This technical section may be skipped in a first reading. It shows that simplicial (co)homology may be defined using larger sets of (co)chains, based on ordered simplexes. This will be used for comparisons between simplicial and singular (co)homology (see Open image in new window 17) and to define the cup and cap products in Chap. 4.

Let

\(K\) be a simplicial complex. Define

$$ \hat{\mathcal S}_m(K) = \{(v_0,\dots ,v_m)\in V(K)^{m+1} \mid \{v_0,\dots ,v_m\}\in {\mathcal S}(K) \} \, . $$

Observe that

\(\dim \{v_0,\dots ,v_m\}\le m\) and may be strictly smaller if there are repetitions amongst the

\(v_i\)’s. An element of

\(\hat{\mathcal S}_m(K)\) is an

*ordered* \(m\) *-simplex*
of

\(K\).

The definitions of ordered (co)chains and (co)homology are the same those for the simplicial case (see Sect. 2.2), replacing the simplexes by the ordered simplexes. We thus set

The set of ordered \(m\)-cochains of \(K\) is denoted by \(\hat{C}^m(K)\) and that of ordered \(m\)-chains by \(\hat{C}_m(K)\). As in Sect. 2.2, Definition 2.10.1 are equivalent to

Definition 2.10.2 endow \(\hat{C}^m(K)\) and \(\hat{C}_m(K)\) with a structure of a \({\mathbb Z}_2\)-vector space. The singletons provide a basis of \(\hat{C}_m(K)\), in bijection with \(\hat{\mathcal S}_m(K)\). Thus, Definition 2.10.2.b is equivalent to

We consider the graded

\({\mathbb Z}_2\)-vector spaces

\(\hat{C}_*(K)=\oplus _{m\in {\mathbb N}}\hat{C}_m(K)\) and

\(\hat{C}^*(K)=\oplus _{m\in {\mathbb N}}\hat{C}^m(K)\). The

*Kronecker pairing* on ordered (co)chains

$$ \hat{C}^m(K)\times \hat{C}_m(K) \xrightarrow {\langle \, ,\,\rangle } {\mathbb Z}_2 $$

is defined, using the various above definitions, by the equivalent formulae

$$\begin{aligned} \begin{array}{rcll} \langle a ,\alpha \rangle &{}=&{} \sharp (a\cap \alpha ) \ (\mathrm{mod\,} 2) &{} \small \mathrm {using\;Definition\;2.10.1a\;and\;b}\\ &{}=&{} \sum \nolimits _{\sigma \in \alpha } a(\sigma ) &{} \small \mathrm {using\;Definitions\;2.10.1a\;and\;2.10.2b}\\ &{}=&{} \sum \nolimits _{\sigma \in S_m(K)} a(\sigma ) \alpha (\sigma ) &{} \small \mathrm {using\;Definitions\;2.10.2a\;and\;b}. \end{array} \end{aligned}$$

(2.10.1)

As in Lemma

2.2.4, we check that the map

\(\mathbf{k}\):

\(\hat{C}^m(K)\rightarrow \hat{C}_m(K)^\sharp \), given by

\(\mathbf{k}(a)=\langle a ,\,\rangle \), is an isomorphism.

The

*boundary operator* \(\hat{\partial }\):

\(\hat{C}_m(K)\rightarrow \hat{C}_{m-1}(K)\) is the

\({\mathbb Z}_2\)-linear map defined, for

\((v_0,\dots ,v_m)\in \hat{\mathcal S}_m(K)\) by

$$\begin{aligned} \hat{\partial }(v_0,\dots ,v_m) = \sum _{i=0}^m (v_0,\dots ,\hat{v}_i,\dots ,v_m) \, , \end{aligned}$$

(2.10.2)

where

\((v_0,\dots ,\hat{v}_i,\dots ,v_m)\in \hat{\mathcal S}_{m-1}\) is the

\(m\)-tuple obtained by removing

\(v_i\). The

*coboundary operator* \(\hat{\delta }:C^m(K)\rightarrow C^{m+1}(K)\) is defined by the equation

$$\begin{aligned} \langle \hat{\delta }a ,\alpha \rangle = \langle a ,\hat{\partial }\alpha \rangle \,. \end{aligned}$$

(2.10.3)

With these definition,

\((\hat{C}_*(K),\hat{\partial },\hat{C}^*(K),\hat{\delta },\langle \, ,\rangle )\) is a Kronecker pair. We define the vector spaces of

*ordered cycles* \(\hat{Z}_*(K)\),

*ordered boundaries* \(\hat{B}_*(K)\),

*ordered cocycles* \(\hat{Z}^*(K)\),

*ordered coboundaries* \(\hat{B}^*(K)\),

*ordered homology* \(\hat{H}_*(K)\) and

*ordered cohomology* \(\hat{H}^*(K)\) as in Sect.

