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

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manifolds

The Morse Lemma

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