Chapter 02 - Simplicial Complexes and Homology

Simplicial complexes

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”

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Theory - Simplices and complexes

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 ${\mathbb{R}}^n$.

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Illustration

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):

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Torus as simplicial complex

Torus as simplicial complex

It is easy to create a torus by gluing the edges of a square:

But it is not this easy to represent a torus as a simplicial complex. Initial attempts fail:

One can do it with a little more work and complexity:

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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 $K_{3,3}$:

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Theory - Subdivision and triangulation

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.

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Simplicial homology

Introduction

Simplicial homology can be defined with “coefficients in $\mathbb{Z}$” or with “coefficients in $\mathbb{R}$” (or another ring). Using $\mathbb{Z}$ gives abelian groups, while using $\mathbb{R}$ gives vector spaces. Vector spaces are nicer to work with, but some information is lost. (Generators of finite order are lost, for example.)

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Theory - Boundaries, chain complexes

Boundaries, chain complexes

Fix a simplicial complex $\mathcal{K}$ which is $n$-dimensional. (Highest dimension of a simplex in $\mathcal{K}$ is $n$.)

Chain groups

For each $k\in \mathbb{N}$, $k\le n$, define the $k^{\text{th}}$ chain group $C_k(\mathcal{K})$:

$$C_k(\mathcal{K})=\begin{array}{c}\text{free abelian group}\\ \text{generated by}\\ \text{k-simplices in K}\end{array}$$

In other words:

$$C_k=\left\{\sum _{i\in I\text{finite}}{n}_i{\sigma }_i|n_i\in \mathbb{Z},{\sigma }_i\in \mathcal{K},\dim ({\sigma }_i)=k\right\}$$

An element of the $k$-chain group, written generically as $n_1{\sigma }_1+\cdots +n_{\ell }{\sigma }_{\ell }$, is called a $k$-chain.

• Sometimes $\mathcal{K}$ is dropped and we write just $C_k$ when the complex is clear from context.

By allowing combinations of simplices with coefficients, we can define a boundary operator with alternating signs. This alternation leads to cancellation when boundaries of boundaries are taken. The cancellation in turn enables the definition of homology, which is generated by cycles of simplices that are not boundaries.

Boundary operator

The boundary operator ${\partial }_k$ maps simplices to their boundaries. It is defined on simplices and extended linearly to chains:

$$\begin{array}{rl}{\partial }_k:C_k& ⟶C_{k-1}\\ {\partial }_k:[v_0,\dots ,v_k]& ⟼\sum _{i=0}^n(-1{)}^i[v_0,\stackrel{\begin{array}{c}{i}^{\text{th}}\text{ omitted}\\ \end{array}}{\dots ,\widehat{v_i},\dots ,}{v}_k]\end{array}$$

Examples:

• A $1$-simplex:

$$\partial ([v_0,v_1])\gg \gg [\widehat{v_0},v_1]-[v_0,\widehat{v_1}]\gg \gg v_1-v_0$$

• A $2$-simplex:

$$\partial ([v_0,v_1,v_2])\gg \gg [v_1,v_2]-[v_0,v_2]+[v_0,v_1]$$

Orientation and vertex ordering

A simplex was initially defined as the convex closure of a set of points. The points were not ordered! The simplex was not ordered!

The boundary map was defined for an ordered list of points (“$i^{\text{th}}$ omitted”). These definitions are not compatible!

Let us upgrade both definitions to employ oriented sets of points.

Hereafter, the notation $[v_0,\dots ,v_n]$ will indicate an ordered list of vertex points up to the equivalence relation that identifies lists which are related by an even permutation.

For example: $[v_0,v_1,v_2]=[v_1,v_2,v_0]$, but: $[v_1,v_0,v_2]\ne [v_0,v_1,v_2]$.

• With this convention, the boundary map is well-defined.
• Notice that the orientation of the vertex list determines the sign of its boundary:

$$\begin{array}{rl}\partial ([v_0,v_1])& \gg \gg v_0-v_1\\ & \\ \text{but:}\partial ([v_1,v_0])& \gg \gg v_1-v_0\end{array}$$

• Therefore: one may interpret the sign of a coefficient in $C_k$ as indicating the orientation of the simplex it is attached to.
• Thus the boundary of a 2-simplex (filled triangle) is an oriented path around the triangle:

$$\begin{array}{rl}\partial ([v_0,v_1,v_2])& \gg \gg [v_1,v_2]-[v_0,v_2]+[v_0,v_1]\\ & \\ & \gg \gg [v_0,v_1]+[v_1,v_2]+[v_2,v_0]\end{array}$$

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LEMMA: Boundary of boundary is empty