2.3. By Proposition

2.3.5, the pairing on (co)chain descends to a pairing

$$ H^m(K)\times H_m(K) \xrightarrow {\langle \, ,\,\rangle } {\mathbb Z}_2 $$

so that the map

\(\mathbf{k}\):

\(\hat{H}^m\rightarrow \hat{H}_m^\sharp \), given by

\(\mathbf{k}(a)=\langle a ,\,\rangle \), is an isomorphism (

*ordered Kronecker duality*).

Let

\(f\):

\(L\rightarrow K\) be a simplicial map. We define

\(\hat{C}_*f\):

\(\hat{C}_*(L)\rightarrow \hat{C}_*(K)\) as the degree

\(0\) linear map such that

$$ \hat{C}_*f(v_0,\dots ,v_m) = (f(v_0),\dots ,f(v_m)) $$

for all

\((v_0,\dots ,v_m)\in \hat{\mathcal S}(L)\). The degree

\(0\) linear map

\(\hat{C}^*f\):

\(\hat{C}^*(K)\rightarrow \hat{C}^*(L)\) is defined by

$$\begin{aligned} \langle \hat{C}^*f(a) ,\alpha \rangle =\langle a ,\hat{C}_*f(\alpha )\rangle \, . \end{aligned}$$

By Lemma

2.3.6,

\((\hat{C}^*f,\hat{C}_*f)\) is a morphism of Kronecker pairs.

We now construct a functorial isomorphism between the ordered and non-ordered (co)homologies, its existence being suggested by the previous examples. Define

\(\psi _*\):

\(\hat{C}_*(K) \rightarrow C_*(K)\) by

$$ \psi _* ((v_0,\dots ,v_{m})) = {\left\{ \begin{array}{ll} \{v_0,\dots ,v_{m}\} &{} \text {if } v_i\ne v_j \ \hbox {for all} \ i\ne j\\ 0 &{} \text {otherwise.} \end{array}\right. } $$

We check that

\(\psi \) is a morphism of chain complexes. We define

\(\psi ^* : C^*(K) \rightarrow \hat{C}^*(K)\) by requiring that the equation

\(\langle {\psi }^*(a) ,\alpha \rangle =\langle a ,{\psi }_*(\alpha )\rangle \) holds for all

\(a\in C^*(K)\) and all

\(\alpha \in \hat{C}_*(K)\). By Lemma

2.3.6,

\(\psi ^*\) is a morphism of cochain complexes and

\((\psi _*,\psi ^*)\) is a morphism of Kronecker pairs between

\((\hat{C}_*(K),\hat{C}^*(K))\) and

\((C_*(K),C^*(K))\). It thus defines a morphism of Kronecker pairs

\((H_*{\psi },H^*{\psi })\) between

\((\hat{H}_*(K),\hat{H}^*(K))\) and

\((H_*(K),H^*(K))\).

To define a morphism of Kronecker pairs in the other direction, choose a simplicial order

\(\le \) on

\(K\) (see 2.1.8). Define

\({\phi _\le }_*\):

\(C_*(K) \rightarrow \hat{C}_*(K)\) as the unique linear map such that

$$\begin{aligned} {\phi _\le }_* (\{v_0,\dots ,v_m\}) = (v_0,\dots ,v_m) \ , \end{aligned}$$

where

\(v_0\le v_1\le \cdots \le v_m\). We check that

\({\phi _\le }_*\) is a morphism of chain complexes and define

\({\phi _\le }^*\):

\(\hat{C}^*(K) \rightarrow C^*(K)\) by requiring that the equation

\(\langle {\phi _\le }^*(a) ,\alpha \rangle =\langle a ,{\phi _\le }_*(\alpha )\rangle \) holds for all

\(a\in \hat{C}^*(K)\) and all

\(\alpha \in C_*(K)\). By Lemma

2.3.6,

\(({\phi _\le }_*,{\phi _\le }^*)\) is a morphism of Kronecker pairs between

\((C_*(K),C^*(K))\) and

\((\hat{C}_*(K), \hat{C}^*(K))\). It then defines a morphism of Kronecker pairs

\((H_*{\phi _\le },H^*{\phi _\le })\) between

\((H_*(K),H^*(K))\) and

\((\hat{H}_*(K),\hat{H}^*(K))\).

