Morse and Cerf Theory

2012-02-15

Note that the assertion that $A_p^y$ and $D_q^y$, the ascending and descending spheres for critical points $p$ and $q$ in an intermediate regular level $y$, meet at a single point, is equivalent to the assertion that there is a unique gradient flow line from $q$ down to $p$.

In this entry we want to sketch the proof of the following:

Theorem: If $f: X \to [0,1]$ is Morse with exactly two critical points $p$ and $q$ with a unique gradient flow line from $q$ down to $p$, with ascending and descending manifolds meeting transversely, then there is a generic homotopy $f_t$ from $f_0 = f$ to $f_1$ which cancels $p$ and $q$.

Sketch of proof: Here's what I send in lecture, but actually it's subtly wrong: Find an arc $A \cong [0,1]$ embedded in $X$ containing this unique flow line as its middle third $[1/3,2/3]$, with $q$ at $1/3$ and $p$ at $2/3$, and with $[0,2/3)$ contained in the descending manifold for $q$ and $(1/3,1]$ contained in the ascending manifold for $p$. Thus, up to reparametrization, $f|_A$ looks like $f(x) = x^3 - x$. Now we claim that there is a tubular neighborhood $\nu$ of $A$, with coordinate $x_{k+1}$ on $A$ and coordinates $x_1, \ldots, x_k, x_{k+2}, \ldots, x_n$, where $k$ is the index of $p$, such that $f|_\nu = -x_1^2 - \ldots -x_k^2 + x_{k+1}^3 - x_{k+1} + x_{k+2}^2 + \ldots + x_n^2$. The idea is that, along the given flow line, $A_p$ and $D_q$ intersect transversely, so that the descending coordinates $x_1, \ldots, x_k$ come from $D_q$ and the ascending coordinates $x_{k+2}, \ldots, x_n$ come from $A_p$. Once we have this local model, we can cancel the critical points using $x_{k+1}^3 - t x_{k+1}$. This is illustrated below:

CancellationLemma1

So what is wrong with this argument? The first problem is that, yes, one may find a local patch (in this case a tubular neighborhood of an arc) in which there is a certain local model (that much is correct in the above argument), but then one cannot blithely apply a polynomial perturbation because polynomials are not compactly supported, and we should be constructing a homotopy which is constant outside the given patch. Thus one should cut if off with a bump function. But then, when cutting things off with a bump function, one has the potential to accidentally create new critical points, as illustrated in this picture:

CancellationLemma2

So I owe a proper sketch of this proof - the point is that one really does need to work with the full descending manifold for $q$ and the full ascending manifold for $p$. $\Box$

2012-02-13

We have been a bit unclear, when discussing handle slides, about whether we are thinking of families $(f_t,V_t)$ of Morse functions paired with gradient-like vector fields or of a fixed Morse function $f$ with a family of vector fields $V_t$. I claim that this distinction is not important due to the following fact:

Lemma: If $f_t : X \to [0,1]$ is Morse for all $t$ (with distinct critical values for all $t$), with $t \in [0,1]$, then there exist isotopies $\phi_t : X \to X$ and $\psi_t: [0,1] \to [0,1]$, with $\phi_0$ and $\psi_0$ identity maps, such that $f_t = \psi_t \circ f_0 \circ \phi_t$.

Thus, as long as there are no crossings of critical values, we may pull everything back by $\phi_t$ and $\psi_t$ to treat $f_t$ as constant in $t$. When there are crossings, in which case we have no hope of making the function constant in time, we can arrange that no handle slides occur in a short time interval around the crossing.

It is obvious that two Morse functions cannot in general be connected by a path of Morse functions, since critical points remain discrete and therefore the number of critical points would need to be constant. However, they can be connected by a generic homotopy, which we define as follows:

Definition: a generic homotopy is a homotopy $f_t : X \to [0,1]$, $t \in [0,1]$, between Morse functions $f_0$ and $f_1$ such that, near every point $p \in X$ and time $t_0$, there exist coordinates $\tau$ around $t_0 \in [0,1]$ and $\tau$-dependent coordinates $x_1^\tau, \ldots, x_n^\tau$ around $p \in X$, and $\tau$-dependent coordinates $y^\tau$ around $f_{t_0}(p)$, with respect to which $f_t$ has one of the following three local models (in which we suppress the dependence of the $x_i$'s and $y$ on $\tau$):

  1. $(x_1, \ldots, x_n) \mapsto x_1$; i.e. there is no singularity here.
  2. $(x_1, \ldots, x_n) \mapsto -x_1^2 - \ldots -x_k^2 + x_{k+1}^2 + \ldots + x_n^2$; i.e. $p$ is an index $k$ Morse singularity for $f_{\tau_0}$ and there is a path $p_t$, $t \in (t_0-\epsilon,t_0+\epsilon)$, with $p=p_0$ such that $p_t$ is an index $k$ Morse singularity for $f_t$.
  3. $(x_1, \ldots, x_n) \mapsto -x_1^2 - \ldots - x_k^2 + x_k+1^3 - \tau x_{k+1} + x_{k+2}^2 + \ldots x_n^2$; i.e. a birth or death of a pair of critical points of index $k$ and $k+1$ occurs at $p$ at time $t_0$.

