Implicit function theorem without manifolds (Steve Smale article)?

Question

I don't know if this is the right place for this question, if not then please let me know and I will delete it.

In 1974, Steve Smale published an article in the first issue of Journal of Mathematical Economics with the short title "Global Analysis and Economics" which bears on my current research interests. In it, he applies variants of the inverse function theorem/implicit function theorem to a set that is not a manifold and explains that these results "make sense" even in that context. Now please understand, I know who Steve Smale is so I am not saying he's wrong, but this is unlike anything I have seen in my research so I thought someone could help with some thoughts.

In brief, he defines the set $\mathscr{S} \equiv (\bar{P})^m \times S_+$, where $\bar{P}$ is the non-negative orthant of $R^l$ and $S_+$ is the non-negative part of the $l-1$-dimensional sphere $S$. Though his use of the inverse function theorem occurs in a number of places, I will quote one part of the article which should highlight my difficulty:

"We recall that $\psi_u : \mathscr{S} \rightarrow (S)^{m+1}$ is said to be transversal to $\Delta \subset (S)^{m+1}$ provided that for each $s$ with $\psi_u(s) = y \in \Delta$, the image of $D\psi_u(s)$ and $T_y(\Delta)$ (as subspaces of $T_y((S)^{m+1})$) span $T_y((S)^{m+1})$…. Note that although $\mathscr{S}$ is not exactly [italics mine] a manifold, the notion of transversality still makes sense in our context".

"It follows from the implicit function theorem that if $\psi_u$ is transversal to $\Delta$, then $\psi^{-1}_u(\Delta)$ is a submanifold of $\mathscr{S}$ with $\operatorname{codimension} \psi_u^{-1}(\Delta) = \operatorname{codimension} \Delta$. We note also that the notion 'submanifold' of $\mathscr{S}$ is a slight extension of the usual notion. A precise definition would be as follows: Take an open set $\mathscr{O}$ of $\mathscr{S}$ in $(R^l)^m \times S$. Then a 'submanifold' of $\mathscr{S}$ is a subset of $\mathscr{S}$ of the form $V \cap \mathscr{S}$ where $V$ is a submanifold of $\mathscr{O}$." (again, italics mine)."

I hope I haven't violated any copyright, but this article is hard to get hold of online. I have it only because I have hard copies of the first several volumes of JME in my office.

Again, I am posting because I would like to be able to use the IFT et al. on sets like $(\bar{P})^m$, but cannot for the life of me figure out why this works (according to Smale, at least). I realize I haven't yet posed a question, so let me do so now:

Question: In the above passage, the hypothesis is that the function $\psi_u$ is transverse to $\Delta$ only on the "non-manifold" $\mathscr{S}$, yet Smale is saying the corresponding theorem can be applied as though $\mathscr{S}$ were a manifold. Why? At first I thought Smale was making an implicit assumption that $\mathscr{S}$ was contained in some open set $\mathscr{O}$ that would be a manifold and on which $\psi_u$ would be transverse to $\Delta$. Then you could apply the IFT theorems to $\mathscr{O}$ and simply take the intersection of the resulting submanifold with $\mathscr{S}$. But it seems that Smale doesn't require that $\psi_u$ satisfy the transversality condition anywhere on $\mathscr{O}$, just on $\mathscr{S}$.

I understand I may have misunderstood something simple, also that I may have left out vital information (just ask), but I can't for the life of me figure out what is going on.

Answer 1

Having reviewed the things you wrote the right frame for the question should be manifolds with corners (sets whose differentiable structure is locally modelled on open subsets of the orthants in euclidean space).

Hence your needs should be covered by

J. Margalef-Roig, E. Outerelo Dominguez: Differential Topology 1992, 1

I don't think that there is a public version of the book. It develops first the theory of manifolds with corners (and differentiable mappings between them) in the setting of possibly infinite dimensional Banach manifolds. The implicit function theorem which should cover your needs is then

Theorem 2.3.11 in 1.

Let $X,Y,Z$ be differentiable manifolds of class $p \in \mathbb{N}_0 \cup \{\infty\}$ $f\colon X\times Y \rightarrow Z$ a map of class $p$ and $(a,b)\in X\times Y$ a point. Suppose that $T^2_{(a,b)} f \colon T_b Y \rightarrow T_{f(a,b)}Z$ is a linear homeomorphism ($T^2$ is their notation for the derivative with respect to the second component, not the iterated tangent map!) and suppose that there are open neighborhoods $V^a$ of $a$ and $V^b$ of $b$ such that $f(V^a\times (V^b\cap \partial Y))\subseteq \partial Z$.

Then there are open neighborhoods $W^a$ of $a$ , and $W^b$ of $b$ and a unique map $q \colon W^a \rightarrow W^b$ such that $f(x,q(x))=f(a,b)$ for $x\in W^a$. Furthermore, $q(a)=b$; $q$ is of class $p$ on $W^a$; for every $x\in W^a$, $T^2_{(x,q(x))}f$ is a linear homeomorphism and $T_xq = - (T^2_{(x,q(x)}f)^{-1}\circ T^1_{(x,q(x))}f$

There is actually a bit more information on the tangent spaces here which I did not repeat as it references earlier numbers in the book. The proof is what you would expect taking into account the boundary conditions (note in particular the added requirements of mapping the boundary into the boundary beyond the usual condition of the tangent maps).