I have a question about the Equivariant Submersion Theorem (Proposition 2.7) in the article Equivariant control data and neighborhood deformation retractions by Markus J. Pflaum and Graeme Wilkin. The statement is as follows:
Proposition 2.7: Let $G$ be a compact Lie group and let $f:M \to P$ be a $G$-equivariant submersion between smooth $G$-manifolds $M$ and $P$. Let $x\in M$ be a point, $H=G_x$ be the isotropy group of $x$ and $K=G_{f(x)} \supset H$ the isotropy group of $f(x)$. Then there exist:
- a finite dimensional orthogonal $K$-representation space $N$,
- a finite dimensional orthogonal $H$-representation space $W$,
- a $K$-invariant open convex neighborhood of the origin $B\subset N$,
- an $H$-invariant open convex neighborhood of the origin $C\subset W$,
- $G$-equivariant open embeddings $\Theta: G\times_H (B \times C) \hookrightarrow M$ and $\Psi: G\times_K B \hookrightarrow P$
such that $\Theta \big( [e,0]_{G\times_H (B \times C)} \big) = x$, $\Psi \big( [e,0]_{G\times_K B} \big) = f(x)$ and such that the following diagram commutes, where $\pi : B \times C \to C$ is projection onto the first factor and $\overline{\mathrm{id}_G \times \pi}$ maps $[g,(v,w)]_{G \times_H (B\times C)}$ to $[g,v]_{G\times_K B}$: $\require{AMScd}$ \begin{CD} G\times_H (B \times C) @>{\Theta}>> M\\ @V{\overline{\mathrm{id}_G \times \pi}}VV @VV{f}V\\ G\times_KB @>{\Psi}>> P. \end{CD}
Question: Why can't we take $B = N$ and $C = W$?
It seems to me that we can. Indeed, in the proof Pflaum and Wilkin give, the open convex subspaces $B$ and $C$ are actually constructed as open balls inside $N$ resp. $W$. In particular, by scaling the radius using the function $r \mapsto \frac{r}{1-r^2}$ they are even diffeomorphic to $N$ and $W$, and these diffeomorphisms are $G$-equivariant as the $G$-actions on $N$ and $W$ are orthogonal. So can't we just identify $B$ and $C$ with $N$ and $W$ using these diffeomorphisms?