its cohomology. In other words, the Floer invariant should take the form of a *stable homotopy type*.

In this paper we construct moves that change a framed flow category without changing the associated stable homotopy type. These are inspired by moves that can be performed in the Morse–Smale situation without altering the underlying smooth manifold. We posit that if two framed flow categories represent the same stable homotopy type then a finite sequence of these moves is sufficient to connect the two categories. This is directed towards the goal of reducing the study of framed flow categories (and their associated stable homotopy types) to combinatorics.

We show in Theorem \[thm:SNF\] that after a finite sequence of our moves, we can arrange for the associated cochain complex to the framed flow category to be in primary Smith normal form. This immediately gives Corollary \[cor:moore\]. This corollary implies, for example, that if two framed flow categories each represent the same Moore space, then the categories are indeed connected by a finite sequence of moves.

In a later paper we shall go further, developing the combinatorics of the moves and proving results based now on the classifications due to Chang and Baues–Hennes of 