$fr(I_{B,A})=\varepsilon_B$.

Finally, assume that $|a|=|x|$. If $a\not=x$, then ${\mathcal{M}}(a',b')={\mathcal{M}}(a,b)$ and the framings do not change. If $a=x$, we get ${\mathcal{M}}(x',b')={\mathcal{M}}(x,b)\sqcup {\mathcal{M}}(y,b)$ and the framing of ${\mathcal{M}}(x,b)$ does not change. If we perform a $(+)$–slide, the framing of ${\mathcal{M}}(y,b)$ does not change either. In the case of a $(-)$–slide, the first coordinate of ${\mathbb{R}}^{d_{i-1}}$ is reflected. This means that the second coordinate of the framing of an interval is changing sign. However, flipping the sign of the second coordinate maps standard paths to standard paths, compare also [@MR3252965 Lemma 3.3], although only half of this statement is proven there. Hence the framing does not change either.

The extended Whitney trick in framed flow categories {#sec:whitney}
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In [@MR1781277 Theorem 3.1.5], Laures extends the Pontryagin–Thom Theorem to the setting of manifolds with corners using a suitably defined framed cobordism category. As the cell attachment maps in the Cohen–Jones–Segal construction are defined by a version of the Pontryagin–Thom collapse map, the homotopy classification of these attaching maps (hence the eventual homeomorphism type of the CW-complex) is closely related to the framed cobordism classes of the moduli spaces in a flow category.

In this section we will re-encode this relationship by defining a new 