Editing Handle decomposition (section)

==Motivation==
Consider the standard [[CW-complex|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 <math>S^n</math> from the eyes of this decomposition—in particular the smooth structure near the ''0''-cell depends on the behavior of the characteristic map <math>\chi : D^n \to S^n</math> in a neighbourhood of <math>S^{n-1} \subset D^n</math>.

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 neighborhood|tubular neighbourhood theorem]].  Given a point ''p'' in an <math>m</math>-manifold ''M'', its closed tubular neighbourhood <math>N_p</math> is [[Diffeomorphism|diffeomorphic]] to <math>D^m</math>, thus we have decomposed ''M'' into the disjoint union of <math>N_p</math> and <math>M \setminus \operatorname{int}(N_p)</math> glued along their common boundary.  The vital issue here is that the gluing map is a diffeomorphism.  Similarly, take a smooth embedded arc in <math>M \setminus \operatorname{int}(N_p)</math>, its tubular neighbourhood is diffeomorphic to <math>I \times D^{m-1}</math>.  This allows us to write <math>M</math> as the union of three manifolds, glued along parts of their boundaries:

(1) <math>D^m</math>,

(2) <math>I \times D^{m-1}</math>, and

(3) the complement of the open tubular neighbourhood of the arc in <math>M \setminus \operatorname{int}(N_p)</math>. 

Notice all the gluing maps are smooth maps—in particular when we glue <math>I \times D^{m-1}</math> to <math>D^m</math> the equivalence relation is generated by the embedding of <math>(\partial I)\times D^{m-1}</math> in <math>\partial D^m</math>, which is smooth by the [[tubular neighborhood|tubular neighbourhood theorem]].

Handle decompositions are an invention of [[Stephen Smale]].<ref>S. Smale, "On the structure of manifolds" Amer. J. Math. , 84 (1962) pp. 387–399</ref>  In his original formulation, '''the process of attaching a ''j''-handle to an ''m''-manifold ''M'' ''' assumes that one has a smooth embedding of <math>f : S^{j-1} \times D^{m-j} \to \partial M</math>. Let <math>H^j = D^j \times D^{m-j}</math>. The manifold <math>M \cup_f H^j</math> (in words, ''' ''M'' union a ''j''-handle along ''f'' ''') refers to the disjoint union of <math>M</math> and <math>H^j</math> with the identification of <math>S^{j-1} \times D^{m-j}</math> with its image in <math>\partial M</math>, i.e.,
<math display="block"> M \cup_f H^j = \left( M \sqcup (D^j \times D^{m-j}) \right) / \sim</math>
where the [[Quotient space (topology)|equivalence relation]] <math>\sim</math> is generated by <math>(p,x) \sim f(p,x)</math> for all <math>(p,x) \in S^{j-1} \times D^{m-j} \subset D^j \times D^{m-j}</math>.

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]].