Editing Handle decomposition (section)

==Morse theoretic viewpoint==

Given a [[Morse theory|Morse function]] <math>f : M \to \R</math> on a compact boundaryless manifold ''M'', such that the [[critical point (mathematics)|critical points]] <math>\{p_1, \ldots, p_k\} \subset M</math> of ''f'' satisfy <math>f(p_1) < f(p_2) < \cdots < f(p_k) </math>, and provided
<math display="block">t_0 < f(p_1) < t_1 < f(p_2) < \cdots < t_{k-1} < f(p_k) < t_k ,</math>
then for all ''j'', <math>f^{-1}[t_{j-1},t_{j}]</math> is diffeomorphic to <math>(f^{-1}(t_{j-1}) \times [0,1]) \cup H^{I(j)}</math> where ''I''(''j'') is the index of the critical point <math>p_{j}</math>. The ''index'' ''I(j)'' refers to the dimension  of the maximal subspace of the tangent space <math>T_{p_j}M</math> where the [[Hessian matrix|Hessian]] is negative definite.

Provided the indices satisfy <math>I(1) \leq I(2) \leq \cdots \leq I(k)</math>  this is a handle decomposition of ''M'', moreover, every manifold has such Morse functions, so they have handle decompositions.  Similarly, given a cobordism <math>W</math> with <math> \partial W = M_0 \cup M_1</math> and a function <math> f: W \to \R</math> which is Morse on the interior and constant on the boundary and satisfying the increasing index property, there is an induced handle presentation of the cobordism ''W''.

When ''f'' is a Morse function on ''M'', -''f'' is also a Morse function.  The corresponding handle decomposition / presentation is called the '''dual decomposition'''.