$ by its iterated barycentric subdivision.{{citation|surname1=Ralph Stöcker, Heiner Zieschang|title=Algebraische Topologie|edition=2. überarbeitete|publisher=B.G. Teubner|publication-place=Stuttgart|at=p. 81|isbn=3-519-12226-X|language=German

}} 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. This data is computed as follows: Suppose that $X$ and $Y$ are topological spaces that admit finite triangulations. A continuous map $f:\; X\backslash rightarrow\; Y$ induces homomorphisms $f\_i:\; H\_i(X,K)\backslash rightarrow\; H\_i(Y,K)$ between its simplicial homology groups with coefficients in a field $K$. These are linear maps between $K$- vectorspaces, so their trace $tr\_i$ can be determined and their alternating sum

$L\_K(f)=\; \backslash sum\_i(-1)^itr\_i(f)\; \backslash in\; K$

is called the Lefschetz number of $f$. If $f\; =\; id$, this number is the Euler characteristic of $K$. The fixpoint theorem states that whenever $L\_K(f)\backslash neq\; 0$, $f$ 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.

Now, Brouwer's fixpoint theorem is a special case of this statement. Let $f:\backslash mathbb\{D\}^n\; \backslash rightarrow\; \backslash mathbb\{D\}^n$ is an endomorphism of the unit-ball. For $k\; \backslash geq\; 1$ all its homology groups $H\_k(\backslash mathbb\{D\}^n)$ vanish, and $f\_0$ is always the identity, so $L\_K(f)\; =\; tr\_0(f)\; =\; 1\; \backslash neq\; 0$, so $f$ has a fixpoint.{{citation|surname1=Bredon, Glen E.|editor-surname1= Springer Verlag|title=Topology and Geometry|publication-place=Berlin/ Heidelberg/ New York|at=pp. 254 f|isbn=3-540-97926-3|language=German

}}

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:

Let $X\; =\; A\; \backslash cup\; B$ an open cover of the topological space $X$ .

There is an exact sequence

: $\backslash cdots\backslash to\; H\_\{n+1\}(X)\backslash ,\backslash xrightarrow\{\backslash partial\_*\}\backslash ,H\_\{n\}(A\backslash cap\; B)\backslash ,\backslash xrightarrow\{(i\_*,j\_*)\}\backslash ,H\_\{n\}(A)\backslash oplus\; H\_\{n\}(B)\; \backslash ,\; \backslash xrightarrow\{k\_*\; -\; l\_*\}\backslash ,\; H\_\{n\}(X)\backslash ,\; \backslash xrightarrow\{\backslash partial\_*\}\backslash ,\; H\_\{n-1\}\; (A\backslash cap\; B)\backslash to\; \backslash cdots$

: $$

\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad

\cdots \to H_0(A)\oplus H_0(B)\,\xrightarrow{k_* - l_*}\,H_0(X)\to 0.

where we consider singular homology groups, $i:\; A\; \backslash cap\; B\; \backslash hookrightarrow\; A,\; \backslash ;\; j:\; A\; \backslash cap\; B\; \backslash hookrightarrow\; B,\; \backslash ;\; k:\; A\; \backslash hookrightarrow\; X,\; \backslash ;\; l:\; B\; \backslash hookrightarrow\; X$ are embeddings and $\backslash oplus$ denotes the direct sum of abelian groups.

For the construction of singular homology groups one considers continuous maps defined on the standard simplex $\backslash sigma:\; \backslash Delta\; \backslash rightarrow\; X$. An obstacle in the proof of the theorem are maps $\backslash sigma$ such that their image is nor contained in $A$ neither in $B$. This can be fixed using the subdivision operator: By considering the images of such maps as the sum of images of smaller simplices, lying in $A$ or $B$ one can show that the inclusion $C\_n(A)\backslash oplus\; C\_n(B)\backslash hookrightarrow\; C\_n(X)$ induces an isomorphism on homology which is needed to compare the homology groups.{{sfnp|Hatcher|2001|page=149}}

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:

Let $X$ be a topological space and let $Z\; \backslash subset\; A\; \backslash subset\; X$ be subsets, where $Z$ is closed such that $Z\; \backslash subset\; A^\{\backslash circ\}$. Then the inclusion $i:(X\; \backslash setminus\; Z,\; A\; \backslash setminus\; Z)\; \backslash hookrightarrow\; (X,\; A)$ induces an isomorphism $H\_k(X\; \backslash setminus\; Z,\; A\; \backslash setminus\; Z)\; \backslash rightarrow\; H\_k(X,A)$ for all $k\; \backslash geq\; 0.$

Again, in singular homology, maps $\backslash sigma:\; \backslash Delta\; \backslash rightarrow\; X$ 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.{{sfnp|Hatcher|2001|page=119}}

{{Reflist}}

{{DEFAULTSORT:Barycentric Subdivision}}

Category:Algebraic topology

Category:Geometric topology

Category:Triangulation (geometry)

Category:Simplicial homology