For all $k\in \mathbb{N}$, we have:

$${\partial }_{k-1}\circ {\partial }_k=0$$

Proof of LEMMA

Apply the double boundary to a generator of $C_k$:

$$\begin{array}{c}{C}_k\ni [v_0,\dots ,v_k]\stackrel{{\partial }_k}{⟼}\sum _{i=0}^k(-1{)}^i[v_0,\stackrel{\begin{array}{c}{i}^{\text{th}}\text{ omitted}\\ \end{array}}{\dots ,\widehat{v_i},\dots ,}{v}_k]\\ \\ \stackrel{{\partial }_{k-1}}{⟼}\sum _{i=0}^k\sum _{j<i}(-1{)}^i(-1{)}^j[v_0,\dots ,\widehat{v_j},\dots ,\widehat{v_i},\dots ,v_k]+\\ \\ +\sum _{i=0}^k\sum _{j>i}(-1{)}^i(-1{)}^{j-1}[v_0,\dots ,\widehat{v_i},\dots ,\widehat{v_j},\dots ,v_k]\\ \\ =0\end{array}$$

The second summation duplicates the terms of the first summation, but has one fewer power of $-1$, so the whole sum vanishes. Both summations cover all $(k-2)$-simplices omitting some two vertices.

Using this fact we construct the chain complex of group homomorphisms:

Chain complex

Given a simplicial complex $\mathcal{K}$ of maximal dimension $n$, we define the chain complex of $\mathcal{K}$ as the sequence of morphisms:

$$0\to C_n\stackrel{{\partial }_n}{⟶}{C}_{n-1}\stackrel{{\partial }_{n-1}}{⟶}{C}_{n-2}⟶\cdots ⟶C_1\stackrel{{\partial }_1}{⟶}{C}_0⟶0$$

This sequence is a complex because the Lemma ensures that

$$\mathrm{Im}({\partial }_k)\subset \mathrm{Ker}({\partial }_{k-1})\text{for every}k.$$

The image and kernel groups have names:

• $B_k:=\mathrm{Im}({\partial }_{k+1})$ — called the boundary chains
• $Z_k:=\mathrm{Ker}({\partial }_k)$ — called the cycles

Homology

The $k^{\text{th}}$ homology of $\mathcal{K}$ is defined as the quotient group of cycles modulo boundary chains:

$$H_k(\mathcal{K})=\frac{\mathrm{Ker}({\partial }_k)}{\mathrm{Im}({\partial }_{k+1})}$$

Notice that $k$ in $H_k$ appears in the numerator. So $H_k$ is generated by chains of $k$-simplices.

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Illustration

Homology of a triangle

Homology of a triangle

Consider the 1D simplicial complex consisting of a triangle:

The chain complex is:

$$\begin{array}{rl}0& ⟶C_1⟶C_0⟶0\\ & \\ 0& ⟶{\mathbb{Z}}^3\stackrel{{\partial }_1}{⟶}{\mathbb{Z}}^3⟶0\end{array}$$

The group $C_1$ is generated by the 1-simplices:

$$C_1=⟨[v_0,v_1],[v_1,v_2],[v_0,v_2]⟩$$

The group $C_0$ is generated by the 0-simplices:

$$C_0=⟨v_0,v_1,v_2⟩$$

The boundary operator:

$$\begin{array}{rl}[v_0,v_1]& \stackrel{{\partial }_1}{⟶}{v}_1-v_0\\ & \\ [v_1,v_2]& \stackrel{{\partial }_1}{⟶}{v}_2-v_1\\ & \\ [v_0,v_2]& \stackrel{{\partial }_1}{⟶}{v}_2-v_0\end{array}$$

The kernel of ${\partial }_1$ is freely generated by $[v_0,v_1]+[v_1,v_2]-[v_0,v_2]$. The image of ${\partial }_1$ is isomorphic to $\mathbb{Z}$ via the map determined by $v_0\mapsto 1$.

So we have $H_1\cong \mathbb{Z}$ and $H_0\cong \mathbb{Z}$. Higher homology groups vanish.

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Homology counts holes by dimension

Homology counts holes by dimension

Consider the space $X$, the wedge sum of three copies of ${\mathbb{S}}^1$ and one copy of ${\mathbb{S}}^2$ obtained by gluing all spaces at a single point:

The homology of this space is:

$$H^i(X)=\{\begin{array}{ll}0& i>3\\ {\mathbb{Z}}^2& i=2\\ {\mathbb{Z}}^3& i=1\\ \mathbb{Z}& i=0\end{array}$$

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Homology independent loops in a graph

Homology independent loops in a graph

The “nullity” of a connected graph is the number of independent cycles in the graph. If $H^1(\Gamma )={\mathbb{Z}}^c$ for $\Gamma$ a graph considered as a simplicial complex, then $c$ is the nullity.

So $H^1(\Gamma )=6$ for $\Gamma$ the following graph:

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