$\newcommand{\d}{\mathrm{d}}$I am casually reading Guillemin and Pollack's book "differential topology", which tries its best to present the core theorems with a minimum of machinery. There is a multi-step guided proof of the Jordan-Brouwer separation theorem, stated thusly:
If $X$ is a compact connected submanifold of $\Bbb R^n$ of dimension $n-1$, $n\in\Bbb N$, then $\Bbb R^n\setminus X$ has exactly two connected components.
For these authors, a "manifold" is a topological space $Y$ which is a subspace of $\Bbb R^m$ and for which there is a constant $k\in\Bbb N_0$ such that $Y$ is locally diffeomorphic with $\Bbb R^k$. Note this is not quite the usual definition via charts. For them, a diffeomorphism $f:\Bbb R^k\to K$ where $K$ is some not necessarily open subset of $\Bbb R^m$ will be a bijective map for which $\Bbb R^k\to K\to\Bbb R^m$ is smooth (in the elementary sense) and for which there is some open $U\subseteq\Bbb R^m$ containing $K$ and a smooth $F:U\to\Bbb R^m$ which restricts to $f^{-1}$ on $K$. Given $x\in X$ they define the tangent space $T_x(X)\subseteq\Bbb R^m$ to be the image of $\d\phi_0$ where $\phi:\Bbb R^k\to V$ is a diffeomorphism, $\phi(0)=x$ and $V$ is an open neighbourhood of $x$ (in $X$).
I'm stuck verifying one of the steps. We pick $z\in\Bbb R^n\setminus X$ and $v\in S^{n-1}$ and consider the ray $r:(0,\infty)\to\Bbb R^n$, $t\mapsto z+tv$, evidently a smooth map. I need to check that $r$ is transversal to $X$ if and only if $v$ is a regular value of the direction map $u_z:X\to S^{n-1}$, $x\mapsto\frac{x-z}{\|x-z\|}$. The given hint is to consider $g:\Bbb R^n\setminus\{z\}\to S^{n-1}$ the usual retraction, so that $u$ is $X\hookrightarrow\Bbb R^n\overset{g}{\to}S^{n-1}$ and to consider that $v$ is a regular value of $u$ iff. $u$ is transversal to $\{v\}$ iff. $X$ is transversal to $g^{-1}\{v\}$ within $\Bbb R^n$. The final iff. is intuitive but rigorously unclear to me:
I don't know how to see that $T_x(X)+T_x(g^{-1}(v))=\Bbb R^n$ iff. $\mathrm{d}u_x(T_x(X))=\mathrm{d}u_x(T_x(X))+T_v(\{v\})=\Bbb R^{n-1}$ for all $x\in g^{-1}(v)$.
Leaving that aside, it is clear $r$ is transversal to $X$ if and only if $v\Bbb R+T_x(X)=\Bbb R^n$ wherever $x$ is in the image of $r$ and it is clear this holds if and only if $v\notin T_x(X)$. The book mentions something about $(\d g_x)^{-1}(T_v(\{v\}))=T_x(g^{-1}\{v\})$ following from a general result, but it is unclear whether or not I need to check $v$ is a regular value of $g$ first. If this equation holds, it is clear that $T_x(g^{-1}\{v\})=0$ but then I need to show $T_x(X)=\Bbb R^n$ which is definitely false for dimension reasons, so I am very confused.
Have the authors given a mistaken hint or have I been butchering the definitions?
As a final remark, I tried checking whether or not $g$ is a submersion i.e. if $v$ is always a regular value. Using the stereographic projection $S^{n-1}\to\Bbb R^{n-1}$ gives an explicit formula for $g$ in local coordinates which one can explicitly compute the Jacobian of, but I'll be damned if I can figure out when that matrix has full rank; its entries are rather complex-looking (maybe I just didn't try hard enough). Is there an easier way?
First, the retraction $g\colon\Bbb R^n-\{0\} \to S^{n-1}$ is a submersion. You do not want to compute this by parametrizing $S^{n-1}$. (You almost never want to do such geometric computations in horrid local coordinates.) You want to proceed directly: Clearly $\text{Span}(x)\subset T_x\Bbb R^n$ is in $\ker dg_x$, and the orthogonal complement maps surjectively to $T_{g(x)}S^{n-1}$. You can verify this officially by noting that if $v\cdot x = 0$, then \begin{align*} dg_x(v) &= \lim_{t\to 0} \frac{g(x+tv)-g(x)}t = \lim_{t\to 0} \frac1t\left(\frac{x+tv}{\|x+tv\|} - \frac x{\|x\|}\right) \\ &= \lim_{t\to 0} \frac{x+tv-x\sqrt{1+t^2\|v\|^2/\|x\|^2}}{t\|x+tv\|} \\ &= \lim_{t\to 0} \frac{x+tv-x(1+\frac12t^2\|v\|^2/\|x\|^2+\dots)}{t\|x+tv\|} = \frac v{\|x\|}. \end{align*} (You can see this quite plainly geometrically by projecting a right triangle with legs $x$ and $v$ radially to the similar right triangle with legs $g(x)=x/\|x\|$ and $v/\|x\|$.)
This explicit computation makes it clear that the ray is transverse to $X$ at $x$ if and only if $dg_x$ maps surjectively to $T_{g(x)}S^{n-1}$.
By the way, I'm not sure it's required here, but the result of Exercise 7 on p. 33 is highly useful.