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