In the terminology of simplicial complexes, an [*$i$-cochain*]{} is a function $W:X(i) \rightarrow \{0,1\}$. For us it is convenient to view an $i$-cochain as defining a subset of $i$-faces, where $\sigma \in W \Leftrightarrow W(\sigma) = 1$. The space of all the $i$-cochains is denoted by $C^i(X)$.

The [*degree*]{} of a face $\sigma \in X$ is the number of faces of maximal dimension which contain $\sigma$. Again, we assume for simplicity that the complex is regular, i.e., for any $0 \le i < d$ and any two $i$-faces $\sigma, \tau \in X(i)$ it holds that $\deg(\sigma)=\deg(\tau)$ (we deal with non-regularity at Section \[non-regular-section\]). For any $i$-cochain $W \in C^i(X)$ we define its norm to be the fraction of $i$-faces which are in $W$ (this definition will be refined at Section \[non-regular-section\]), i.e., $${\left\lVert W \right\rVert} = \frac{|W|}{|X(i)|}.$$

The [*link*]{} of a face $\sigma \in X$, denoted by $X_\sigma$, is the complex obtained by taking all the faces which contain $\sigma$, and removing $\sigma$ from all of them. Intuitively, it is the local view of $X$ “from the eyes of $\sigma$”. Formally it is given by $X_\sigma = \{\tau \setminus \sigma \;|\; \tau \in X,\, \tau \supseteq \sigma\}$. If $X$ is 