Theory

Relative homology

Suppose a simplicial complex $\mathcal{K}$ is embedded within another simplicial complex $\mathcal{L}$. The homology of $\mathcal{L}$ relative to $\mathcal{K}$ is given by zeroing (quotienting out) the cycles fully within $\mathcal{K}$. Only cycles that inescapably involve $\mathcal{L}$ outside of $\mathcal{K}$ contribute to the relative homology.

More precisely, suppose that $\mathcal{K}\subset \mathcal{L}$, so vertices and simplices of $\mathcal{K}$ are also vertices and simplices of $\mathcal{L}$.

Then observe that chains in $\mathcal{K}$ are automatically chains in $\mathcal{L}$, and thus:

$$\mathcal{K}\subset \mathcal{L}⟹C_i(\mathcal{K})<C_i(\mathcal{L})\text{for all}i$$

(A subgroup: because a sum of chains in $\mathcal{K}$ is still a chain in $\mathcal{K}$.)

Relative chain complex

Suppose $\mathcal{K}\subset \mathcal{L}$ is an inclusion of simplicial complexes.

Define $C_i(\mathcal{L},\mathcal{K})$, the relative chain group, as the quotient:

$$C_i(\mathcal{L},\mathcal{K})=\frac{C_i(\mathcal{L})}{C_i(\mathcal{K})}$$

Proposition: Boundary maps descend to relative chain groups

The boundary maps $\partial :C_i(\mathcal{L})\to C_{i-1}(\mathcal{L})$ send chains in $\mathcal{K}$ to chains in $\mathcal{K}$ and thus descend to the quotient groups, giving a relative chain complex:

$$0\to C_n(\mathcal{L},\mathcal{K})\stackrel{{\partial }^{\mathcal{L},\mathcal{K}}}{⟶}\cdots \stackrel{{\partial }^{\mathcal{L},\mathcal{K}}}{⟶}{C}_0(\mathcal{L},\mathcal{K})\to 0$$

(Here ${\partial }^{\mathcal{L},\mathcal{K}}$ is just the map induced by $\partial$ on the quotient groups.)

Proof of Proposition: Boundary maps descend to relative chain groups

It is sufficient to establish that boundaries from $\mathcal{K}$ must also reside in $\mathcal{K}$, i.e. that:

$$\partial (C_i(\mathcal{K}))\subset C_{i-1}(\mathcal{K})$$

But this is clear because $\mathcal{K}$ is a simplicial complex, so it is closed under taking boundaries.

Relative homology

Suppose $\mathcal{K}\subset \mathcal{L}$ is an inclusion of simplicial complexes.

Define $H_k(\mathcal{L},\mathcal{K})$, the relative homology group, as the homology group of the relative chain complex:

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

Remark: Relativizing before homology

It is tempting to try relativizing the homology groups directly.

The inclusion $\mathcal{K}\hookrightarrow \mathcal{L}$ does induce maps $H_k(\mathcal{K})\to H_k(\mathcal{L})$. We could study the groups $H_k(\mathcal{L})/H_k(\mathcal{K})$.

These quotient groups are not what relative homology is after! See the illustrations.

Cochain complex

Recall that the chain group $C_k(\mathcal{K})$ is the free abelian group of formal linear combinations of $k$-simplices in $\mathcal{K}$:

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

One should think of the chains $\sum n_i{\sigma }_i$ as assignments of integers $n_i$ to the various $k$-simplices in $\mathcal{K}$.

Thinking in this way, the boundary map ${\partial }_k:C_i\to C_{i-1}$ is given on chains by simple linear extension of its definition for individual simplices. Thus ${\partial }_k\left(n_i{\sigma }_i\right)=n_i{\partial }_k({\sigma }_i)$.

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Cohomology is created by interpreting the integers $n_i$ not as coefficients, but as the outputs of linear functions $C_k\to \mathbb{Z}$.

