2012-02-08

We have seen that, in a $1$-parameter family $V_t$ of gradient-like vector fields for a fixed Morse function $f$, we should generically expect isolated times at which ascending and descending manifolds of critical points of the same index intersect in intermediate regular levels. It is not hard to generalize this to the case where the function varies as well, in which case we have a pair $(f_t,V_t)$. As long as $f_t$ remains Morse and critical values do not cross, we can apply all the same transversality arguments from before, letting $\mathcal{A}_p$ be the descending manifold in $[0,1] \times X$ of an arc of critical points labelled $p$.

We also identified these isolated times as handle slides and showed one example where the total dimension is $n=2$ and the critical points have index $k=1$. We want to investigate this more generally.

The first point to make is that, in calling these events "handle slides", we are really describing a particulation operation on handle attaching maps (framed embedded spheres), and claiming that this operation is exactly how the handle attaching maps change from before one of these isolated time events to after the event. So first I will attempt to describe this operation.

Exercise: Generalize the following examples to an operation that makes sense for any dimension $n>2$ and any index $k$ with $1<k<n$. We exclude $0$ and $n$ because you need some ascending and some descending manifold to get the discussion started. We exclude $1$ because we have already discussed it and because it is hard to make sense of many smooth operations on $0$-manifolds; e.g. what is the connected sum of two $S^0$'s? The operation we are looking for should take two framed $S^{k-1}$'s, $K_p$ and $K_q$, in a $(n-1)$-manifold, and produce a new framed $S^{k-1}$ $K_q'$ which results from sliding $q$ over $p$.

Example: n=3, k=2 : Here $K_p$ and $K_q$ are framed $S^1$'s in a surface $M^2$, in which case there is only one framing so we ignore the framing completely. The resulting $K_q'$ is an embedded $S^1$ in $M$ such that $K_p \cup K_q \cup K_q'$ together bound a pair of pants. This is illustrated below:

3D2HandleSlide

In this lecture I then proceeded to describe the 4-dimensional version $n=4$, $k=2$, but once again the exposition improved with the review in the next lecture, so I'll save it for the next post.