Handle decomposition

Summary

A handle decomposition builds a manifold from a ball, or from a given boundary, by attaching handles $D^k \times D^{n-k}$ one after another. Each handle is glued along part of its boundary. It is the smooth analogue of a CW structure: a $k$-handle plays the role of a $k$-cell, thickened to full dimension. Every compact smooth manifold has a handle decomposition, obtained from a Morse function, with one $k$-handle for each critical point of index $k$. Handle decompositions are the main tool in the proof of the h-cobordism theorem and in surgery theory. In dimensions $3$ and $4$ they underlie Kirby calculus.