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I'm trying to study a proof of the h-cobordism theorem from Scorpan's "The wild world of 4-manifolds". Given a handle decomposition for the cobordism, the Whitney trick is used to eliminate every pair of intersection points with opposite sign between the belt sphere of a $k$-handle and the attaching sphere of a $k+1$-handle attached to it.

Essentially the statement is: given two transversally intersecting submanifold $P^p$, $Q^q$ of complementary dimensions in a simply connected smooth manifold $M^{p+q}$, suppose also that $n\ge 2, m\ge 3$. If $x',x''$ are two points in the intersection with opposite signs then there exists an isotopy $\varphi_t$ of $\text{Id}_M$ such that $\varphi_1(P)\cap Q=P\cap Q\setminus \{x',x''\}$

We have two arcs respectively in $P$ and $Q$ connecting the points $x',x''$. Together they form a 2-dimentional disc $D$ and the idea of the trick is to "push" $P$ along this disc. Rigorously, Scorpan says that all we want is to show that the normal bundle $N_{D/M}$ of $D$ in $M$ splits in one part tangent to $P$ and normal to $Q$ and another normal to $P$ and tangent to $Q$. Can anyone explain why we want that? I think I can understand the remaining part of the proof of the trick.

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