embedding $(\iota,{\varphi})$, but modifying $(\iota,{\varphi})$ by an isotopy results in a homeomorphic cell complex (see [@MR3230817 Lemma 3.25]). Modifying the other choices involved in constructing $|{\mathscr{C}}|$, including possibly increasing ${\textbf{d}}$ to ${\textbf{d}}'\geq{\textbf{d}}$, will result, up to homotopy, in some precise number of reduced suspensions of $|{\mathscr{C}}|$ (see [@MR3230817]). The following is a well-defined topological object arising from a framed flow category (and independent of choice of ${\textbf{d}}$).

For a framed flow category $({\mathscr{C}}, \iota, {\varphi})$ and any object $x$ of ${\mathscr{C}}$, let $C$ be the difference between the dimension of the associated cell $\mathcal{C}(x)\subset |{\mathscr{C}}|_{\iota,{\varphi}}$ and the grading of $x$. The *stable homotopy type of $({\mathscr{C}}, \iota, {\varphi})$* is the desuspended suspension spectrum $\mathcal{X}({\mathscr{C}}):=\Sigma^{-C}\Sigma^\infty|{\mathscr{C}}|_{\iota,{\varphi}}$, considered up to homotopy of spectra.

Handle slides in framed flow categories {#sec:handleslide}
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In this section, we will define the framed flow category analogue for the process in handlebody theory of sliding one $i$–handle over a second $i$–handle. This results in a new framed flow category giving rise to the same stable homotopy type.

\[thm:handleslide\] If $({\mathscr{C}}_S,\iota_S,{\varphi}_S)$ is the result of a *handle slide* in $({\mathscr{C}}, \iota, {\varphi})$, then there is a homotopy equivalence $$\mathcal{X}({\mathscr{C}}_S) \simeq \mathcal{X}({\mathscr{C}}) {\rm .}$$

We first describe a method for 