The example I gave last time was awful I’ve realized. I need something a little more complicated to better motivate why we’d believe some of these things, and to illustrate what happens in certain situations.
So let’s take a surface embedded in given by the equation . It is a “mountain landscape”:
It might be hard to tell, but there is the one peak, and it forever decreases to the general left, and forever increases to the general right.
We have a global chart to work with. Our Morse function will again be the “height function”. So . We have two critical points. One will occur when we reach the “saddle point” at and one when we reach the peak of the mountain at . Since this is a conceptual example, I won’t go through all the technical stuff to show that this is actually a Morse function.
Now as we stated before, at a non-critical value, i.e. a regular value, the level set is an embedded submanifold. Thus if , then is something that vaguely looks like:
This is because it is below the saddle point. As the height increases to 0, our level set starts to close in, and when we reach , we get:
This is not an embedded submanifold, because the point of intersection is not locally Euclidean. Continuing up, we get that will look something like:
Then we hit the critical value :
It doesn’t show up on the graph, but there is a point at which is why this one is not a manifold. There would be no well-defined dimension since it is a 0-dimensional object union a 1-dimensional object. Then everything above 4/27 looks like the last picture but without the dot.
Let’s analyze a little bit. Between critical values all of our embedded submanifolds seemed to be diffeomorphic to each other (you may be able to guess the proof of this even if you haven’t seen it). But when we cross a critical value we don’t even maintain homotopy type.
If anyone actually worked out the math behind this example, then they would see that we also now have an example of an index 1 critical point at the saddle. The top of the mountain is index 2, which still fits with the local min/max conjecture from last time.
I may post again later today on actual Morse theory, but I decided that I really needed a better example to reference once we got going.