Handle decomposition

In mathematics, a handle decomposition of an m-manifold M is a union $${\displaystyle \emptyset =M_{-1}\subset M_{0}\subset M_{1}\subset M_{2}\subset \dots \subset M_{m-1}\subset M_{m}=M}$$ where each ${\displaystyle M_{i}}$ is obtained from ${\displaystyle M_{i-1}}$ by the attaching of ${\displaystyle i}$-handles. A handle decomposition is to a manifold what a CW-decomposition is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of smooth manifolds. Thus an i-handle is the smooth analogue of an i-cell. Handle decompositions of manifolds arise naturally via Morse theory. The modification of handle structures is closely linked to Cerf theory.