non-negative integers. Suppose $x,y$ are objects of ${\mathscr{C}}$ with grading $i$. To mimic the geometric process above, we will introduce a pair of cancelling objects $e$ and $f$ in degrees $i$ and $i+1$ respectively. Morphisms will be introduced so that cancelling $f$ against $e$ returns ${\mathscr{C}}$, and cancelling $f$ against $x$ returns a new category ${\mathscr{C}}_S$ which we will call the *effect of sliding $x$ over $y$*. It is proved in [@JLSmorse Theorem 2.17] that handle cancellation in flow categories preserves the stable homotopy type of a framed flow category, and hence the stable homotopy types $\mathcal{X}({\mathscr{C}})$ and $\mathcal{X}({\mathscr{C}}_S)$ will agree.

The main task of this section therefore will be to correctly describe, embed, and frame the *intermediate flow category* obtained by introducing $e$ and $f$. This is carried out in Subsection \[subsec:inter\], with Theorem \[thm:handleslide\] deduced in Subsection \[subsec:sliding\]. We end with a discussion of how handle sliding affects the framings of $1$-dimensional moduli spaces in Subsection \[subsec:framings\].

The intermediate flow category {#subsec:inter}
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We begin with a small lemma that we will use to embed and frame moduli spaces later. Suppose for now that $M^m$ is a smooth $m$–dimensional $\langle n\rangle$–manifold. Then $M\times[0,1]$ has the structure of a 