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.

enter image description here

enter image description here


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.