manifolds

## Handle Decomposition

Today I'll just prove that a Morse function will give a handle decomposition of a closed manifold. Let's use all the notation already set up (meaning critical points, values, attaching maps, dimension, Morse function, gradient-like vector field, etc). We just induct on the subscripts of critical points. We've already done the base case (it is… Continue reading Handle Decomposition

manifolds

## Handlebodies III

I keep naming my posts "handlebodies", so I think it is officially time to define what one is. A handlebody is a manifold obtained from $latex D^m$ by attaching various $latex \lambda$-handles successively. Thus a general handlebody will look like $latex D^m\cup D^{\lambda_1}\times D^{m-\lambda_1}\cup \cdots \cup D^{\lambda_n}\times D^{m-\lambda_n}$. If you're familiar with how to construct… Continue reading Handlebodies III

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

manifolds

## Handlebodies I

We now come to the main point of all these Morse theory posts. We want to somehow figure out what a closed manifold looks like based a Morse function that it admits (who knows how long I'll develop this theory, maybe we'll even get to how Smale proved the Poincare Conjecture in dimensions greater than… Continue reading Handlebodies I