# On the proof of the Whitney trick (from Scorpan's book)

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.