to be a **regular face** of a cell $\sigma^{(i+1)}$ if, given a characteristic map $\Phi_\sigma:D^{i+1} \rightarrow Y$ for $\sigma^{(i+1)}$, $\Phi_{\sigma}^{-1}(\tau)$ is a closed ball, and the map $\Phi_{\sigma}|_{\Phi_\sigma^{-1}(\tau)}$ is a homeomorphism.\
A **discrete vector field** $X$ on $Y$ is a collection of partially defined functions $X_i:{\mathcal{K}}_i \rightarrow {\mathcal{K}}_{i+1}$ satisfying the following three conditions for each $i$:

1.  $X_i$ is injective;

2.  the image of $X_i$ is disjoint from the domain of $X_{i+1}$;

3.  for any $\sigma^{(i)}$ in the domain of $X_i$, $\sigma^{(i)}$ is a regular face of $X_i(\sigma^{(i)})$.\

Given a CW complex $Y$ equipped with a discrete vector field $X$, a **cellular path** between two cells $\alpha^{(i)}$ and $\beta^{(i)}$ is a finite sequence of $i$-cells $$\alpha^{(i)} = \alpha_0^{(i)}, \alpha_1^{(i)}, \ldots, \alpha_{l-1}^{(i)}, \alpha_l^{(i)} = \beta^{(i)}$$ such that $\alpha_{j+1}^{(i)}$ is a face of $X_i(\alpha_{j}^{(i)})$. We say that the path is **closed** if $\alpha^{(i)} = \beta^{(i)}$, and we say it is **trivial** if $\alpha_j^{(i)} = \alpha_{k}^{(i)}$ for all $j,k$.\
A discrete vector field $V$ is said to be a **discrete gradient vector field** if it admits no non-trivial closed cellular paths.\
If $X$ is a discrete gradient vector field on a CW complex $X$, then we call a cell $\sigma$ of $X$ **redundant** 