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Construction (The Fundamental Chain of a Cube). Let $I$ be a finite set of cardinality $n$, and let $\operatorname{\raise {0.1ex}{\square }}^{I} = \prod _{i \in I} \Delta ^1$ denote the associated cube (Notation, which we will identify with the nerve of the partially ordered set of all subsets of $I$. Using this identification, we obtain a bijective correspondence

\[ \{ \text{Linear orderings of $I$} \} \simeq \{ \text{Nondegenerate $n$-simplices of $\operatorname{\raise {0.1ex}{\square }}^{I}$} \} , \]

which carries a linear ordering $\{ i_1 < i_2 < \cdots < i_ n \} $ to the chain of subsets

\[ \emptyset \subset \{ i_1 \} \subset \{ i_1, i_2 \} \subset \cdots \subset \{ i_1, \ldots , i_{n-1} \} \subset I. \]

In particular, the symmetric group $\Sigma _{I}$ of permutations of $I$ acts simply transitively on the set of nondegenerate $n$-simplices of $\operatorname{\raise {0.1ex}{\square }}^{I}$.

Fix a linear ordering of $I$, corresponding to a nondegenerate $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\raise {0.1ex}{\square }}^{I}$. We let $[ \operatorname{\raise {0.1ex}{\square }}^{I} ]$ denote the alternating sum $\sum _{\pi \in \Sigma _{I} } (-1)^{\pi } \pi (\sigma )$, which we regard as an $n$-chain of the normalized chain complex $\mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I}; \operatorname{\mathbf{Z}})$. We will refer to $[ \operatorname{\raise {0.1ex}{\square }}^{I} ]$ as the fundamental chain of the cube $\operatorname{\raise {0.1ex}{\square }}^{I}$. We will be particularly interested in the special case where $I$ is the set $\{ 1, 2, \cdots , n \} $, endowed with its usual ordering; in this case, we denote the cube $\operatorname{\raise {0.1ex}{\square }}^{I}$ by $\operatorname{\raise {0.1ex}{\square }}^{n}$ and its fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{I} ]$ by $[ \operatorname{\raise {0.1ex}{\square }}^{n} ]$.