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For Exercise 5 section 5 chapter 2 of Guillemin & Pollak: Set $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ and assume that g is transversal to a submanifold $W\subset Z$. Show $f\pitchfork g^{-1}(W)$ iff $g\circ f\pitchfork W$.

First for $f(x)=y\in g^{-1}(W)$ and since $g\pitchfork W$ we have \begin{equation} d_{f(x)} g(T_{f(x)} Y)+T_{g(f(x))}W=T_{g(f(x))}Z \end{equation} And for the prove put $y=f(x)$. If $g\circ f\pitchfork W$ then for $x\in (g\circ f)^{-1}(W)$ $$d_x(g\circ f)T_x X+T_{g\circ f(x)}W=T_{g\circ f(x)}Z$$ And if $f\pitchfork g^{-1}(W)$ $$d_x f(T_x X)+T_{f(x)}(g^{-1}(W))=T_{f(x)}Y$$ I have the fact that for transversality $T_{f(x)}(g^{-1}(W))=(d_{f(x)}g)^{-1}(T_{g(f(x))}W)$ and trivialy $d_x(g\circ f)T_x X=d_{f(x)}g\circ d_x f(T_x X)$.

I dont know if it's just a matter of developing, this was only the breakdown and not know what else to do, not how to attack. A hint or another way to solve it.

Donyarley
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Hint: In these situations, it can be helpful (if not concise) to work directly with tangent vectors. Fix a point $y \in g^{-1}(W)$. Since $g$ is transversal to $W$, any vector $a \in T_{g(y)}Z$ can be written as a sum $a=a'+d_y g(b)$ for $a' \in T_{g(y)} W$ and $b\in T_y Y$. Now further suppose that $y=f(x)$ for some $x \in X$.
For the "$\Rightarrow$" direction: If $f \pitchfork g^{-1}(W)$, then we can write $b$ as $b'+d_x f(c)$ for $b' \in T_y g^{-1}(W)$ and some $c \in T_x X$. Then we have $$a=a'+d_y g(b'+d_x f(c))=(a'+d_y g(b')) + d_x (g \circ f)(c).$$ Where do each of the above tangent vectors live? And what does that say about $(g \circ f)$ and $W$? A similar approach works for the "$\Leftarrow$" direction.

Unless this is homework, I'm happy to explain more.

Kyle
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