An Introduction to Morse Theory Gianmarco Molino UConn Sigma - - PowerPoint PPT Presentation

An Introduction to Morse Theory

Gianmarco Molino

UConn Sigma Seminar

27 July, 2017

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A quick introduction to Differential Geometry

Geometry is the study of shape, size, relative position of figures, and the properties of space, and has been historically one of the major motivating reasons for the field of mathematics. In applying the methods of calculus to this, we arrive at the modern field of Differential Geometry. Our primary objects of interest are manifolds.

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A quick introduction to Differential Geometry

A manifold is a generalization of Euclidean geometry, defined as

• bjects that “locally look like” Euclidean Rn.

The first examples of a nontrivial manifold are surfaces in R3, such as spheres, tori, and surfaces of revolution. Some 2-dimensional manifolds can’t be embedded in R3, like the Klein bottle. We can also consider higher-dimensional manifolds, but it can be very hard to visualize these.

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A quick introduction to Differential Geometry

Moreover, since we are considering differential geometry, we want to consider smooth manifolds. Simply put, smooth manifolds are manifolds on which calculus can be done; there can be no sharp corners.

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A quick introduction to Differential Geometry

One idea to keep in mind: when working with manifolds, you have to jump back and forth between local and global properties. Locally (that is, in a small area around any point,) manifolds look just like Rn. We have coordinate systems, and most of the ideas from multivariable calculus can be carried forward in the way you expect. Globally (that is, on the whole manifold,) most of these ideas don’t usually work.

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What is Morse Theory?

In the following, let M be a closed smooth manifold of dimension n. Initiated by Marston Morse, 1920-1930. Study of critical points of smooth functions f : M → R. Attempts to recover topological (not dependent on calculus) information about M.

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1

Definitions

2

Motivating Example

3

First Results

4

Morse Inequalities

5

Existence Results

6

Applications and Further Reading

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Definitions

A smooth manifold M is a topological manifold with compatible smooth atlas. A critical point p ∈ M of a smooth function f : M → R is a zero of the differential df . The Hessian Hp(f ) of f at a critical point p ∈ M is the matrix of second derivatives. (Independent of coordinate system at critical points.)

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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 ).

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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.

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

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Morse Lemma

Nondegeneracy of critical points is a generalization of non-vanishing

• f the second derivative of functions f : R → R.

◮ Remember, the 2nd derivative test lets you decide if a critical point is a

local maximum or minimum if the 2nd derivative is nonzero.

We thus expect to be able to describe M in relation to these points.

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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.

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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.

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The gradient of f induces a local 1-parameter family of diffeomorphisms φt : M → M away from critical points. Thus allowing the points of Ma to flow along these gives the desired deformation retract. Remark: The condition that f −1([a, b]) be compact cannot be relaxed.

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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 ǫ > 0 such that f −1([c − ǫ, c + ǫ]) is compact and contains no critical points of f other than p. Then for all sufficiently small ǫ, Mc+ǫ has the homotopy type of Mc−ǫ with a λ-cell attached.

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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−ǫ along the (y1, . . . , yλ) axis, Mc−ǫ ∪ eλ ∼ = Mc+ǫ.

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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.

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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.

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Differential Forms

Recall the space Ωk(M) of differential k-forms over M, and the exterior derivative d : Ωk → Ωk+1, which gives rise to the deRham co-chain complex 0 → · · · d − → Ωk(M) d − → Ωk+1(M) d − → · · · → 0

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Betti Numbers

The associated cohomology group is the deRham cohomology group Hk

dR(M) = ker d : Ωk → Ωk+1

im d : Ωk−1 → Ωk and further we define the k-th Betti number of M, βk = dim Hk

dR(M).

This cohomology encodes topological information about the manifold algebraically, and is the starting point for fields such as Hodge Theory and Index Theory.

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The Betti numbers are topological invariants. They are related to the classical Euler characteristic χ(M) by χ(M) =

n

• k=0

(−1)kβk. Which is an explicit expression for the following lemma from Index Theory:

Lemma

Let D = d + δ be the Dirac operator for the Hodge Laplacian ∆ = D2 = dδ + δd. Then χ(M) = index(D) where index(D) = dim ker(D) − dim coker(D) denotes the analytic index.

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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. Note, the largest nonzero Betti number in a n-dimensional manifold M is βn.

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The Betti numbers have many other properties, and are of wide interest for many applications. They can also be defined for general topological spaces, not just smooth manifolds. Unfortunately, the Betti numbers can be remarkably difficult to compute directly. This is where Morse Theory provides a solution.

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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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Torus Example

The Weak Morse Inequalities give good estimates on the the Betti

• 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.

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Witten’s Proof

We sketch the idea of Edward Witten’s remarkable proof: By a result from Hodge Theory, βk = dim ker ∆: Ωk → Ωk. Let f be Morse. Then we define the ‘twisted exterior derivative’ dt = e−tf detf from which we can construct the ‘Witten Laplacian’ ∆t = dtδt + δtdt : Ωk → Ωk,

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There is an induced co-chain complex 0 → · · · dt − → Ωk(M) dt − → Ωk+1(M) d − → · · · → 0 which is isomorphic to the deRham complex, so that βk = dim ker ∆k = dim ker ∆k

t .

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But this is a remarkable improvement, leading to the conclusion that as t → ∞ the elements of the kernel of ∆t will concentrate around the critical points of f . Computations can then be approximated in local coordinates, leading to the Morse Inequalities.

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Strong Morse Inequalities

We can make the inequalities sharper.

Theorem (Strong 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 for any 0 ≤ k ≤ n, βk − βk−1 + · · · ± β0 ≤ Mk − Mk−1 + · · · ± M0

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Polynomial Morse Inequalities

Define the Poincar´ e Polynomial Pt = n

i=0 βiti and the Morse Polynomial

Mt = n

i=0 Miti.

Theorem (Polynomial Morse Inequalities)

Assumptions as before. For t ∈ R there exist some non-negative integers Qi such that Mt − Pt = (1 + t)

n−1

• i=0

Qiti

Lemma (Banyaga)

The Strong Morse Inequalities and the Polynomial Morse Inequalities are equivalent.

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Raoul Bott writes (Morse Theory Indomitable): “The (1 + t) term on the right gives this inequality much more power than it would have without it. The (1 + t) term feeds back information from the critical points of f to the topology of M.”

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Existence of Morse Functions

So, given a manifold M and a Morse function f we have nice results, but can we actually find Morse functions?

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• Yes. In fact, there is an ‘easy’ construction:

Theorem (Milnor)

Let M be a compact smooth manifold, and ι: M → RN be an embedding

• f M into RN. For p ∈ RN, define Lp : M → R by

Lp(q) = p − ι(q)2 where · is the standard Euclidean norm on RN. Then Lp is Morse for almost every p ∈ RN.

Corollary

On any compact smooth manifold M there exists a Morse function, for which each Ma is compact.

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Theorem (Milnor)

Let M be a smooth manifold, K ⊂ M compact, and k ≥ 0 an integer. Any bounded smooth function f : M → R can be uniformly approximated by a Morse function g. Furthermore, for 1 ≤ i ≤ k it is possible to choose g such that the i-th derivatives of g on K uniformly approximate the corresponding derivatives of f .

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Applications

There are a number of important applications, including Classification of compact 2-manifolds h-cobordism Theorem Lefschetz Hyperplane Theorem Yang-Mills Theory

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Openings

There is much active research deriving from Morse Theory: Index Theory Witten Helffer-Sj¨

• strand Theory

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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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