Handle decomposition
In mathematics, a handle decomposition of an m-manifold M is a union
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.
Consider the standard CW-decomposition of the n-sphere, with one zero cell and a single n-cell.
From the point of view of smooth manifolds, this is a degenerate decomposition of the sphere, as there is no natural way to see the smooth structure of
from the eyes of this decomposition—in particular the smooth structure near the 0-cell depends on the behavior of the characteristic map
The problem with CW-decompositions is that the attaching maps for cells do not live in the world of smooth maps between manifolds.
The germinal insight to correct this defect is the tubular neighbourhood theorem.
Given a point p in a manifold M, its closed tubular neighbourhood
The vital issue here is that the gluing map is a diffeomorphism.
Similarly, take a smooth embedded arc in
as the union of three manifolds, glued along parts of their boundaries: 1)
and 3) the complement of the open tubular neighbourhood of the arc in
Notice all the gluing maps are smooth maps—in particular when we glue
the equivalence relation is generated by the embedding of
, which is smooth by the tubular neighbourhood theorem.
Handle decompositions are an invention of Stephen Smale.
[1] In his original formulation, the process of attaching a j-handle to an m-manifold M assumes that one has a smooth embedding of
One says a manifold N is obtained from M by attaching j-handles if the union of M with finitely many j-handles is diffeomorphic to N. The definition of a handle decomposition is then as in the introduction.
Thus, a manifold has a handle decomposition with only 0-handles if it is diffeomorphic to a disjoint union of balls.
A connected manifold containing handles of only two types (i.e.: 0-handles and j-handles for some fixed j) is called a handlebody.
is sometimes called the framing of the attaching sphere, since it gives trivialization of its normal bundle.
A manifold obtained by attaching g k-handles to the disc
Whereas handle decompositions are the analogue for manifolds what cell decompositions are to topological spaces, handle presentations of cobordisms are to manifolds with boundary what relative cell decompositions are for pairs of spaces.
on a compact boundaryless manifold M, such that the critical points
The index I(j) refers to the dimension of the maximal subspace of the tangent space
this is a handle decomposition of M, moreover, every manifold has such Morse functions, so they have handle decompositions.
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.
