AN INVITATION TO MORSE THEORY

ZHENGJUN LIANG

The following work is dedicated to Shizi Liu.

Abstract. The following report serves as a tangible proof of work for Math197 of University of California, Los Angeles, mentored by professor Kefeng Liu.

The official course name of Math 197 is Individual Studies. Primarily based

on Morse Theory by John Milnor, the report investigates the local behavior offunctions around non-degenerate critical points (Lemma of Morse), and studies

homotopy type changes with respect to critical values. Eventually the report

gives some application of Morse Theory in determining homeomorphism, andgives concrete examples of functions with no degenetate critical points.

Contents

1. Introduction 22. Definitions and Lemmas 22.1. Lemma of Morse 22.2. 1-Parameter Group of Diffeomorphisms 53. Homotopy Type 63.1. A Prerequisite on CW-Complex 63.2. Homotopy Type in Terms of Critical Values 74. Applications 115. Manifolds in Euclidean Space 126. Reflections 14Acknowledgments 14References 15

Date: December 14, 2018.

1

2 ZHENGJUN LIANG

1. Introduction

Morse theory is a generalization of calculus of variations which depicts the rela-tionship between the critical points of a smooth real-valued function on a manifoldand the global topology of the manifold.

In this report, we give an introduction to Morse theory by primarily analyzingconcepts presented in John Milnor’s classic Morse Theory. As other books byMilnor, this book bears several appealing features:

(1) It is concise. The whole book has only 153 pages, and a large proportion ofthe volume is devoted to introducing necessary prerequisites. I personallyenjoy concise books, since I will have more opportunities to compemplateand interact with the text. Verifying the occasionally omitted details inthe proofs is also a great exercise to me. In fact, many inspirations of thereport come when I rewrite the proofs with more details.

(2) It builds up the theory beautifully. The book first introduce necessary toolsin Morse theory, which are quite elementary for the most part, and thenprove beautiful results based on them. For example, using basic resultsin calculus and linear algebra, the book proves Morse lemma, which is apowerful result that fully depicts the behavior of smooth functions aroundnon-degenerate critical points. Also, the theorems developed in homotopytype changes around critical points leads to the elegant Reeb’s theorem.

Due to limitation of time and ability, this report focuses on Part I: Non-DegenerateSmooth Functions on a Manifold of the reference text. The parts II and III of thebook are more geometrically flavored by developing Morse theory in the context ofRiemannian geometry. The last part of the book gives applications in Lie Groupsand Symmetric Spaces, all of which we shall revisit in the future.

2. Definitions and Lemmas

Definition 2.1 (Attaching k-cell). Let Y be a topological space, ek = {x ∈ Rk :|x| ≤ 1}, Sk−1 = {x ∈ Rk : |x| = 1}, and g : Sk−1 → Y be a continuous mapping.Then Y ∪g ek, Y with a k-cell attached by g, is obtained by first taking the disjointunion of Y and ek, and then identify each x ∈ Sk−1 with g(x) ∈ Y . This can beobserved in the following graph.

Figure 1. Attaching a 1-Cell

2.1. Lemma of Morse.

Definition 2.2 (Critical Point). A critical point p is called non-degenerate if andonly if the matrix

(∂2f

∂xi∂xj(p))i,j

AN INVITATION TO MORSE THEORY 3

is non-singular.

Remark 2.3. It can be verified that non-degeneracy is coordinate-independent.

Definition 2.4 (Hessian). If p is a critical point of f we define a symmetric bilinearfunctional f∗∗ on TMp, the tangent space at p, called the Hessian of f at p. If v, w ∈TMp then v and w have extensions v and w to vector fields (here extension to vectorfields just means any vector field v such that vp = v). We let f∗∗(v, w) = vp(w(f))(where w(f) means the directional derivative of f in the direction w )

Proposition 2.5. f∗∗ is well-defined, i.e.

(1) f∗∗ is a symmetric bilinear form.(2) f∗∗ is independent of the extensions chosen.

Proof. (1) vp(w) − wp(v(f)) = [v, w]p(f), where [v, w] is the Lie bracket of vand w. For any smooth vector fields X and Y on an n-manifold M , [X,Y ]is again a smooth vector field on M . Then [v, w](f) is just directionalderivative of f at the direction [v, w], and [v, w]p(f) = 0 since p is a criticalpoint of f . Thus we finish the first statement.

