Let’s think back to our example to model our -handle (where is not a max or min). Well, it was a “saddle point”. So it consisted of a both a downward arc and upward arc. If you got close enough, it would probably look like .
Well, generally this will fit with our scheme. An n-handle looked like … or better yet , and a 0-handle looked like , so maybe it is the case that a -handle looks like . Let’s call the core of the handle, and the co-core.
By doing the same trick of writing out what our function looks like at a critical point of index in some small enough neighborhood using the Morse lemma, we could actually prove this, but we’re actually more interested now in how to figure out what happens with as crosses this point.
By that I mean, it is time to figure out what exactly it is to “attach a -handle” to the manifold.
Suppose as in the last post that is a critical value of index . Then I propose that is diffeomorphic to (sorry again, recall my manifold is actually m-dimensional with n critical values).
I wish I had a good way of making pictures to get some of the intuition behind this across. I’ll try in words. A 1-handle for a 3-manifold, will be , i.e. a solid cylinder. So we can think of this as literally a handle that we will bend the cylinder into, and attach those two ends to the existing manifold. This illustration is quite useful in bringing up a concern we should have. Attaching in this manner is going to create “corners” and we want a smooth manifold, so we need to make sure to smooth it out. But we won’t worry about that now, and we’ll just call the smoothed out , say .
Let’s use our gradient-like vector field again. Let’s choose small enough so that we are in a coordinate chart centered at such that is in standard Morse lemma form.
Let’s see what happens on the core . At the center, it takes the critical value and it decreases everywhere from there (as we move from 0, only the first coordinates change). This decreasing goes all the way to the boundary where it is . Thus it is the upside down bowl (of dimension ). Likewise, the co-core goes from the critical value and increases (as in the right side up bowl) to the boundary of a disk at a value (where ).
Let's carefully figure out the attaching procedure now. If we think of our 3-manifold for intuition, we want to attach to by pasting along .
So I haven't talked about attaching procedures in this blog, but basically we want a map and then forming the quotient space of the disjoint union under the relation of identifying with . Sometimes this is called an adjunction space.
So really is a smooth embedding of a thickened sphere , since . And the dimensions in which it was thickened is . Think about the "handle" in the 3-dimensional 1-handle case. We gave the two endpoints of line segment (two points = ) a 2-dimensional thickening by a disk.
Now it is the same old trick to get the diffeo. The gradient-like vector field, , flows from to , so just multiply by a smooth function that will make match after some time. This is our diffoemorphism and we are done.