A fact about finite-dimensional manifolds I fear does not hold for Frechet manifolds (what's new?)

Question

Let $M$ be a manifold equipped with a pair of surjective submersions $N_1 \stackrel{p_1}{\leftarrow} M \stackrel{p_2}{\rightarrow} N_2$ where $dim N_1 = dim N_2 = n$. Then we can find, for any point $m\in M$, a chart $U_1$ around $p_1(m)$ and a local section $s\colon U_1 \to M$ of $p_1$ such that $s(p_1(m)) = m$ and $s(U_1)$ is transverse to the fibres of $p_2$. In thinking about this we can clearly reduce to the case $N_1 = N_2 = \mathbb{R}^n$. We can even restrict attention to $M = \mathbb{R}^m$, and then it becomes a problem of linear algebra, namely finding a basis on $\mathbb{R}^m$ for which the submersions are both projections onto $n$ coordinates (we already know this separately).

I suspect that for infinite-dimensional vector spaces, and Frechet spaces in particular (really anything above Hilbert spaces in the usual hierarchy) this sort of result will not hold, and so for Frechet manifolds one cannot construct the analogous local section.

In more detail, I'm fairly sure that given a diagram of Frechet spaces

$$V_1 \stackrel{pr_1}{\leftarrow} V_1 \times F_1 \simeq W_1 \simeq V_2 \times F_2 \stackrel{pr_1}{\to} V_2$$

where $V_1$ is known to be isomorphic to $V_2$, one cannot in general find a (nonlinear?) map $F\colon V_1 \to V_2\times F_2$ (which is a section, and passes through the origin) such that $pr_1\circ F\colon V_1 \to V_2$ is injective an isomorphism. However, I'd like to see a counterexample (or, if I'm wrong, a proof that we can do it!). (EDIT: the map might need be be non-linear in this case because the definition of submersions doesn't use tangent spaces like in the finite-dimensional case. But linear would be good.)

VERSION 2: well it turns out that I have, in addition to the $V_i$s being isomorphic, I have $F_1$ isomorphic to $F_2$. Kudos to Andrew for guessing this would be the case for the application I have in mind.

Answer 1

(I'm strongly tempted to leave this as a comment, as it's a rather trivial counter-example and it may result in the question being changed to avoid it, but I want to take advantage of the better formatting of answers.)

Here's a trivial counter-example. Let $V$ be a space which is isomorphic to one of its hyperplanes, in that we have an isomorphism $V \cong V \oplus \mathbb{R}$ (see shift space on the nLab). There are lots of these around and most of the "usual suspects" for locally convex topological vector spaces have this property (especially including spaces like $C^\infty(\mathbb{R},\mathbb{R}^n)$). We take $V_1 = V_2 = W_1 = V$, $F_1 = \lbrace 0\rbrace$, and $F_2 = \mathbb{R}$. The isomorphisms as $W_1 = V_1$ on the left, and $W_1 = V \cong \mathbb{R} \oplus V = \mathbb{R} \oplus V_2$ on the right. Then $\operatorname{pr}_1 \colon W_1 \to V_1$ is the identity and $\operatorname{pr}_1 \colon W_1 \to V_2$ is the composition $W_1 = V \cong \mathbb{R} \oplus V \to V$.

The only option for the section is the identity, since $W_1 \to V_1$ is an isomorphism, whence the induced map $V_1 \to V_2$ is the map $V_1 = V \cong \mathbb{R} \oplus V \to V$ which is not injective.