(2) Notice that vp(w(f)) = v(w(f)) is independent of the extension v of v,while wp(v(f)) is independent of the extension w of w. Then suppose thereis some v′, w′ extensions of v and w, then

v′p(w′(f))− vp(w(f)) = v(w′(f))− v(w(f)) (1)

w′p(v′(f))− wp(v(f)) = w(v′(f))− w(v(f)) (2)

Now

(1) + (2) = v(w′(f))− v(w(f)) + w(v′(f))− w(v(f))

= (v(w′(f))− w(v(f)) + (w(v′(f))− v(w(f))

= (vp(w′(f))− w′p(v(f)) + (wp(v

′(f))− v′p(w(f)) = 0

and similarly we can verify that (1)− (2) = 0 as well. Then (1) = (2) = 0and we show that the definition of f∗∗ is independent of the extensions vand w.

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If (x1, ..., xn) is a local coordinate system and v =∑ai

∂

∂xi

∣∣∣∣p

, w =∑ai

∂

∂xj

∣∣∣∣p

,

we can take w =∑bj

∂

∂xjwhere bj is a constant. Then

f∗∗(v, w) = v(w(f))(p) = v(∑

bj∂f

∂xj) =

∑i,j

aibj∂2f

∂xi∂xj(p)

Then the matrix (∂2f

∂xi∂xj(p)) represents the bilinear functional f∗∗ at p with respect

to the basis

ai∂

∂x1

∣∣∣∣p

, ..., ai∂

∂xn

∣∣∣∣p

Definition 2.6. The index of a bilinear functinoal H on a vector space V is definedto be the maximal dimension of a subspace of V such that H is negative definite;the nullity is the dimension of the null space, i.e. the subspace consisting of allv ∈ V such that H(v, w) = 0 for every w ∈ V .

4 ZHENGJUN LIANG

Remark 2.7. The point p is obviously a non-degenerate critical point of f if andonly if f∗∗ on TMp has nullity 0.

Usually the index of f∗∗ on TMp is referred to simply as the index of f at p. Alemma by Morse shows that the behavior of f at p can be completely characterizedby the index at p.

We first prove a useful lemma:

Lemma 2.8. Let f be a C∞ function in a convex neighborhood V of 0 in Rn withf(0) = 0. Then

f(x1, .., xn) =

n∑i=1

xigi(x1, ..., xn)

for some suitable C∞ functions gi defined in V , with gi(0) =∂f

∂xi(0).

Proof. Observe that

f(x1, ..., xn) =

∫ 1

0

df(tx1, ..., txn)

dtdt =

∫ 1

0

n∑i=1

∂f

∂xi(tx1, ..., txn) · xi dt

Interchanging the order of the integral and summation, we can let

gi(x1, ..., xn) =

∫ 1

0

∂f

∂xi(tx1, ..., txn) · xi dt

and we finish our proof. �

Lemma 2.9 (Morse). Let p be a non-degenerate critical point for f . Then thereis a local coordinate system (y1, ..., yn) in a neighborhood U of p with y1(p) = 0 forall i and such that the identity

f = f(p)− (y1)2 − ...− (yλ)2 + (yλ+1)2 + ...+ (yn)2

holds throughout U , where λ is the index of f at p.

Proof. The proof is divided into two steps: we first show that if f has such anexpression, then λ is the index of f at p, and then show that a suitable coordinatesystem (y1, ..., yn) exists.

Step 1: For any coordinate system (z1, ..., zp), if

f(q) = f(p)− (z1(q))2 − ...− (zλ(q))2 + (zλ+1(q))2 + ...+ (zn(q))2

Then we have

∂2f

∂z1∂zj(p) =

−2 i = j ≤ λ2 i = j > λ0 otherwise

which shows that f∗∗ with respect to the basis∂

∂z1|p, ...,

∂

∂zn|p is

−2 . . . 0 0 . . . 0...

. . ....

.... . .

...0 . . . −2 0 . . . 00 . . . 0 2 . . . 0...

. . ....

.... . .

...0 . . . 0 0 . . . 2

AN INVITATION TO MORSE THEORY 5

Therefore there is a subspace of TMp of dimension λ where f∗∗ is negativedefinite, and a subspace of dimension n−λ such that f∗∗ is positive definite.If a subspace of TMp has dimension > λ, then it must intersect V , whichis clearly impossible. Then λ is the index of f∗∗.

