Parametric transversality

Because $\mathcal{F}⋔Z$, [L1] makes $W:={\mathcal{F}}^{-1}(Z)\subseteq M\times S$ an embedded submanifold. Let ${\pi }_S:W\to S$ be the restriction of the second projection.

Fix $(p,s)\in W$ and write $z:=\mathcal{F}(p,s)$. The fibre of ${\pi }_S$ over $s$ is ${\pi }_S^{-1}(s)=\{(q,s)\in M\times S:\mathcal{F}(q,s)\in Z\},$ which identifies with ${\mathcal{F}}_s^{-1}(Z)$ by [F1]. A pair $(u,w)\in T_{p}M\times T_{s}S$ lies in $T_{(p,s)}W$ exactly when $d{\mathcal{F}}_{(p,s)}(u,w)\in T_{z}Z$, so $d({\pi }_S{)}_{(p,s)}$ is surjective exactly when every $w\in T_{s}S$ admits some $u\in T_{p}M$ with $d{\mathcal{F}}_{(p,s)}(u,w)\in T_{z}Z$.

If ${\mathcal{F}}_s⋔Z$ at $p$, then [F2] gives $d{\mathcal{F}}_{(p,s)}(T_{p}M\times \{0\})+T_{z}Z=T_{z}N\mathrm{.}$ For any $w\in T_{s}S$, choose $u\in T_{p}M$ so that $d{\mathcal{F}}_{(p,s)}(u,0)+d{\mathcal{F}}_{(p,s)}(0,w)\in T_{z}Z$. Then $(u,w)\in T_{(p,s)}W$, so step 2.1 makes ${\pi }_S$ a submersion at $(p,s)$.

Conversely, assume ${\pi }_S$ is a submersion at $(p,s)$. Given $\xi \in T_{z}N$, the transversality of $\mathcal{F}$ in [F2] gives $(u,w)\in T_{p}M\times T_{s}S$ and $\eta \in T_{z}Z$ with $\xi =d{\mathcal{F}}_{(p,s)}(u,w)+\eta$. By step 2.1 choose $u^{\mathrm{\prime }}\in T_{p}M$ with $(u^{\mathrm{\prime }},w)\in T_{(p,s)}W$, so $d{\mathcal{F}}_{(p,s)}(u^{\mathrm{\prime }},w)\in T_{z}Z$. Then $\xi =d{\mathcal{F}}_{(p,s)}(u-u^{\mathrm{\prime }},0)+(\eta +d{\mathcal{F}}_{(p,s)}(u^{\mathrm{\prime }},w))\in d{\mathcal{F}}_{(p,s)}(T_{p}M\times \{0\})+T_{z}Z,$ which is exactly the transversality condition for ${\mathcal{F}}_s$ at $p$.

Therefore $s$ is a regular value of ${\pi }_S$ if and only if the slice ${\mathcal{F}}_s$ is transverse to $Z$ at every point of its fibre. Applying the Sard statement in [L1] to ${\pi }_S$ shows that the bad parameters form a null subset of $S$.