Applying Kronecker duality to Proposition 2.10.7 gives the following

We shall see in Sect.

4.1 that

\(H^*\psi \) and

\(H^*{\phi _{\scriptscriptstyle \le }}\) are isomorphisms of graded

\({\mathbb Z}_2\)-algebras. We now prove that they are also natural with respect to simplicial maps. Let

\(f\):

\(L\rightarrow K\) be a simplicial map. Let

\(\hat{C}_*f\):

\(\hat{C}_*(L)\rightarrow \hat{C}_*(K)\) be the unique linear map such that

$$ \hat{C}_*f((v_0,\dots ,v_m))=(f(v_0),\dots ,f(v_m)) $$

for each

\((v_0,\dots ,v_m)\in \hat{\mathcal S}_m(K)\). Doing this for each

\(m\in {\mathbb N}\) produces a

\(\mathbf{GrV}\)-morphism

\(\hat{C}_*f\):

\(\hat{C}_*(L)\rightarrow \hat{C}_*(K)\). The formula

\(\hat{\partial }{\scriptstyle \circ } \hat{C}_*f = \hat{C}_*f {\scriptstyle \circ } \hat{\partial }\) is straightforward (much easier than that for non-ordered chains). Hence, we get a

\(\mathbf{GrV}\)-morphism

\(\hat{H}_*f\):

\(\hat{H}_*(L)\rightarrow \hat{H}_*(K)\). A

\(\mathbf{GrV}\)-morphism

\(\hat{C}^*f\):

\(\hat{C}^*(K)\rightarrow \hat{C}^*(L)\) is defined by the equation

\(\langle \hat{C}^*f(a) ,\alpha \rangle =\langle a ,\hat{C}_*f(\alpha )\rangle \) required to hold for all

\(a\in \hat{C}^m(L)\),

\(\alpha \in \hat{C}_m(K)\) and all

\(m\in {\mathbb N}\). It is a cochain map and induces a

\(\mathbf{GrV}\)-morphism

\(\hat{H}^*f\):

\(\hat{H}^*(K)\rightarrow \hat{H}^*(L)\), Kronecker dual to

\(H_*f\).

The above isomorphism results also work in relative ordered (co)homology. Let

\((K,L)\) be a simplicial pair. Denote by

\(i\):

\(L\hookrightarrow K\) the simplicial inclusion. We define the

\({\mathbb Z}_2\)-vector space of

*relative ordered* (

*co*)

*chain* by

$$ \hat{C}^m(K,L)=\ker \big (\hat{C}^m(K)\xrightarrow {\hat{C}^*i} \hat{C}^m(L)\big ) $$

and

$$\begin{aligned} \hat{C}_m(K,L)=\mathrm{coker\,}\big (i_*: \hat{C}_m(L)\hookrightarrow \hat{C}_m(K)\big ) \, . \end{aligned}$$

These inherit (co)boundaries

\(\hat{\delta }: \hat{C}^*(K,L)\rightarrow \hat{C}^*(K,L)\) and

\(\hat{\partial }=\hat{C}_*(K,L)\rightarrow \hat{C}_{*-1}(K,L)\) which give rise to the definition of

*relative ordered* (

*co*)

*homology* \(\hat{H}^*(k,L)\) and

\(\hat{H}_*(K,L)\). Connecting homomorphisms

\(\hat{\delta }_*\):

\(\hat{H}^*(L)\rightarrow \hat{H}^{*+1}(K,L)\) and

\(\hat{\partial }_*\):

\(\hat{H}_*(K,L)\rightarrow \hat{H}_{*-1}(L)\) are defined as in Sect.

2.7, giving rise to long exact sequences. Our homomorphisms

\(\psi _*\):

\(\hat{C}_*(K) \rightarrow C_*(K)\) and

\({\phi _\le }_*\):

\(C_*(K) \rightarrow \hat{C}_*(K)\) satisfy

\(\psi _*(\hat{C}_*(L))\subset C_*(L)\) and

\({\phi _\le }_*(C_*(L)\subset \hat{C}_*(L)\), giving rise to homomorphisms on relative (co)chains and relative (co)homology

\(H_*\psi \):

\(\hat{H}_*(K,L)\rightarrow H-*(K,L)\),

*etc*. Proposition

2.10.7 and Corollary

2.10.8 and their proofs hold in relative (co)homology. Hence, as for Corollaries

2.10.9 and

2.10.10, we get