Step 2: Without generality we can assume p is the origin of Rn and thatf(p) = f(0) = 0. By previous lemma we can write

f(x1, ..., xn) =

n∑j=1

xjgj(x1, ..., xn)

for (x1, ..., xn) in some neighborhood of 0. Since 0 is assumed to be a critical

point, we have gj(0) =∂f

∂xj(0) = 0. Therefore, applying previous lemma

to gj again we have gj(x1, ..., xn) =∑ni=1 xihij(x1, ..., xn) for smooth hij .

It follows that

f(x1, ..., xn) =

n∑i,j=1

xixjhij(x1, ..., xn)

We can assume that hij = hji, since we can write hij =1

2(hij + hji), and

then hij = hji. Moreover, the matrix (hij(0)) = (1

2

∂2f

∂x1∂xj(0)), which is

non-singular since p is a non-degenerate critical point.We can then show there is a non-singular transformation of the coor-

dinate functions which gives us the desired expression for f in a perhapssmaller neighborhood of 0 by imitating the usual diagonalization proof forquadratic forms. (i.e. principle axis theorem). Then we complete our proof.

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Corollary 2.10. Non-denegerate critical points are isolated.

Proof. Suppose p is a non generate critical point of f . Then the lemma of Morsetells us that there is a local coordinate system (y1, ..., yn) in a neighborhood U ofp with yi(p) = 0 and

f = f(p)− (y1)2 − ...− (yλ)2 + (yλ+1)2 + ...+ (yn)2

Then for any x ∈ U that is not p, there is some i such that yi(x) 6= 0 and thus∂f

∂yi(x) 6= 0 and thus x is not critical point. Then we finish the proof. �

2.2. 1-Parameter Group of Diffeomorphisms.

Definition 2.11. A 1-parameter group of diffeomorphism of a manifold M is aC∞ map such that

(1) for each t ∈ R, ϕt : M →M defined by ϕt(q) = ϕ(t, q) is a diffeomorphismof M onto itself.

(2) for all t, s ∈ R we have ϕt+s = ϕt ◦ ϕsDefinition 2.12. Given a 1-parameter group ϕ of diffeomorphism of M we definea vector field X on M as follows: for every smooth real valued function f let

Xq(f) = limh→0

f(ϕh(q))− f(q)

h

6 ZHENGJUN LIANG

This field X is said to generate the group ϕ.

Lemma 2.13. A smooth vector field on M which vanishes outside of a compactset K ⊂M generates a unique 1-parameter group of diffeomorphism of M .

3. Homotopy Type

3.1. A Prerequisite on CW-Complex.

Definition 3.1 (Cell). An n-dimensional closed cell is the image of an n-dimensionalclosed ball under an attaching map; an n-dimensional open cell is a topological spacethat is homeomorphic to the n-dimensional open ball.

Definition 3.2 (CW-Complex). A CW-complex is a Hausdorff space X togetherwith a partition of X into open cells (of perhaps varying dimension) that satisfiestwo additional properties:

(1) For each n-dimensional open cell C in the partition of X, there is a contin-uous f from an n-dimensional closed ball to X such that• the restriction of f to the interior of the closed ball is a homeomorphism

onto the cell C• the image of the boundary of a closed ball is contained in the union of

a finite number of elements of the partition, each having cell dimension< n.

(2) A subset of X is closed if and only if it meets the closure of each cell in aclosed set.

Remark 3.3. In the definition CW-complex, C stands for ”closure finite”, meaningthat each closed cell is covered by a finite union of open cells, and W stands forweak topology.

Remark 3.4. CW-complex can be constructed by iteratively attaching k-cells. Adetailed explanation is seen in standard algebraic topology text such as AlgebraicTopology, Hatcher.

Definition 3.5 (Skeleton). If the largest dimension of any of the cells is n, then theCW-complex is said to have dimension n. If there is no bound of such dimensions,the CW-complex is said to be infinite dimensional. An n-skeleton of a CW complexis the union of the cells whose dimension is at most n.

Example 3.6. We give some common examples of CW-complex:

(1) The standard CW structure on the real numbers has as 0-skeleton theintegers Z and as 1-cells the intervals {[n, n + 1] : n ∈ Z}. Similarly, thestandard CW structure on Rn has cubical cells that are products of the 0and 1-cells from R. This is the standard cubic lattice cell structure on Rn.

(2) A polyhedron is naturally a CW complex.(3) A graph is a 1-dimensional CW complex.

Definition 3.7 (Cellular Map). If X and Y are CW-complexes, and f : X → Yis a continuous map, then f is said to be cellular if f takes the n-skeleton of X tothe n-skeleton of Y for all n, i.e. if f(Xn) ⊆ Y n for all n.

