Handle Decomposition - Morse Theoretic Viewpoint

Morse Theoretic Viewpoint

Given a Morse function on a compact boundaryless manifold M, such that the critical points of f satisfy, and provided

,

then for all j, is diffeomorphic to where I(j) is the index of the critical point . The index I(j) refers to the dimension of the maximal subspace of the tangent space where the Hessian is negative definite.

Provided the indices satisfy this is a handle decomposition of M, moreover, every manifold has such Morse functions, so they have handle decompositions. Similarly, given a cobordism with and a function 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.