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{{short description|Analyzes the topology of a manifold by studying differentiable functions on that manifold}}
{{redirect|Morse function|anharmonic oscillators|Morse potential}}

In [[mathematics]], specifically in [[differential topology]], '''Morse theory''' enables one to analyze the [[Topological space|topology]] of a [[manifold]] by studying [[differentiable function]]s on that manifold. According to the basic insights of [[Marston Morse]], a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find [[CW complex|CW structures]] and [[handle decomposition]]s on manifolds and to obtain substantial information about their [[Homology (mathematics)|homology]].

Before Morse, [[Arthur Cayley]] and [[James Clerk Maxwell]] had developed some of the ideas of Morse theory in the context of [[topography]]. Morse originally applied his theory to [[geodesic]]s ([[Critical point (mathematics)|critical points]] of the [[Hamiltonian mechanics|energy]] [[Functional (mathematics)|functional]] on the space of paths). These techniques were used in [[Raoul Bott]]'s proof of his [[Bott periodicity theorem|periodicity theorem]].

The analogue of Morse theory for complex manifolds is [[Picard–Lefschetz theory]].

==Basic concepts==
[[Image:Saddle point.png|thumb|right|A saddle point]]

To illustrate, consider a mountainous landscape surface <math>M</math> (more generally, a [[manifold]]). If <math>f</math> is the [[Function (mathematics)|function]] <math>M \to \mathbb{R}</math> giving the elevation of each point, then the [[inverse image]] of a point in <math>\mathbb{R}</math> is a [[contour line]] (more generally, a [[level set]]). Each [[Connected component (topology)|connected component]] of a contour line is either a point, a [[closed curve|simple closed curve]], or a closed curve with [[Singular point of a curve|double point(s)]].  Contour lines may also have points of higher order (triple points, etc.), but these are unstable and may be removed by a slight deformation of the landscape.  Double points in contour lines occur at [[saddle points]], or passes, where the surrounding landscape curves up in one direction and down in the other.

[[Image:Saddle contours.svg|thumb|left|Contour lines around a saddle point]]
Imagine flooding this landscape with water. When the water reaches elevation <math>a</math>, the underwater surface is <math>M^a \,\stackrel{\text{def}}{=}\, f^{-1}(-\infty, a]</math>, the points with elevation <math>a</math> or below. Consider how the topology of this surface changes as the water rises.  It appears unchanged except when <math>a</math> passes the height of a [[Critical point (mathematics)|critical point]], where the [[gradient]] of <math>f</math> is <math>0</math> (more generally, the [[Jacobian matrix]] acting as a [[linear map]] between [[Tangent space|tangent spaces]] does not have maximal [[Rank (linear algebra)|rank]]). In other words, the topology of <math>M^a</math> does not change except when the water either (1) starts filling a basin, (2) covers a saddle (a [[mountain pass]]), or (3) submerges a peak.

[[Image:3D-Leveltorus.png|thumb|right|The torus]]
To these three types of [[Critical point (mathematics)|critical points]]{{Em dash}}basins, passes, and peaks (i.e. minima, saddles, and maxima){{Em dash}}one associates a number called the index, the number of [[Linear independence|independent]] directions in which <math>f</math> decreases from the point. More precisely, the index of a non-degenerate critical point <math>p</math> of <math>f</math> is the [[Dimension (vector space)|dimension]] of the largest subspace of the [[tangent space]] to <math>M</math> at <math>p</math> on which the [[Hessian matrix|Hessian]] of <math>f</math> is negative definite. The indices of basins, passes, and peaks are <math>0, 1,</math> and <math>2,</math> respectively.

Considering a more general surface, let <math>M</math> be a [[torus]] oriented as in the picture, with <math>f</math> again taking a point to its height above the plane. One can again analyze how the topology of the underwater surface <math>M^a</math> changes as the water level <math>a</math> rises.