Theorem 3.8 (Cellular Approximation Theorem). Any continuous map f : X →Y between CW-complexes X and Y is homotopic to a cellular map, and if f isalready cellular on a subcomplex A of X, then we can furthermore choose the ho-motopy to be stationary on A.

AN INVITATION TO MORSE THEORY 7

Since it is not the main focus of this report, a detailed proof will not be given.

3.2. Homotopy Type in Terms of Critical Values. In this section we shallassume f to be a real valued function on a manifold M and let Ma := f−1(−∞, a] ={p ∈ M : f(p) ≤ a}. We now introduce a theorem that relates critical value andhomotopy type changes.

Theorem 3.9. Let f be a smooth real valued function on a manifold M . Let a < band suppose that the set f−1[a, b] is compact and contains no critcal point of f .Then Ma is diffeomorphic to M b. Furthermore, Ma is a deformation retract ofM b, so that the inclusion map Ma →M b is a homotopy equivalence.

The idea of the proof is to push M b down to Ma along the orthogonal trajectoriesof the hypersurfaces f = constant.

Figure 2. The Idea of the Proof

Proof. Part 1: Choose a Riemannian metric on M and let 〈X,Y 〉 denote the innerproduct of two tangent vectors as determined by this metric. The gradient of f isthe vector field grad f on M which is characterized by the identity

〈X, grad f〉 = X(f)

where X(f) is the directional derivative of f along X. This vector field grad fvanishes precisely at the critical points of f . Now suppose c : R → M is a curve

with velocity vectordc

dt, by chain rule we have

〈dcdt, grad f〉 =

d(f ◦ c)dt

Then let ρ : M → R be a smooth function which is equal to1

〈grad f, grad f〉throughout the compact set f−1[a, b]; and which vanishes outside of a compactneighborhood of this set. Then the vector field X, defined by

Xq = ρ(q)(grad f)q

satisfies the condition of lemma 2.6. Then X generates a 1-parameter group ofdiffeomorphisms

ϕt : M →M

For fixed q ∈M consider the function t→ f(ϕt(q)). Then

df(ϕt(q))

dt= 〈dϕt(q)

dt, grad f〉 =

d(f ◦ ϕt)dt

= 〈X, grad f〉 = ρ(q)〈grad f, grad f〉 ≤ 1 (1)

8 ZHENGJUN LIANG

and equality holds when ϕt(q) ∈ f−1[a, b]. We now claim that ϕb−a : M → Mcarries Ma diffeomorphically onto M b.

By (1), for any d ≥ 0 and m ∈M ,

0 ≤ f(ϕd(m))− f(m) =

∫ b

0

df(ϕd(m))

dtdt ∈ [0, d] (2)

Then observe that ϕd(Mh) ⊂ Mh+d, ϕ−d(M

h) ⊂ Mh, and therefore ϕb−a(Ma) ⊂M b and ϕa−b(f

−1[a, b]) ⊂ Ma. Pick m ∈ f−1[a, b], we let m = a + δ for 0 ≤ δ ≤b−a. Then we want to show that f(ϕ−s(m)) = a+δ−s for all s ∈ [0, δ]. To begin,we notice that we can use a similar formula to (2) to show that f(ϕ−s(m))−f(m) ∈[−s, 0], and thus

f(m)− s ≤ f(ϕ−s(m)) ≤ f(m) ≤ bfor any s > 0. In particular, when s ∈ [0, δ],

a ≤ f(m)− s ≤ f(ϕ−s(m)) ≤ f(m) ≤ b

so f(ϕs(m)) ∈ [a, b]. Now by equality condition of (1) we know thatd

dtf(ϕ−s(m)) =

1 and

f(ϕ−s(m)− f(m) =

∫ −s0

d

dtf(ϕ−s(m)) = −s

implying that f(ϕ−s(m) = f(m) − s = a + δ − s, which is what we intended toshow. In particular, f(ϕ−δ(m)) = f(m)− δ = a. Now notice that

ϕ−(b−a−δ) ◦ ϕ−δ(m) = ϕa−b(m) ∈Ma

and ϕa−b(Mb) ⊂Ma since ϕa−b(M

a) ⊂Ma and ϕa−b(f−1[a, b]) ⊂Ma and M b =

Ma∪f−1[a, b]. Notice that this implies that ϕb−a(Ma) ⊃M b. However we alreadyknow that ϕb−a(Ma) ⊂M b, so we show the conclusion that ϕb−a(Ma) = M b.

