handle sliding, on which we base our flow category analogue. Given a handlebody $H$, which has $i$–handles $h_1$ and $h_2$, form a new handlebody $H'$ by attaching an $i$–handle $h_3$ and a cancelling $(i+1)$–handle $g$, attached as follows. The attaching sphere for $h_3$ will be glued along the sphere embedded in $H$ formed by taking an embedded connected sum of slightly pushed-off copies of the attaching spheres of $h_1$ and $h_2$. The attaching sphere of $g$ will be glued to intersect the belt spheres of each of our three $i$–handles precisely once positively each (and with no other $i$–handle interactions).

There are then exactly three $i$–handles against which $g$ can be cancelled. The handlebodies which are the effects of cancelling $g$ against $h_1$, $h_2$ and $h_3$ are homeomorphic to, respectively, the effect of $h_1$ slid over $h_2$, the effect of $h_2$ slid over $h_1$, and the original handlebody $H$. The reason for describing handle sliding with such an emphasis on handle cancellation is that it will allow us to build on the flow category handle cancellation technique described in the earlier paper [@JLSmorse].

Let ${\mathscr{C}}$ denote a framed flow category $({\mathscr{C}}, \iota, {\varphi})$ with respect to an $(n+1)$–tuple $\textbf{d}=(d_k,\dots,d_{n+k})$ of 