manifolds

## Handlebodies II

Let's think back to our example to model our $latex \lambda$-handle (where $latex \lambda$ 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 $latex D^1\times D^1$. Well, generally this will… Continue reading Handlebodies II

Today we prove what is known as The Morse Lemma. It tells us exactly what our Morse function looks like near its critical points. Let $latex p\in M$ be a non-degenerate critical point of $latex f:M\to \mathbb{R}$. Then we can choose coordinates about p, $latex (x_i)$, such that in these coordinates $latex f=-x_1^2-x_2^2-\cdots -x_\lambda^2+x_{\lambda+1}^2+\cdots +x_n^2+f(p)$.… Continue reading The Morse Lemma