Motivation
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 in a neighbourhood of .
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 is diffeomorphic to, thus we have decomposed M into the disjoint union of and glued along their common boundary. The vital issue here is that the gluing map is a diffeomorphism. Similarly, take a smooth embedded arc in, its tubular neighbourhood is diffeomorphic to . This allows us to write as the union of three manifolds, glued along parts of their boundaries: 1) 2) 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 to the equivalence relation is generated by the embedding of in, which is smooth by the tubular neighbourhood theorem.
Handle decompositions are an invention of Stephen Smale. In his original formulation, the process of attaching a j-handle to an m-manifold M assumes that one has a smooth embedding of . Let . The manifold (in words, M union a j-handle along f ) refers to the disjoint union of and with the identification of with its image in, i.e.:
where the equivalence relation is generated by for all .
One says a manifold N is obtained from M by attaching j-handles, if M union 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.
Read more about this topic: Handle Decomposition
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