Morse and Cerf Theory


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:


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.


We spent today's class with students presenting solutions and/or half-baked ideas about exercises. We had a complete solution to all of the various $S^1 \times S^2$ problems. The problem of showing that the space of metrics adapted to a fixed Morse function is connected (by which I really meant path-connected) was reduced to the following question:

Let $f$ be a standard Morse model function $f = \sum \pm x_i^2$ on $\mathbb{R}^n$ and let $\phi : \mathbb{R}^n \to \mathbb{R}^n$ be any orientation-preserving diffeomorphism sending $0$ to $0$ and respecting $f$, i.e. $f \circ \phi = f$. Show that $\phi$ is isotopic to the identity through a $1$-parameter family of maps $f_t$ with $f_t(0)=0$ and $f_t \circ \phi = f_t$.

A suggestion for showing that $f(x)=x^2$ is stable was to use the fact that any function (in particular, a $1$-parameter family $f_t$) can be approximated by polynomials. Another approach suggested was to show that $f_t$, for small $t$, has "the same kind of singularity" that $f$ has, where "same kind" means $f'(x_t)=0$ and $f''(x_t)>0$.

That's it.


Consider two critical points $p,q \in X$ of a Morse function $f$ on $X^n$, of indices $k$ and $l$, respectively, with $f(p) < f(q)$ and no critical values in between. We want to investigate conditions under which we can assume that their ascending and descending manifolds, $A_p$ and $D_q$, can be assumed to be disjoint. Of course, $A_p$ and $D_q$ are not defined until we choose a metric $g$ or, at least, a gradient-like vector field $V$.

Note first that, in between $f^{-1}(f(p))$ and $f^{-1}(f(q))$, everything is a product and is determined by behavior in $f^{-1}(y)$ for some regular $y \in (f(p),f(q))$. Thus we look at $A_p^y = A_p \cap f^{-1}(y) \cong S^{n-k-1}$ and $D_q^y = D_q \cap f^{-1}(y) \cong S^{l-1}$, all inside the $(n-1)$-dimensional manifold $f^{-1}(y)$.

Next we note that any isotopy of $A_p^y$ (resp. $D_q^y$) can be realized by homotoping the vector field, and thus the metric $g$, inside $f^{-1}[y-2\epsilon,y-\epsilon]$ (resp. $f^{-1}[y+\epsilon,y+2\epsilon]$). Again, this uses the product structure on $f^{-1}[y-2\epsilon,y-\epsilon]$ and just spreads the isotopy out across this product. Furthermore, any homotopy of $g$ or $V$ moves $A_p^y$ and $D_q^y$ by (independent) isotopies in $f^{-1}(y)$. Thus we can apply the transversality theorem to say that $g$ (or $V$) can be homotoped to make $A_p^y \cap D_q^y$ transverse in $f^{-1}(y)$ and that, if they are transverse, a small perturbation of $g$ or $V$ will keep them transverse.

So now we assume that $A_p^y$ and $D_q^y$ intersect transversely in $f^{-1}(y)$ and now we count the dimension of their intersection. Recall that, for transverse intersections, the mantra is "codimensions add". $A_p^y$ has dimension $n-k-1$ in the $(n-1)$-manifold, hence codimension $k$. $D_q^y$ has dimension $l-1$, hence codimension $n-l$. Thus $A_p^y \cap D_q^y$ has codimension $n+k-l$, hence dimension $n-1-(n+k-l) = l-k-1$. This is negative if $l < k+1$ or $l \leq k$. Thus we can assume that $A_p \cap D_q = \emptyset$ as long as $l \leq k$.

As a corollary, if a cobordism $X$ has a Morse function with all critical points of the same index, then $X$ can be built as a handlebody with all the handles attached at once to the bottom level.

Note that if $l = k+1$ then $A_p^y \cap D_q^y$ has dimension $0$, i.e. points, in which case we have isolated flow lines from $q$ down to $p$. In terms of handles, the handle for $q$ "goes over" the handle for $p$; we have seen many examples of this when $k=1$ and $l=2$.

Now we want to consider what sorts of intersections between $A_p$ and $D_q$ to expect as we move through a $1$-parameter family of Morse functions and gradient-like vector fields. The first case to consider is where $f$ stays fixed, but the vector field varies as $V_t$, $t \in [0,1]$. Now consider $\mathcal{A}_p$ and $\mathcal{D}_q$ in $[0,1] \times X$, defined by $\mathcal{A}_p \cap \{t\} \times X = \{t\} \times \mathcal{A}_{p,t}$, where $\mathcal{A}_{p,t}$ is the ascending manifold for $p$ with respect to $V_t$ (and similary for $\mathcal{D}_q$. A similar argument to the preceding case shows that, if we want to move $\mathcal{A}_p$ through an isotopy in $[0,1] \times f^{-1}(y)$ (remaining transverse to the slices $\{t\} \times X$), we can do this by homotoping the homotopy $V_t$ in a slab $f^{-1}[y-2\epsilon,y-\epsilon]$ (and comparable statement for $\mathcal{D}_q$. And, similarly, any homotopy of the homotopy $V_t$ moves these manifolds by isotopies. Thus, again, transversality applies and we can assume $\mathcal{A}_p \cap \mathcal{D}_q$ is transverse.

Now when we count dimensions we discover that this intersection, if transverse, should be empty if $k < l$. Thus, for example, in a $1$-parameter family we do not expect a critical point of index $1$ to suddenly develop a flow line down to a critical point of index $2$. But, if $k=l$, then we expect $\mathcal{A}_p \cap \mathcal{D}_q \cap ([0,1] \times f^{-1}(y))$ to have dimension $0$. This means that at isolated times, there will be a single point of intersection between $A_{p,t}^y$ and $D_{q,t}^y$ or, equivalently, a single flow line from $q$ down to $p$. Such events are called handle slides. Below is a simple example that justifies this term; we will discuss handle slides more carefully next time.