Editing Handle decomposition (section)

In [[mathematics]], a '''handle decomposition''' of an ''m''-[[manifold]] ''M'' is a union
<math display="block">\emptyset = M_{-1} \subset M_0 \subset M_1 \subset M_2 \subset \dots \subset M_{m-1} \subset M_m = M</math>
where each <math>M_i</math> is obtained from <math>M_{i-1}</math> by the attaching of <math>i</math>-'''handles'''.   A handle decomposition is to a manifold what a [[CW complex|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 manifold]]s.  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]].

[[Image:Sphere with three handles.png|right|thumb|A 3-ball with three 1-handles attached.]]