Editing Handle decomposition (section)

==Terminology==

When forming ''M'' union a ''j''-handle <math>H^j</math>
<math display="block"> M \cup_f H^j = \left( M \sqcup (D^j \times D^{m-j}) \right) / \sim</math>

<math>f(S^{j-1} \times \{0\}) \subset M</math> is known as the '''attaching sphere'''.

<math>f</math> is sometimes called the '''framing''' of the attaching sphere, since it gives [[vector bundle|trivialization]] of its [[normal bundle]].

<math>\{0\}^j \times S^{m-j-1} \subset D^j \times D^{m-j} = H^j</math> is the '''belt sphere''' of the handle <math>H^j</math> in <math> M \cup_f H^j</math>.

A manifold obtained by attaching ''g'' ''k''-handles to the disc <math>D^m</math> is an ''' ''(m,k)''-handlebody of genus ''g'' '''.