Theory

Introduction

A simplicial complex is an assembly of simplices, which generalize triangles and tetrahedra. Simplices provide “building block” objects in each dimension that may be glued together to form a complex.

Simplices provide a combinatorial method of representing a large class of topological spaces, since the ‘assembly data’ encoding the gluing of simplices into a complex is combinatorial in nature.

Simplicial homology is a homology theory (topological invariant) that is naturally defined in terms of a given simplicial complex. It is easy to find this homology for a space when a representation as simplicial complex is known.

Simplicial homology is isomorphic to other homology theories.

• Singular homology: abstract ‘naturality’; proving invariance is easier
• Simplicial homology: combinatorial; build complexes by gluing faces ‘directly’; stricter gluing rules
• Delta homology: combinatorial, like simplicial; more relaxed gluing rules
• CW homology: combinatorial, like simplicial but gluing cells (balls) not simplices; build CW complexes inductively; very relaxed gluing rules

For smooth manifolds, we also have:

• De Rham cohomology: uses differential forms “closed but not exact”

Simplices and complexes

$n=0$:

• A $0$-simplex is a point (a vertex).

$n=1$:

• A $1$-simplex is a line segment.

$n=2$:

• A $2$-simplex is a triangle (interior included).

$n=3$:

• A $3$-simplex is a tetrahedron.

Geometric simplex

Vertices: $v_0,v_1,\dots ,v_n\in {\mathbb{R}}^{n+1}$, required to be affinely independent.

Geometric simplex:

$$\begin{array}{c}\sigma =[v_0,v_1,\dots ,v_n]\\ \\ =\{x\in {\mathbb{R}}^{n+1},x=\sum t_i{v}_i|\forall i,t_i\ge 0\text{and}\sum t_i=1\}\end{array}$$

• Thus $\sigma$ is the “convex hull” of $v_0,\dots ,v_n$. In other words: all weighted averages of the vertices.
• Affinely independent means: $v_0,\dots ,v_n$ do not lie in any hyperplane (dimension $n$) of ${\mathbb{R}}^{n+1}$; equivalently that $v_1-v_0,\dots ,v_n-v_0$ are linearly independent, or that they generate a vector space of dimension $n$.
• A simplex failing this criterion is called degenerate.

• Standard $n$-simplex ${\sigma }_n$: vertices are the unit vectors in ${\mathbb{R}}^{n+1}$.

For example, with $n=1$ the standard simplex is:

$${\sigma }_1=\{\vec{v}\in {\mathbb{R}}^2|\vec{v}=t_0{\mathbf{e}}_x+(1-t_0){\mathbf{e}}_y,0\le t_0\le 1\}$$

This defines the line segment from $(1,0)$ to $(0,1)$.

Similarly, with $n=2$ we find that ${\sigma }_2$ is parametrized by $\vec{v}=t_0{\mathbf{e}}_x+t_1{\mathbf{e}}_y+(1-t_0-t_1){\mathbf{e}}_z$.

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A simplicial complex can be characterized in several ways:

• Combinatorially
• Geometrically
• Topologically

Simplicial complex - Combinatorial

A combinatorial simplical complex $K$ with a set of ‘vertices’ $\mathcal{V}=\{v_0,v_1,\dots ,v_N\}$ is a collection of subsets of $\mathcal{V}$, called ‘simplices’, satisfying:

• All faces included: If $\sigma \in K$, then all faces of $\sigma$ are in $K$.
  (And thus faces of faces, all the way down to vertices.)
• Touching only at a face: If $\sigma ,\tau \in K$, and $\sigma \cap \tau =\varnothing$, then $\sigma \cap \tau$ is a face of $\sigma$ and a face of $\tau$.

For $\sigma =\{w_1,\dots ,w_n\}\subset \mathcal{V}$, the faces of $\sigma$ are the subsets of $\sigma$ lacking a single vertex of $\sigma$.

A simplicial complex has a topology. This is given by gluing the disjoint union of constituent simplices along the faces that intersect. (The quotient topology.)

If the simplicial complex is given a geometric representation in ${\mathbb{R}}^n$, then the glued/quotient topology should match the one inherited by the complex as a subspace of ${\mathbb{R}}^n$.

Illustration - faces, subcomplexes and intersections

For example, given a simplex $[v_0,v_1,v_2]$, the faces are $[v_0,v_1]$ and $[v_1,v_2]$ and $[v_0,v_2]$:

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A simplicial complex $\mathcal{K}$ with $3$D subcomplex ${\sigma }^3$ and $2$D subcomplex ${\tau }^2$ and $1$D subcomplex ${\gamma }^1$.

Simplices of dimension $0$ are called vertices, simplices of dimension $1$ are called edges, and simplices of dimension $n-1$ in an $n$D complex are called faces.

The simplices of top dimension are sometimes called facets (i.e. the biggest ones are called ‘little faces’).

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The intersection of two sub-simplices is allowed to be a single face, not two faces (vertices, in this case):

Illustration - graphs and geometric representation

Every finite simple undirected graph is a simplex.

• No “parallel edges”
• No “one-loops”

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In a geometric representation of a simplicial complex, edges should not cross, etc.

Kuratawki: Graph planarity

The graph planarity theorem of Kuratowski says that a graph is planar (can be drawn in ${\mathbb{R}}^2$ without self intersection) if and only if it does not contain a copy of $K_5$ or $K_{3,3}$:

Subdivision and triangulation

A single topological space (up to homeomorphism) can be represented in many different ways as a simplicial complex:

One can perform “barycentric subdivision” upon a given simplicial complex. This produces a simplicial complex that is homeomorphic to the original one, but the edges are shorter and areas of faces smaller.

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A triangulation is a simplicial complex that is homeomorphic to a given smooth manifold. A triangulated manifold is a simplicial complex together with a homeomorphism to a given smooth manifold.

For example, the tetrahedron obtained by gluing four copies of the standard $2$-simplex ${\sigma }_2$ is homeomorphic to ${\mathbb{S}}^2$ and is therefore a triangulation of ${\mathbb{S}}^2$.

Triangulations of spheres

For ${\mathbb{S}}^2$, all possible triangulation types have been found by Tutte.

For ${\mathbb{S}}^3$, there is not a known enumeration of triangulation types.