Editing Morse theory (section)

=== Fundamental theorems ===
A smooth real-valued function on a manifold <math>M</math> is a '''Morse function''' if it has no degenerate critical points.  A basic result of Morse theory says that almost all functions are Morse functions.  Technically, the Morse functions form an open, dense subset of all smooth functions <math>M \to \R</math> in the <math>C^2</math> topology.  This is sometimes expressed as "a typical function is Morse" or "a [[Generic property|generic]] function is Morse".

As indicated before, we are interested in the question of when the topology of <math>M^a = f^{-1}(-\infty, a]</math> changes as <math>a</math> varies.  Half of the answer to this question is given by the following theorem.

:'''Theorem.''' Suppose <math>f</math> is a smooth real-valued function on <math>M,</math> <math>a < b,</math> <math>f^{-1}[a, b]</math> is [[Compact space|compact]], and there are no critical values between <math>a</math> and <math>b.</math>  Then <math>M^a</math> is [[diffeomorphic]] to <math>M^b,</math> and <math>M^b</math> [[deformation retract]]s onto <math>M^a.</math> 

It is also of interest to know how the topology of <math>M^a</math> changes when <math>a</math> passes a critical point.  The following theorem answers that question.

:'''Theorem.'''  Suppose <math>f</math> is a smooth real-valued function on <math>M</math> and <math>p</math> is a non-degenerate critical point of <math>f</math> of index <math>\gamma,</math> and that <math>f(p) = q.</math>  Suppose <math>f^{-1}[q - \varepsilon, q + \varepsilon]</math> is compact and contains no critical points besides <math>p.</math> Then <math>M^{q + \varepsilon}</math> is [[homotopy equivalent]] to <math>M^{q - \varepsilon}</math> with a <math>\gamma</math>-cell attached.

These results generalize and formalize the 'rule' stated in the previous section.

Using the two previous results and the fact that there exists a Morse function on any differentiable manifold, one can prove that any differentiable manifold is a CW complex with an <math>n</math>-cell for each critical point of index <math>n.</math>  To do this, one needs the technical fact that one can arrange to have a single critical point on each critical level, which is usually proven by using [[gradient-like vector field]]s to rearrange the critical points.