2012-02-03

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.

HandleSlide1