[[Image:3D-Cylinder and disk with handle.png|thumb|left|A cylinder (upper right), formed by <math>M^a</math> when <math>f(q)<a<f(r)</math>, is homotopy equivalent to a 1-cell attached to a disk (lower left).]]
[[Image:3D-Cylinder with handle and torus with hole.png|thumb|right|A torus with a disk removed (upper right), formed by <math>M^a</math> when <math>f(r)<a<f(s)</math>, is homotopy equivalent to a 1-cell attached to a cylinder (lower left).]]
Starting from the bottom of the torus, let <math>p, q, r,</math> and <math>s</math> be the four critical points of index <math>0, 1, 1,</math> and <math>2</math> corresponding to the basin, two saddles, and peak, respectively. When <math>a</math> is less than <math>f(p) = 0,</math> then <math>M^a</math> is the empty set.  After <math>a</math> passes the level of <math>p,</math> when <math>0 < a < f(q),</math> then <math>M^a</math> is a [[Disk (mathematics)|disk]], which is [[homotopy equivalent]] to a point (a 0-cell) which has been "attached" to the empty set. Next, when <math>a</math> exceeds the level of <math>q,</math> and <math>f(q) < a < f(r),</math> then <math>M^a</math> is a cylinder, and is homotopy equivalent to a disk with a 1-cell attached (image at left). Once <math>a</math> passes the level of <math>r,</math> and <math>f(r) < a < f(s),</math> then <math>M^a</math> is a torus with a disk removed, which is homotopy equivalent to a [[Cylinder (geometry)|cylinder]] with a 1-cell attached (image at right).  Finally, when <math>a</math> is greater than the critical level of <math>s,</math> <math>M^a</math> is a torus, i.e. a torus with a disk (a 2-cell) removed and re-attached.

This illustrates the following rule: the topology of <math>M^{a}</math> does not change except when <math>a</math> passes the height of a critical point; at this point, a <math>\gamma</math>-cell is attached to <math>M^{a}</math>, where <math>\gamma</math> is the index of the point. This does not address what happens when two critical points are at the same height, which can be resolved by a slight perturbation of <math>f.</math> In the case of a landscape or a manifold [[Embedding|embedded]] in [[Euclidean space]], this perturbation might simply be tilting slightly, rotating the coordinate system.

One must take care to make the critical points non-degenerate. To see what can pose a problem, let <math>M = \R</math> and let <math>f(x) = x^3.</math> Then <math>0</math> is a critical point of <math>f,</math> but the topology of <math>M^{a}</math> does not change when <math>a</math> passes <math>0.</math> The problem is that the second derivative is <math>f''(0) = 0</math>{{Em dash}}that is, the [[Hessian matrix|Hessian]] of <math>f</math> vanishes and the critical point is degenerate. This situation is unstable, since by slightly deforming <math>f</math> to <math>f(x) = x^3 +\epsilon x</math>, the degenerate critical point is either removed (<math>\epsilon>0</math>) or breaks up into two non-degenerate critical points (<math>\epsilon<0</math>).

==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 : M \to R.</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]].

==Morse–Bott theory==
The notion of a Morse function can be generalized to consider functions that have nondegenerate manifolds of critical points. A '''{{visible anchor|Morse–Bott function|Morse–Bott function}}''' is a smooth function on a manifold whose [[critical set]] is a closed submanifold and whose Hessian is non-degenerate in the normal direction. (Equivalently, the kernel of the Hessian at a critical point equals the tangent space to the critical submanifold.) A Morse function is the special case where the critical manifolds are zero-dimensional (so the Hessian at critical points is non-degenerate in every direction, that is, has no kernel).

The index is most naturally thought of as a pair
<math display="block">\left(i_-, i_+\right),</math>
where <math>i_-</math> is the dimension of the unstable manifold at a given point of the critical manifold, and <math>i_+</math> is equal to <math>i_-</math> plus the dimension of the critical manifold. If the Morse–Bott function is perturbed by a small function on the critical locus, the index of all critical points of the perturbed function on a critical manifold of the unperturbed function will lie between <math>i_-</math> and <math>i_+.</math>

Morse–Bott functions are useful because generic Morse functions are difficult to work with; the functions one can visualize, and with which one can easily calculate, typically have symmetries. They often lead to positive-dimensional critical manifolds. [[Raoul Bott]] used Morse–Bott theory in his original proof of the [[Bott periodicity theorem]].

