4 Def 1: Boundary and Coboundary Maps
Let \(G = (V, E)\) be a finite simple graph. We define \(\mathbb {Z}_2\)-linear maps that form the chain and cochain maps of the graph viewed as a cell complex over \(\mathbb {F}_2\).
The boundary map \(\partial : \mathbb {Z}_2^E \to \mathbb {Z}_2^V\) is the \(\mathbb {Z}_2\)-linear map defined as follows. For \(\gamma \in \mathbb {Z}_2^E\) and a vertex \(v \in V\),
The coboundary map \(\delta : \mathbb {Z}_2^V \to \mathbb {Z}_2^E\) is the \(\mathbb {Z}_2\)-linear map defined as follows. For \(f \in \mathbb {Z}_2^V\) and an edge \(e = \{ a, b\} \in E\),
The coboundary map \(\delta \) is the transpose of the boundary map \(\partial \) with respect to the standard \(\mathbb {Z}_2\) inner product. That is, for all \(f \in \mathbb {Z}_2^V\) and \(\gamma \in \mathbb {Z}_2^E\),
We unfold the definitions of \(\delta \) and \(\partial \). On the left-hand side, we distribute the multiplication into the sum, obtaining
On the right-hand side, we similarly distribute:
We exchange the order of summation on the right-hand side so that both sides sum first over edges \(e\) and then over vertices. It suffices to show equality for each edge \(e = \{ a, b\} \). By the definition of \(\delta \), we have \((\delta \, f)_e = f(a) + f(b)\). Since \(G\) is loopless, \(a \neq b\). We expand \((\delta \, f)_e \cdot \gamma _e = f(a)\, \gamma _e + f(b)\, \gamma _e\). On the other side, for vertex \(x\), the indicator \([x \in \{ a,b\} ]\) is \(1\) if \(x = a\) or \(x = b\) and \(0\) otherwise. Therefore
splits as a sum of two terms: \(f(a)\, \gamma _e\) (from \(x = a\)) and \(f(b)\, \gamma _e\) (from \(x = b\)), with all other terms vanishing. By splitting the sum using \(\sum _x (\cdots ) = \sum _x (\text{if } x=a \text{ then } \cdots \text{ else } 0) + \sum _x (\text{if } x=b \text{ then } \cdots \text{ else } 0)\) and evaluating via \(\sum _x [x = a] \cdot (\cdots ) = (\cdots )|_{x=a}\), we obtain equality by ring arithmetic.
Let \(C\) be a finite type of “cycles” (or plaquettes), and let each \(c \in C\) be associated with a set of edges \(\operatorname {cycles}(c) \subseteq E\). The second boundary map \(\partial _2 : \mathbb {Z}_2^C \to \mathbb {Z}_2^E\) is the \(\mathbb {Z}_2\)-linear map defined by: for \(\sigma \in \mathbb {Z}_2^C\) and an edge \(e \in E\),
The second coboundary map \(\delta _2 : \mathbb {Z}_2^E \to \mathbb {Z}_2^C\) is the \(\mathbb {Z}_2\)-linear map defined by: for \(\gamma \in \mathbb {Z}_2^E\) and a cycle \(c \in C\),
The second coboundary map \(\delta _2\) is the transpose of the second boundary map \(\partial _2\). That is, for all \(\gamma \in \mathbb {Z}_2^E\) and \(\sigma \in \mathbb {Z}_2^C\),
We unfold the definitions of \(\delta _2\) and \(\partial _2\). On the left-hand side, we distribute the multiplication of \(\sigma _c\) into the inner sum, obtaining
On the right-hand side, we distribute \(\gamma _e\) into the inner sum:
We exchange the order of summation on the left-hand side. It then suffices to show that for each pair \((e, c)\), the summands agree. If \(e \in \operatorname {cycles}(c)\), both summands equal \(\gamma _e \cdot \sigma _c\) by ring arithmetic. If \(e \notin \operatorname {cycles}(c)\), both summands are \(0\).
For a single edge \(e \in E\) and a vertex \(v \in V\), the boundary of the indicator function \(\mathbf{1}_e\) satisfies
By the definition of the boundary map, \((\partial \, \mathbf{1}_e)(v) = \sum _{e'} [v \in e']\, (\mathbf{1}_e)_{e'}\). We rewrite each summand: if \(v \in e'\), then \((\mathbf{1}_e)_{e'} = [e' = e]\) by the definition of the indicator function; if \(v \notin e'\), the summand is \(0\). Thus the sum reduces to \(\sum _{e'} [e' = e] \cdot [v \in e'] = [v \in e]\), as desired.
For a vertex \(v \in V\) and an edge \(e = \{ a, b\} \in E\), the coboundary of the indicator function \(\mathbf{1}_v\) satisfies
By the definition of the coboundary map, \((\delta \, \mathbf{1}_v)(e) = \mathbf{1}_v(a) + \mathbf{1}_v(b)\) where \(e = \{ a, b\} \). Since \(G\) is loopless, \(a \neq b\). We consider three cases. If \(v = a\), then \(\mathbf{1}_v(a) = 1\) and \(\mathbf{1}_v(b) = 0\) (since \(a \neq b\)), so the result is \(1\), and indeed \(v \in \{ a, b\} \). If \(v = b\), then \(\mathbf{1}_v(a) = 0\) and \(\mathbf{1}_v(b) = 1\), so the result is \(1\), and indeed \(v \in \{ a, b\} \). If \(v \neq a\) and \(v \neq b\), then \(v \notin \{ a, b\} \), and \(\mathbf{1}_v(a) = \mathbf{1}_v(b) = 0\), so the result is \(0\).
For a cycle \(c \in C\) and an edge \(e \in E\), the second boundary map applied to the indicator \(\mathbf{1}_c\) satisfies
By the definition of the second boundary map, \((\partial _2\, \mathbf{1}_c)(e) = \sum _{c'} [e \in \operatorname {cycles}(c')]\, (\mathbf{1}_c)_{c'}\). We rewrite each summand: if \(e \in \operatorname {cycles}(c')\), then \((\mathbf{1}_c)_{c'} = [c' = c]\); otherwise the summand is \(0\). The sum thus reduces to \([e \in \operatorname {cycles}(c)]\).
For an edge \(e \in E\) and a cycle \(c \in C\), the second coboundary map applied to the indicator \(\mathbf{1}_e\) satisfies
By the definition of the second coboundary map, \((\delta _2\, \mathbf{1}_e)(c) = \sum _{e'} [e' \in \operatorname {cycles}(c)]\, (\mathbf{1}_e)_{e'}\). We rewrite each summand: if \(e' \in \operatorname {cycles}(c)\), then \((\mathbf{1}_e)_{e'} = [e' = e]\); otherwise the summand is \(0\). The sum reduces to \([e \in \operatorname {cycles}(c)]\).