Theory

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.)

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.