[[Round function]]s are examples of Morse–Bott functions, where the critical sets are (disjoint unions of) circles.

[[Morse homology]] can also be formulated for Morse–Bott functions; the differential in Morse–Bott homology is computed by a [[spectral sequence]].  Frederic Bourgeois sketched an approach in the course of his work on a Morse–Bott version of symplectic field theory, but this work was never published due to substantial analytic difficulties.

==See also==

{{Div col|colwidth=20em}}
* {{annotated link|Almgren–Pitts min-max theory}}
* {{annotated link|Digital Morse theory}}
* {{annotated link|Discrete Morse theory}}
* {{annotated link|Jacobi set}}
* {{annotated link|Lagrangian Grassmannian}}
* {{annotated link|Lusternik–Schnirelmann category}}
* {{annotated link|Morse–Smale system}}
* {{annotated link|Mountain pass theorem}}
* {{annotated link|Sard's lemma}}
* {{annotated link|Stratified Morse theory}}
{{Div col end}}

==References==

{{reflist|group=note}}
{{reflist}}

== Further reading ==

* {{cite journal|last=Bott|first=Raoul|author-link=Raoul Bott|year=1988|url=http://www.numdam.org/item?id=PMIHES_1988__68__99_0|title=Morse Theory Indomitable|journal=[[Publications Mathématiques de l'IHÉS]]|volume=68|pages=99–114|doi=10.1007/bf02698544|s2cid=54005577}}
* {{cite journal|last=Bott|first=Raoul|author-link=Raoul Bott|year=1982|title=Lectures on Morse theory, old and new|journal=[[Bulletin of the American Mathematical Society]]|series=(N.S.)|volume=7|issue=2|pages=331–358|doi=10.1090/s0273-0979-1982-15038-8|doi-access=free}}
* {{cite journal|last=Cayley|first=Arthur|year=1859|url=http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf|title=On Contour and Slope Lines|journal=[[Philosophical Magazine|The Philosophical Magazine]]|volume=18|issue=120|pages=264–268}}
* {{cite arXiv|last=Guest|first=Martin|year=2001|eprint=math/0104155|title=Morse Theory in the 1990s}} 
* {{cite book|last=Hirsch|first=M.|title=Differential Topology|year=1994|edition=2nd|publisher=Springer}}
* {{Kosinski Differential Manifolds 2007}} <!-- {{sfn|Kosinski|2007|p=}} -->
* {{Lang Fundamentals of Differential Geometry}} <!-- {{sfn|Lang|1999|p=}} -->
* {{cite book|last=Matsumoto|first=Yukio|year=2002|title=An Introduction to Morse Theory}}
* {{cite journal|last=Maxwell|first=James Clerk|year=1870|url=https://www.maths.ed.ac.uk/~v1ranick/surgery/hilldale.pdf|title=On Hills and Dales|journal=The Philosophical Magazine|volume=40|issue=269|pages=421–427}}
* {{cite book|last=Milnor|first=John|author-link=John Milnor|title=Morse Theory|publisher=Princeton University Press|year=1963|isbn=0-691-08008-9}} A classic advanced reference in mathematics and mathematical physics.
* {{cite book|last=Milnor|first=John|year=1965|title=Lectures on the h-cobordism theorem|url=https://www.maths.ed.ac.uk/~v1ranick/surgery/hcobord.pdf}}
* {{cite book|last=Morse|first=Marston|year=1934|title=The Calculus of Variations in the Large|series=American Mathematical Society Colloquium Publication|volume=18|location=New York}}
* {{cite book|last=Schwarz|first=Matthias|title=Morse Homology|url=https://archive.org/details/morsehomology0000schw|url-access=registration|publisher=Birkhäuser|year=1993|isbn=9780817629045}}

{{Manifolds}}

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[[Category:Smooth functions]]