= Morse handle-attachment theorem
{c}
Crossing one nondegenerate <critical point> of <Morse index> $\lambda$ in a compact sublevel interval attaches a <handle> $D^\lambda\times D^{m-\lambda}$. Between critical levels a suitably normalized <gradient flow> identifies the sublevels. The <Morse lemma> supplies the local quadratic model for the attachment. Retracting each <handle> to its <core disk> gives a <CW complex> model with one $\lambda$-cell per <critical point>. Closed <manifolds> have finite such decompositions; a noncompact <manifold> requires a proper bounded-below exhaustion for the corresponding statement.
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