technique for modifying framed flow categories, called *the extended Whitney trick* (compare the Whitney trick of [@JLSmorse 1]). Suppose $({\mathscr{C}}, \iota, {\varphi})$ is a framed flow category and that for some objects $x\neq y$ of ${\mathscr{C}}$, that ${\mathcal{M}}(x,y)=M$ is a manifold with boundary. Let $W$ be a framed cobordism (rel. boundary) between $M$ and another manifold with boundary $M'$. We will show how to use $W$ to define a new framed flow category $({\mathscr{C}}_W,\iota_W,{\varphi}_W)$ in which $M$ is replaced by $M'$ (and the other moduli spaces in ${\mathscr{C}}$ are modified appropriately). Using the ideas of the Pontryagin–Thom theorem we will deduce the following.

\[thm:whitney\] If $({\mathscr{C}}_W,\iota_W,{\varphi}_W)$ is the result of an *extended Whitney trick* in $({\mathscr{C}}, \iota, {\varphi})$, then there is a homotopy equivalence $$\mathcal{X}({\mathscr{C}}_W) \simeq \mathcal{X}({\mathscr{C}}) {\rm .}$$

Framed cobordism of manifolds with corners rel. boundary
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We will not need Laures’s full machinery [@MR1781277] of cobordism of manifolds with corners in the sequel. The eventual complicated interactions between the moduli spaces in a framed flow category mean that allowing an unrestricted framed cobordism in the sense of [@MR1781277] becomes intractable. Instead we will work with the following.

Suppose $M$ and $M'$ is are $m$–dimensional $\langle n\rangle$–manifolds with $\partial_iM=\partial_iM'$ 