Handle decomposition

In mathematics, a handle decomposition of an m-manifold M is a unionConsider 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 S n {displaystyle S^{n}}   from the eyes of this decomposition—in particular the smooth structure near the 0-cell depends on the behavior of the characteristic map χ : D n → S n {displaystyle chi :D^{n} o S^{n}}   in a neighbourhood of S n − 1 {displaystyle S^{n-1}}  .When forming M union a j-handle H j {displaystyle H^{j}}  A handle presentation of a cobordism consists of a cobordism W where ∂ W = M 0 ∪ M 1 {displaystyle partial W=M_{0}cup M_{1}}   and an ascending unionGiven a Morse function f : M → R {displaystyle f:M o mathbb {R} }   on a compact boundaryless manifold M, such that the critical points { p 1 , … , p k } ⊂ M {displaystyle {p_{1},ldots ,p_{k}}subset M}   of f satisfy f ( p 1 ) < f ( p 2 ) < ⋯ < f ( p k ) {displaystyle f(p_{1})<f(p_{2})<cdots <f(p_{k})}  , and provided