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

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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