To notate this effectively, define indicator functions ${\phi }_i$ for each of the $k$-simplices:

$$\begin{array}{c}{\phi }_i:\{\text{oriented k-simplices in K}\}\to \mathbb{Z}\\ \\ {\sigma }_i\mapsto 1,{\sigma }_j\mapsto 0\text{for}j\ne i\end{array}$$

Cochains

The space of linear maps $C_k\to \mathbb{Z}$ is isomorphic to the free abelian group of formal linear combinations of indicator functions:

$$C^k(\mathcal{K})=\left\{\sum _{i\in I}{m}_i{\phi }_i|m_i\in \mathbb{Z},{\phi }_i\text{indicator of}{\sigma }_i\right\}$$

Elements of this space, written generically as formal sums $m_1{\phi }_1+m_2{\phi }_2+\cdots$, are called cochains.

(For cochains it is not necessary that $I$ be finite.)

Since there is a bijective correspondence of sets between oriented simplices and their indicator functions, we see that $C^k(\mathcal{K})\cong C_k(\mathcal{K})$ as abelian groups; we have simply reinterpreted the chains as assignments of integers $m_i$ to the indicators of $k$-simplices in $\mathcal{K}$.

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In both cases, the data of a chain / cochain consists of the assignment of a number ($n_i$ or $m_i$) to each of the $k$-simplices in $\mathcal{K}$. But the difference in interpretation of this data allows a difference in the construction of boundary operators:

Coboundary operator

Define the coboundary operator thus:

$$\begin{array}{rl}{\delta }^k:C^k(\mathcal{K})& \to C^{k+1}(\mathcal{K})\\ & \\ {\phi }_i& \mapsto {\phi }_i\circ {\delta }^{k+1}(\text{extended linearly})\end{array}$$

In words: the coboundary of a simplex indicator indicates that simplex in the boundary of a higher simplex.

For example, in the triangular 2-simplex:

If ${\phi }_{[v_0,v_2]}$ is the indicator of $[v_0,v_2]$, then the coboundary ${\delta }^1\left({\phi }_{[v_0,v_2]}\right)$ indicates the presence of $[v_0,v_2]$ in the boundaries of 2-simplices. It returns $-1$ for the triangle itself, $[v_0,v_1,v_2]$, since the boundary of the triangle includes the term $-[v_0,v_2]$.

Thus, in general, ${\delta }^k\left(m_i{\phi }_i\right)=m_i({\phi }_i\circ {\partial }_k)$, in contrast to ${\partial }_k\left(n_i{\sigma }_i\right)=n_i{\partial }_k({\sigma }_i)$ that we saw for ${\partial }_k$.

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Witness a simple computation, applying ${\delta }^{k+1}\circ {\delta }^k$ to an arbitrary indicator:

$$\begin{array}{c}({\delta }^{k+1}\circ {\delta }^k)({\phi }_i)\gg \gg {\delta }^{k+1}({\delta }^k({\phi }_i))\\ \\ \gg \gg {\delta }^{k+1}({\phi }_i\circ {\partial }_k)\gg \gg {\phi }_i\circ {\partial }_k\circ {\partial }_{k+1}\gg \gg 0\end{array}$$

Cochain complex

Given a simplicial (or $\Delta$-) complex $\mathcal{K}$ of maximal dimension $n$, we define the cochain complex of $\mathcal{K}$ as the sequence of morphisms:

$$0⟵C_n\stackrel{{\delta }^{n-1}}{⟵}{C}_{n-1}\stackrel{{\delta }^{n-2}}{⟵}{C}_{n-2}⟵\cdots ⟵C_1\stackrel{{\delta }^0}{⟵}{C}_0⟵0$$

This sequence is a complex because:

$${\delta }^{k+1}\circ {\delta }^k=0\text{for every}k$$

The image and kernel groups are named in the obvious way:

• $B_k:=\mathrm{Im}({\delta }^{k-1})$ — called the coboundary cochains
• $Z_k:=\mathrm{Ker}({\delta }^k)$ — called the cocycles

Cohomology

Cohomology

The $k^{\text{th}}$ cohomology of $\mathcal{K}$ is defined as the quotient group of cocycles modulo coboundary chains:

$$H^k(\mathcal{K})=\frac{\mathrm{Ker}({\delta }^k)}{\mathrm{Im}({\delta }^{k-1})}$$

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Cohomology is an upgrade from homology because it has an additional product structure, so cohomology actually gives a collection of rings.

Cup product on cochains, for simplicial complexes

Suppose we are given a simplicial complex $\mathcal{K}$, together with a chosen ordering $O=(v_1,\dots ,v_N)$ of all vertices of $\mathcal{K}$.

