Cobordism Morse function ($f:W\to [0,1]$)

The incoming and outgoing boundaries are the regular levels zero and one, and all critical points lie in the interior. Near the boundary use a collar coordinate as the function. The Morse handle-attachment theorem produces a relative chain group freely generated by the interior critical points. Thus $\chi (W,M_0)$ is the alternating sum of their Morse indices, regardless of how their critical values are ordered.

If a cobordism Morse function on a trivial cobordism has exactly two critical points, its relative Morse chain complex has two generators and must be acyclic. Their indices therefore differ by one. The only differential is multiplication by the signed intersection of the lower point's belt sphere with the higher point's attaching sphere in a common regular level. A two-term complex of copies of $\mathbb{Z}$ is acyclic exactly when that integer is a unit. Algebraic intersection is therefore a necessary condition; it is not by itself a geometric handle-cancellation statement.