Editing Handle decomposition (section)

==Cobordism presentations==

A '''handle presentation of a cobordism''' consists of a cobordism ''W'' where <math>\partial W = M_0 \cup M_1</math> and an ascending union
<math display="block">W_{-1} \subset W_0 \subset W_1 \subset \cdots \subset W_{m+1} = W </math>
where {{math|''M''}} is {{mvar|m}}-dimensional, {{math|''W''}} is ''m+1''-dimensional, <math>W_{-1}</math> is diffeomorphic to <math>M_0 \times [0,1]</math> and <math>W_i</math> is obtained from <math>W_{i-1}</math> by the attachment of ''i''-handles.  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.