Editing Morse theory (section)

===Morse inequalities===
Morse theory can be used to prove some strong results on the homology of manifolds.  The number of critical points of index <math>\gamma</math> of <math>f : M \to \R</math> is equal to the number of <math>\gamma</math> cells in the CW structure on <math>M</math> obtained from "climbing" <math>f.</math> Using the fact that the alternating sum of the ranks of the homology groups of a topological space is equal to the alternating sum of the ranks of the chain groups from which the homology is computed, then by using the cellular chain groups (see [[cellular homology]]) it is clear that the [[Euler characteristic]] <math>\chi(M)</math> is equal to the sum
<math display="block">\sum(-1)^\gamma C^\gamma\, = \chi(M)</math>
where <math>C^{\gamma}</math> is the number of critical points of index <math>\gamma.</math> Also by cellular homology, the rank of the <math>n</math><sup>th</sup> homology group of a CW complex <math>M</math> is less than or equal to the number of <math>n</math>-cells in <math>M.</math> Therefore, the rank of the <math>\gamma</math><sup>th</sup> homology group, that is, the [[Betti number]] <math>b_\gamma(M)</math>, is less than or equal to the number of critical points of index <math>\gamma</math> of a Morse function on <math>M.</math> These facts can be strengthened to obtain the '''{{em|{{visible anchor|Morse inequalities}}}}''':
<math display="block">C^\gamma -C^{\gamma -1} \pm \cdots + (-1)^\gamma C^0 \geq b_\gamma(M)-b_{\gamma-1}(M) \pm \cdots + (-1)^\gamma b_0(M).</math>

In particular, for any
<math display="block">\gamma \in \{0, \ldots, n = \dim M\},</math>
one has
<math display="block">C^\gamma \geq b_\gamma(M).</math>

This gives a powerful tool to study manifold topology. Suppose on a closed manifold there exists a Morse function <math>f : M \to \R</math> with precisely ''k'' critical points. In what way does the existence of the function <math>f</math> restrict <math>M</math>? The case <math>k = 2</math> was studied by [[Georges Reeb]] in 1952; the [[Reeb sphere theorem]] states that <math>M</math> is homeomorphic to a sphere <math>S^n.</math> The case <math>k = 3</math> is possible only in a small number of low dimensions, and ''M'' is homeomorphic to an [[Eells–Kuiper manifold]].
In 1982 [[Edward Witten]] developed an analytic approach to the Morse inequalities by considering the [[de Rham complex]] for the perturbed operator 
<math>d_t = e^{-tf} d e^{tf}.</math><ref>{{cite journal|last=Witten |first=Edward |title=Supersymmetry and Morse theory |journal=[[Journal of Differential Geometry|J. Differential Geom.]] |volume=17 |year=1982 |issue=4 |pages=661–692 |doi=10.4310/jdg/1214437492 |doi-access=free}}</ref><ref>{{cite book|last=Roe|first= John|title= Elliptic Operators, Topology and Asymptotic Method |edition=2nd |series=Pitman Research Notes in Mathematics Series |volume= 395 |publisher= Longman |year= 1998 |isbn= 0582325021}}</ref>