The claim, which we offer without proof or perhaps defer to a later date, is that generic homotopies are generic and stable. (This emphasizes, of course, the poor choice of the adjective "generic" in the definition.) Here stable means that a generic homotopy is still a generic homotopy after a small perturbation, and generic means that any homotopy can be perturbed to generic by an arbitrarily small perturbation.

Thus, between any two Morse functions, we can always find a homotopy which remains Morse except at isolated times when a birth or death occurs (and, if we insist that Morse functions have discrete critical values then we also count critical value crossings as special isolated non-Morse events).

Immediately after a birth has occurred, producing critical points $p$ and $q$ with indices $k$ and $k+1$, respectively, then $f(p) < f(q)$ and there are no critical values in $(f(p),f(q))$. Also, there exists a gradient-like vector field with respect to which there is a single flow line from $q$ down to $p$; i.e. $A_p^y \cap D_q^y$ is a single point in $f^{-1}(y)$ for some $y \in (f(p),f(q))$. In terms of handles, this means that the handle attaching sphere $S^k$ for $q$ "goes over" the handle $B^k \times B^{n-k}$ for $p$ once, intersecting the belt sphere $\{0\} \times S^{n-k-1}$ at one point. We have already examples of such diagrams.

In the next post I'll explain the converse of this, namely that if a $(k+1)$-handle goes over a $k$-handle once then we can cancel them; Gompf and Stipsicz give a non-Morse theory proof which is more basic, but we will construct a generic homotopy cancelling the two critical points.

2012-02-10

Today's goal is to present the handle slide move in dimension $4$, for $2$-handles, and then justify this move in both dimensions $3$ and $4$. I claim that, in both the cases $n=3,k=2$ and $n=4, k=2$, the move can be described as follows (bearing in mind that the framing is irrelevant when $n=3$): Let $K_p, K_q \subset M^{n-1}$ be the framed descending $S^1$'s for index $2$ critical points $p$ and $q$, resp., with $f(p) < f(q)$, and $M$ a level set below $p$ and $q$. Then the result of sliding $q$ over $p$ is that $K_q$ is replaced by $K_q'$, where $K_p \cup K_q \cup K_q' = \partial \Sigma$ and $\Sigma \subset M$ is an embedded pair of pants realizing the given framings of $K_p$ and $K_q$. The framing of $K_q'$ that results from the slide is exactly the framing coming from $\Sigma$. Below is an example when $n=4$, so that the drawing takes place in a $3$-manifold:

4D2HandleSlide

Now we want to simultaneously justify the following two statements: (1) If two handle diagrams are related by a handle slide then they describe diffeomorphic manifolds. (2) If two Morse functions (with gradient-like vector fields) are related by a homotopy in which the function remains Morse, then their corresponding handle diagrams are related by handle slides. To do this, consider two critical points $p$ and $q$ of the same index $k$ with $f(p) < f(q)$ and let us follow their ascending and descending manifolds in two different regular level sets: $f^{-1}(y_0)$ and $f^{-1}(y_1)$, with $y_0 < f(p) < y_1 < f(q)$. We focus on times just before and just after a time $t_0$ at which there is a single point of intersection between $D_q$ and $A_p$ in $f^{-1}(y_1)$ (a transverse intersection between $\mathcal{D}_q$ and $\mathcal{A}_p$). In $f^{-1}(y_1)$, $D_q^y$ moves around by an isotopy, crossing $A_p^y$ (transversely in time) at time $t_0$. But in $f^{-1}(1)$, $D_q$ makes a discrete jump somehow from before $t_0$ to after. As $D_q^y$ crosses $A_p^y$, it sweeps out an annulus punctured once by $A_p^y$. Removing a disk neighborhood of this puncture from the annulus, we get a pair of pants $\Sigma \subset f^{-1}(y_1) \setminus A_p^y$ with boundary the union of $D_q^y$ before $t_0$, $D_q^y$ after $t_0$, and the boundary of the disk we removed from the annulus. Since $\Sigma$ is disjoint from $A_p$, it can flow down to $f^{-1}(y_0)$. The "before" and "after"versions of $D_q^y$ flow down to become "before" and "after" attaching spheres for $q$ (framed by $\Sigma$) while the boundary of the disk we removed flows down to become a parallel push-off of the attaching sphere for $p$.

This demonstrates directly that the singular event in the Morse function movie corresponds to a handle slide in the handle diagram. To go the other way, note that the previous paragraph also provides a construction of a Morse function movie that corresponds to a given hande slide; this, together with the fact that handle diagrams uniquely determine manifolds up to diffeomorphism, shows that, when two diagrams are related by handle slides, then they describe diffeomorphic manifolds.

I claim that the above argument also works in higher dimensions and different indices, but leave that to the reader to sort out. Also, as a suggestion, it might be useful to construct the pair of pants and its higher-dimensional generalizations as a framed cobordism in $[0,1] \times f^{-1}(y_0)$, with a single critical point for the Morse function arising from projection to the $[0,1]$ factor.