Editing Morse theory (section)

==Formal development==
For a real-valued [[smooth function]] <math>f : M \to \R</math> on a [[differentiable manifold]] <math>M,</math> the points where the [[Differential (calculus)|differential]] of <math>f</math> vanishes are called [[Critical point (mathematics)|critical points]] of <math>f</math> and their images under <math>f</math> are called [[Critical value (critical point)|critical value]]s.  If at a critical point <math>p</math> the matrix of second partial derivatives (the [[Hessian matrix]]) is non-singular, then <math>p</math> is called a '''{{em|{{visible anchor|non-degenerate critical point}}}}'''; if the Hessian is singular then <math>p</math> is a '''{{em|{{visible anchor|degenerate critical point}}}}'''.

For the functions
<math display="block">f(x)=a + b x+ c x^2+d x^3+\cdots</math>
from <math>\R</math> to <math>\R,</math> <math>f</math> has a critical point at the origin if <math>b = 0,</math> which is non-degenerate if <math>c \neq 0</math> (that is, <math>f</math> is of the form <math>a + c x ^2 + \cdots</math>) and degenerate if <math>c = 0</math> (that is, <math>f</math> is of the form <math>a + dx^3 + \cdots</math>). A less trivial example of a degenerate critical point is the origin of the [[monkey saddle]].

The '''[[Critical point (mathematics)#Several variables|index]]''' of a non-degenerate critical point <math>p</math> of <math>f</math> is the dimension of the largest subspace of the [[tangent space]] to <math>M</math> at <math>p</math> on which the Hessian is [[Positive-definite matrix#Negative-definite, semidefinite and indefinite matrices|negative definite]]. This corresponds to the intuitive notion that the index is the number of directions in which <math>f</math> decreases.  The degeneracy and index of a critical point are independent of the choice of the local coordinate system used, as shown by [[Sylvester's law of inertia|Sylvester's Law]].

===Morse lemma===
Let <math>p</math> be a non-degenerate critical point of <math>f \colon M \to \reals.</math>  Then there exists a [[Chart (topology)|chart]] <math>\left(x_1, x_2, \ldots, x_n\right)</math> in a [[Neighborhood (topology)|neighborhood]] <math>U</math> of <math>p</math> such that <math>x_i(p) = 0</math> for all <math>i</math> and
<math display="block">f(x) = f(p) - x_1^2 - \cdots - x_{\gamma}^2 + x_{\gamma +1}^2 + \cdots + x_n^2</math>
throughout <math>U.</math> Here <math>\gamma</math> is equal to the index of <math>f</math> at <math>p</math>. As a corollary of the Morse lemma, one sees that non-degenerate critical points are [[Isolated point|isolated]].  (Regarding an extension to the complex domain see [[Method of steepest descent#Complex Morse lemma|Complex Morse Lemma]]. For a generalization, see [[Morse–Palais lemma]]).

=== 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.

===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>

===Application to classification of closed 2-manifolds===
Morse theory has been used to classify closed 2-manifolds up to diffeomorphism. If <math>M</math> is oriented, then <math>M</math> is classified by its genus <math>g</math> and is diffeomorphic to a sphere with <math>g</math> handles: thus if <math>g = 0,</math> <math>M</math> is diffeomorphic to the 2-sphere; and if <math>g > 0,</math> <math>M</math> is diffeomorphic to the [[connected sum]] of <math>g</math> 2-tori. If <math>N</math> is unorientable, it is classified by a number <math>g > 0</math> and is diffeomorphic to the connected sum of <math>g</math> [[real projective space]]s <math>\mathbf{RP}^2.</math> In particular two closed 2-manifolds are homeomorphic if and only if they are diffeomorphic.<ref>{{cite book |last=Gauld |first=David B. |title=Differential Topology: an Introduction |series=Monographs and Textbooks in Pure and Applied Mathematics|volume= 72 |publisher= Marcel Dekker |year= 1982|isbn= 0824717090 |url-access= registration |url= https://archive.org/details/differentialtopo0000gaul }}</ref><ref>{{cite book|last=Shastri|first=Anant R.|url=https://books.google.com/books?id=-BrOBQAAQBAJ|title=Elements of Differential Topology|publisher=CRC Press|year=2011|isbn=9781439831601}}</ref>

===Morse homology===
[[Morse homology]] is a particularly easy way to understand the [[Homology (mathematics)|homology]] of [[smooth manifold]]s. It is defined using a generic choice of Morse function and [[Riemannian metric]].  The basic theorem is that the resulting homology is an invariant of the manifold (that is, independent of the function and metric) and isomorphic to the singular homology of the manifold; this implies that the Morse and singular [[Betti number]]s agree and gives an immediate proof of the Morse inequalities.  An infinite dimensional analog of Morse homology in [[symplectic geometry]] is known as [[Floer homology]].