A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
A Brief Introduction to Morse Theory
Gianmarco Molino
Mathematics Continued ConferenceUniversity of Connecticut
November 9, 2019
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Single Variable Calculus
In the single variable setting y = f (x), we called zeros of thederivative
f ′(a) = 0
critical points and we could test if they were extrema using thesecond derivative test{
f ′′(a) > 0 a is a minimum
f ′′(a) < 0 a is a maximum
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Multivariable Calculus
In the multivariable setting y = f (x1, x2, . . . , xn), we calledzeros of the gradient
∇f = ~0
critical points and we could test if they were extrema byconsidering the determinant of the Hessian
Hess(f ) =
fx1x1 fx1x2 · · · fx1xn
fx2x1. . . fx2xn
.... . .
...fxnx1 fxnx2 · · · fxnxn
and then {
det(Hess(f ))(~a) > 0 ~a is an extremum
det(Hess(f ))(~a) < 0 ~a is a saddle
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Why does this work?
We can consider this in terms of the eigenvalues of theHessian;
If all of the eigenvalues have the same sign, then the spaceis curved “in one direction”.
If some eigenvalues are positive and some are negative,then the space curves “upwards” in some directions and“downwards” in other directions.
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
What is Morse Theory?
Morse theory was initiated by Marston Morse in the 1920s.
The idea is to study manifolds by considering criticalpoints of functions defined on them.
Morse theory generalizes the early calculus ideas of usingthe derivative to detect local properties of a space, andthen to piece that information together to get a globalpicture of a space.
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Morse Functions
A smooth function f : M → R is called Morse if its criticalpoints are nondegenerate (that is, the Hessian of f isnonsingular.)
Remark: Nondegenerate critical points are necessarilyisolated.
The index λ(p) of a critical point p is the dimension of thenegative eigenspace of Hp(f ).
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Torus with height function
Consider the 2-dimensional torus T2 embedded in R3 and atangent plane:
Define f : T2 → R to be the height above the plane.
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Morse Indicies of the Torus
The function f has 4 critical points, a, b, c, d , withλ(a) = 0, λ(b) = λ(c) = 1, λ(d) = 2.
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Morse Lemma
Theorem (Lemma of Morse)
Let f ∈ C∞(M,R), and let p ∈ M be a nondegenerate criticalpoint of f . Then there exists a neighborhood U ⊂ M of p anda coordinate system (y1, . . . , yn) on U such that yi (p) = 0 forall 1 ≤ i ≤ n, and moreover
f = f (p)− (y1)2 − · · · − (yλ)2 + (yλ+1)2 + · · ·+ (yn)2
where λ = λ(p) is the index of p.
Corollary
If p ∈ M is a nondegenerate critical point of f , then it isisolated.
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Half-Space Deformations
Given f : M → R, define the ‘half-space’
Ma = f −1(−∞, a] = {x ∈ M : f (x) ≤ a}.
Theorem (Milnor)
Let f : M → R be C∞. If f −1([a, b]) is compact and containsno critical points of f ,
then Ma is diffeomorphic to Mb and furthermore Ma is adeformation retract of Mb.
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Half-Space Deformations
Theorem (Milnor)
Let f : M → R be C∞ and let p ∈ M be a (nondegenerate,isolated) critical point of f .
Set c = f (p) and λ = λ(p) to be the index of p.
Suppose there exists ε > 0 such that f −1([c − ε, c + ε]) iscompact and contains no critical points of f other than p.
Then for all sufficiently small ε, Mc+ε has the homotopy typeof Mc−ε with a λ-cell attached.
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Attaching λ-cells
The key observation is that when crossing a critical point, theMorse Lemma is applicable. It can be shown that attaching aλ-cell eλ to Mc−ε along the (y1, . . . , yλ) axis,
Mc−ε ∪ eλ ∼= Mc+ε.
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
CW-Complexes
Intuitively then, the manifold can be constructed from cellsdetermined by the indices of the critical points.
Theorem (Milnor)
If f : M → R is Morse and for all a ∈ R it holds that Ma iscompact, then M has the homotopy type of a CW complexwith one cell of dimension λ for each critical point with index λ.
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Simple Results
This is enough to get a few results. For example,
Theorem (Reeb)
Let M be a compact smooth manifold, and let f : M → R beMorse. If f has exactly two critical points, then M ishomeomorphic to a sphere.
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Betti Numbers
The Betti numbers are topological invariants. They are relatedto the classical Euler characteristic χ(M) by
χ(M) =n∑
k=0
(−1)kβk .
The Betti numbers can be interpreted directly:
β0 is the number of connected components of M.
β1 is the number of 1-dimensional “holes” (nontrivialloops) in M.
β2 is the number of 2-dimensional “cavities” (nontrivialspheres) in M.
and so on.
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Weak Morse Inequalities
Let f : M → R be Morse, and define the Morse numbers, Mk ,by
Mk = #{p ∈ M, df (p) = 0, λ(p) = k}
Theorem (Weak Morse Inequalities)
Let M be compact, βi be the Betti numbers of M, f : M → Rbe Morse, and Mk be the Morse numbers of f . Then
βk ≤ Mk
and moreover
χ(M) =n∑
k=0
(−1)kβk =n∑
k=0
(−1)kMk .
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Torus Example
The Weak Morse Inequalities give good estimates on the theBetti numbers. For example, we have for T2
β0 ≤ M0 = 1
β1 ≤ M1 = 2
β2 ≤ M2 = 1
χ(T2) = M0 −M1 + M2 = 1− 2 + 1 = 0,
using the height function from before.
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Applications
There are a number of important applications, including
Classification of compact 2-manifolds
h-cobordism Theorem
Lefschetz Hyperplane Theorem
Yang-Mills Theory
A BriefIntroductionto MorseTheory
GianmarcoMolino
MotivatingIdeas
Morse Theory
MotivatingExample
Attachingλ-cells
MorseInequalitiesand Euler’sNumber
Applicationsand FurtherReading
Further Reading
John Milnor, Morse Theory
Raoul Bott, Morse Theory Indomitable
Edward Witten, Supersymmetry and Morse Theory
Augustin Banyaga, Lectures on Morse Homology
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