vector field by assigning a cell $\sigma \in {\mathcal{K}}_i$ to the (unique) cell $\tau \in {\mathcal{K}}_{i+1}$ containing $\sigma$ for which $f(\sigma) \geq f(\tau)$. The fact that this discrete vector field is actually a discrete gradient vector field, i.e. the fact that it admits no non-trivial closed cellular paths, is exhibited in [@Fo1] and [@Fo2].\
Just as with classical Morse theory, it is the case that the critical cells of a discrete gradient vector field $X$ on a space $Y$ determine the whole of $Y$.\

\[VFdecomp\] Let $Y$ be a CW complex equipped with a discrete gradient vector field $X$. Consider the filtration $$\emptyset = Y_0'' \subseteq Y_0' \subseteq Y_1'' \subseteq Y_1' \subseteq \ldots \subseteq Y_n'' \subseteq Y_n' \subseteq \ldots$$ where $Y_i'$ is the $i$-skeleton of $Y$ with the redundant cells removed, and $Y_i''$ is the $i$-skeleton of $Y$ with both the redundant and critical cells removed. Then:

1.  For any $i$, $Y'_i$ is obtained from $Y''_i$ by attaching $m_i$ $i$-cells to $Y_n''$ along their boundaries, where $m_i$ is the number of critical $i$-cells of the discrete gradient vector field $X$.

2.  For any $i$, $Y_{i+1}''$ deformation retracts onto $Y_i'$.\

The above proposition leads to one notable corollary, which we record 