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A Brief Introduction to Morse Theory - · PDF file Morse Theory Motivating Example Attaching -cells Morse Inequalities and Euler’s Number Applications and Further Reading Morse Functions

Aug 08, 2020

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  • 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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