A difficulty in the proof of partial converse I.(p. 24 in Guillemin & Pollack)

Question

The statement and its proof is given below:

But I am wondering: How can I check that 0 is a regular value for $g$?

Answer 1

Using G&P's hint, take a chart $(\psi, W)$ in $Y$ centered at $y$, which means $\psi(y)=0_{\mathbb{R}^l}$. Then the map $g=\psi \circ f$ takes any point $x\in f^{-1}(y)$ to $0_{\mathbb{R}^l}$. It's just left to prove that the coordinates of this new map $g=(y^1, \dots, y^l)$ are independent functions. Just observe that at any $x\in S=f^{-1}(y)$ we have ker $df_x=T_xS$, so $df_x$ sends any subspace complementary to $T_xS$ in $T_xX$ isomorphically onto $T_yY$. This means that the map $g:f^{-1}(W)\supset S\rightarrow \mathbb{R}^l$ has rank $l$ at every $x\in S$ (because $\psi$ is a diffeomorphism, so its derivative is an isomorphism), which is another way to say that $y^1, \dots, y^l$ are independent in $S$.