There are maps of cochain groups $C^k\times C^{\ell }\to C^{k+\ell }$ defined as follows:

For any pair of cochains $\phi \in C^k$ and $\psi \in C^{\ell }$, and any vertices $w_0<\cdots <w_{k+\ell }$ ordered according to $O$, with $[w_0,\dots ,w_{k+\ell }]$ the $(k+\ell )$-simplex they form, we have:

$$(\phi ⌣\psi )([w_0,\dots ,w_{k+\ell }])=\phi ([w_0,\dots ,w_k])\cdot \psi ([w_k,\dots ,w_{k+\ell }])$$

Note: the oriented simplices here may be written with various orderings, but the splitting into $k$- and $\ell$-simplices must follow the order of $O$.

Cup product using indicators

Let $\phi \in C^k$ be the indicator of an oriented simplex $[u_{i_0},\dots ,u_{i_k}]$, and let $\psi \in C^{\ell }$ be the indicator of another oriented simplex $[v_{i_k},\dots ,v_{i_{k+\ell }}]$.

Let $(w_0,\dots ,w_{k+\ell })$ be the vertices of the total list

$$u_{i_0},\dots ,u_{i_k},v_{i_k},\dots ,v_{i_{k+\ell }}$$

reordered in the ordering of $O$ (allowing repetition), so $w_0\le w_1\le \cdots \le w_{k+\ell }$.

Then $\phi ⌣\psi$ is defined as follows: it is the indicator of $[w_0,\dots ,w_{k+\ell }]$ provided:

• (i) the $w_i$ are all distinct,
• (ii) $u_p\le v_q$ for all $p,q$;

and $\phi ⌣\psi =0$ if (i) and (ii) do not hold.

The definition is extended bilinearly to all of $C^k\times C^{\ell }$.

Vertex orderings and the cup product

Instead of choosing a total ordering $O$ of all vertices of $\mathcal{K}$, it is sufficient to choose a partial ordering of the vertices $v_1,\dots ,v_N$ of $\mathcal{K}$ having the property that $O$ induces a total ordering of each simplex of $\mathcal{K}$.

A set of compatible orderings of the vertices in each simplex is used to define a ‘front half’ and ‘back half’ of each $(k+\ell )$-simplex $\sigma$, thereby cutting $\sigma$ into one $k$-simplex and one $\ell$-simplex.

The cutting method applied to a given simplex must be compatible with the method applied to every other simplex. A global vertex ordering ensures this compatibility.

Cup product on cohomology

The cup product has the following relation to the coboundary operator, a type of signed Leibniz rule:

$$\delta (\phi ⌣\psi )=\delta \phi ⌣\psi +(-1{)}^k\phi ⌣\delta \psi$$

when $\phi \in C^k(\mathcal{K})$.

This formula implies that if $\phi$ and $\psi$ are cocycles (or coboundaries), then $\phi ⌣\psi$ is also a cocyle (coboundary). Therefore the cup product descends to cohomology:

$$H^k(\mathcal{K})\times H^{\ell }(\mathcal{K})\to H^{k+\ell }(\mathcal{K})$$

The cup product on cohomology provides a ring structure to the direct sum ${⨁}_i{H}^i(\mathcal{K})$. The product, and the ring structure, do not depend on the choice of ordering used to define the product on cochains.

(Note: The cup product is always zero when $k+\ell$ is greater than the maximal simplex dimension.)

Coefficients

It is possible to define homology and cohomology with other coefficients. Given an abelian group $G$, one defines the chain groups thus:

$$C_k(\mathcal{K},G)=\left\{\sum _{i\in I\text{finite}}{g}_i{\sigma }_i|g_i\in G,{\sigma }_i\in \mathcal{K},\dim ({\sigma }_i)=k\right\}$$

And the cochain groups thus:

$$C^k(\mathcal{K},G)=\left\{\sum _{i\in I}{g}_i{\phi }_i|g_i\in \mathbb{Z},{\phi }_i\text{indicator of}{\sigma }_i\in \mathcal{K},\dim ({\sigma }_i)=k\right\}$$

Homology and cohomology are defined in the usual way from these chain groups.

The most important cases are $G=\mathbb{R}$ (which eliminates torsion), and $G=\mathbb{Z}/2\mathbb{Z}$ (where coefficients track parity only).