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Aug 08, 2020

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

A Brief Introduction to Morse Theory

Gianmarco Molino

Mathematics Continued Conference University of Connecticut

November 9, 2019

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

Single Variable Calculus

In the single variable setting y = f (x), we called zeros of the derivative

f ′(a) = 0

critical points and we could test if they were extrema using the second derivative test{

f ′′(a) > 0 a is a minimum

f ′′(a) < 0 a is a maximum

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

Multivariable Calculus

In the multivariable setting y = f (x1, x2, . . . , xn), we called zeros of the gradient

∇f = ~0

critical points and we could test if they were extrema by considering 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 Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

Why does this work?

We can consider this in terms of the eigenvalues of the Hessian;

If all of the eigenvalues have the same sign, then the space is 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 Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

What is Morse Theory?

Morse theory was initiated by Marston Morse in the 1920s.

The idea is to study manifolds by considering critical points of functions defined on them.

Morse theory generalizes the early calculus ideas of using the derivative to detect local properties of a space, and then to piece that information together to get a global picture of a space.

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

Morse Functions

A smooth function f : M → R is called Morse if its critical points are nondegenerate (that is, the Hessian of f is nonsingular.)

Remark: Nondegenerate critical points are necessarily isolated.

The index λ(p) of a critical point p is the dimension of the negative eigenspace of Hp(f ).

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

Torus with height function

Consider the 2-dimensional torus T2 embedded in R3 and a tangent plane:

Define f : T2 → R to be the height above the plane.

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

Morse Indicies of the Torus

The function f has 4 critical points, a, b, c, d , with λ(a) = 0, λ(b) = λ(c) = 1, λ(d) = 2.

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

Morse Lemma

Theorem (Lemma of Morse)

Let f ∈ C∞(M,R), and let p ∈ M be a nondegenerate critical point of f . Then there exists a neighborhood U ⊂ M of p and a coordinate system (y1, . . . , yn) on U such that yi (p) = 0 for all 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 is isolated.

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

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 contains no critical points of f ,

then Ma is diffeomorphic to Mb and furthermore Ma is a deformation retract of Mb.

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

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 � > 0 such that f −1([c − �, c + �]) is compact and contains no critical points of f other than p.

Then for all sufficiently small �, Mc+� has the homotopy type of Mc−� with a λ-cell attached.

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

Attaching λ-cells

The key observation is that when crossing a critical point, the Morse Lemma is applicable. It can be shown that attaching a λ-cell eλ to Mc−� along the (y1, . . . , yλ) axis,

Mc−� ∪ eλ ∼= Mc+�.

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

CW-Complexes

Intuitively then, the manifold can be constructed from cells determined by the indices of the critical points.

Theorem (Milnor)

If f : M → R is Morse and for all a ∈ R it holds that Ma is compact, then M has the homotopy type of a CW complex with one cell of dimension λ for each critical point with index λ.

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

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 be Morse. If f has exactly two critical points, then M is homeomorphic to a sphere.

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

Betti Numbers

The Betti numbers are topological invariants. They are related to 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” (nontrivial loops) in M.

β2 is the number of 2-dimensional “cavities” (nontrivial spheres) in M.

and so on.

A Brief Introduction to Morse Theory

Gianmarco Molino

Motivating Ideas

Morse Theory

Motivating Example

Attaching λ-cells

Morse Inequalities and Euler’s Number

Applications and Further Reading

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 → R be 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

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