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:
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:
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 $p$. $\Box$