Moving a sphere off a lower-dimensional submanifold

Choose a product neighbourhood $k:A\times {\mathbb{R}}^d\to U$ of $A$ with $U\subseteq W_0$, write $\pi :A\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ for the projection, and put $B_0:=k^{-1}(B\cap U)\subseteq A\times {\mathbb{R}}^d$. Since $B\cap U$ is open in $B$, the set $B_0$ is an embedded submanifold of dimension $\dim B$; the projection $\pi$ is smooth, hence $C^1$, and $\dim B_0\le \dim B<d$, so $\pi (B_0)$ is a null subset of ${\mathbb{R}}^d$ with dense complement, and an arbitrarily small nonzero $w$ lies outside it.

Fix $r>0$, restrict the choice of $w$ to $0<\mathrm{\mid }w\mathrm{\mid }<r$, and choose a smooth cutoff $\chi :{\mathbb{R}}^d\to [0,1]$ with $\chi =1$ on the closed ball of radius $r$ about $0$ and $\mathrm{supp}\chi$ in the ball of radius $2r$; it exists by [F3] after normalizing any bump for the compact ball inside the larger ball. Define a vector field $X$ on $V$ by $X(k(a,z)):=\chi (z)(0,w)$ in the coordinates of $U$ and $X:=0$ on $V\setminus U$. The field is smooth, for the two definitions agree near $\mathrm{\partial }U$ where $\mathrm{supp}\chi$ is avoided, and its support is contained in $k(A\times \mathrm{supp}\chi )$, a compact subset of $U$; hence it is complete and has a global flow $\mathrm{\Phi }:\mathbb{R}\times V\to V$.

The time-one map $h:={\mathrm{\Phi }}_1$ is a diffeomorphism of $V$ with inverse ${\mathrm{\Phi }}_{-1}$, it is supported in $U\subseteq W_0$, and $t\mapsto {\mathrm{\Phi }}_t$ is a smooth isotopy from the identity to $h$. Because the cutoff is fixed and the field is linear in $w$, the field tends to zero in every coordinate derivative as $w\to 0$. Its flow tends smoothly to the identity: apply The fundamental theorem on flows to the augmented field with $w$ as a constant parameter coordinate, using a parameter cutoff outside a fixed ball. Its support is compact since $A$ is compact, so the flow is defined for the entire time interval. Smooth dependence on $(w,t,x)$ and compactness give convergence of every derivative. For $a\in A$ the trajectory of $k(a,0)$ is $t\mapsto k(a,tw)$, because along the segment from $0$ to $w$ the cutoff equals $1$ and the second coordinate moves linearly; hence $h(k(a,0))=k(a,w)$, that is, $h(A)=k(A\times \{w\})\subseteq U$.

Finally $h(A)\cap B=\mathrm{\varnothing }$: a point of $h(A)\cap B$ would lie in $U$ and equal $k(a,w)$ for some $a\in A$; then $(a,w)\in B_0$, contradicting $w\mathrm{\notin }\pi (B_0)$, since $\pi (a,w)=w$. Together with steps 1.1 and 2.1 this gives a diffeomorphism supported in the prescribed neighbourhood, isotopic to the identity, that moves $A$ off $B$. If $B=\mathrm{\varnothing }$ or $A=\mathrm{\varnothing }$ the identity map already satisfies the conclusion, and the construction above also covers these cases because then $\pi (B_0)$ is empty.