Part 2: Define a 1-parameter family of maps rt : M b →M b by

rt(q) =

{q f(q) ≤ aϕt(a−f(q))(q) a ≤ f(q) ≤ b

Then r0 is the identity, and r1 is the deformation retraction desired. �

Theorem 3.10. Let f : M → R be a smooth function, and let p be a non-degeneratecritical point with index λ. Setting f(p) = c, suppose that f−1[c−ε, c+ε] is compact,and contains no critical point of f other than p, for some ε > 0. Then, for allsufficiently small ε, the set M c+ε has the homotopy type of M c−ε with a λ-cellattached.

Figure 3. The idea of the proof where M is a torus

AN INVITATION TO MORSE THEORY 9

Proof Sketch. The idea of the proof is indicated in the graph for M being a torus,on which M c−ε = f−1(−∞, c − ε] is heavily shaded. We will introduce a newfunction F : M → R defined such that F < f in a small neighborhood of p. Thusthe region F−1(−∞, c−ε] will consist of M c−ε together with some neighborhood Hof p. In the graph H is horizontally shaded. Choosing a suitable eλ ⊂ H, a directargument such as pushing in along the horizontal lines will show that M c−ε ∪ eλ isa deformation retract of M c−ε ∪H. Finally, we apply theorem 3.1 to the functionF and region F−1[c − ε, c + ε] to see that M c−ε ∪ H is a deformation retract ofM c+ε and finish the proof.

Now we formally define the auxiliary smooth function F as follows. We first fixa coordinate system u1, ..., un in a neighborhood of U so that the identity

f = c− (u1)2 − ...− (uλ)2 + (uλ+1)2 + ...+ (un)2

holds in U and ui(p) = 0 for all i, as characterized by Morse lemma. Let

µ : R→ R

be a smooth function satisfying the conditionsµ(0) > εµ(r) = 0 r ≥ 2ε

−1 < µ′(r) :=du

dr≤ 0 ∀ r

Now let F coincide with f outside of U and let

F = f − µ[(u1)2 + ...+ (uλ)2 + 2(µλ+1)2 + ...+ 2(un)2]

and it can be verified that F is a well-defined smooth function throughout M . Wenow finish the proof by proving the following assertions:

Assertion 1: The region F−1(−∞, c + ε] coincides with the region M c+ε =f−1(−∞, c+ ε].

Assertion 2: The critical points of F are the same as those of f , and thereforeF−1[c− ε, c+ ε] contains no critical point.

Assertion 3: The region F−1(−∞, c − ε] is a deformation retract of M c+ε.For convenience, we denote H = F−1(−∞, c− ε]−M c−ε.

Assertion 4: M c−ε ∪ eλ is a deformation retract of M c−ε ∪H.

Then assertion 4 shows that M c−ε ∪ eλ is a deformation retract of F−1(−∞, c− ε).Together with assertion 3, we complete the proof.

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Remark 3.11. More generally suppose that there are k non-degenerate criticalpoints p1, ..., pk with indices λ1, ..., λk in f−1(c). Then a similar proof to abovewill show that M c+ε has the same homotopy type as M c+ε ∪ eλ1 ∪ ... ∪ eλk .

Remark 3.12. A slight modification of the proof above will show that the set M c

is also a deformation retract of M c+ε. In fact M c is a deformation retract ofF−1(−∞, c], which is a deformation retraction of M c+ε. By assertion 3, M c−ε ∪ eλis a deformation retract of F−1(−∞, c − ε], which is a deformation retract of M c.Then M c−ε ∪ eλ is a deformation retract of M c.

Theorem 3.13. If f is a differentiable function on a manifold M with no degen-erate critical points, and if each Ma is compact, then M has the homotopy type ofa CW-complex, with one cell of dimension λ for each critical point of index λ.

10 ZHENGJUN LIANG

Lemma 3.14 (Whitehead). Let ϕ0 and ϕ1 be homotopic maps from the sphere eλ

to X. Then the identity map of X extends to a homotopy equivalence

k : X ∪ϕ0eλ → X ∪ϕ1

eλ

Proof of Lemma. Define k by

k :=

k(x) = x x ∈ Xk(tu) = 2tu 0 ≤ t ≤ 1

2 , u ∈ eλ

k(tu) = ϕ2−2t(u) 12 ≤ t ≤ 1, u ∈ eλ

where ϕt is the homotopy between ϕ0 and ϕ1 and u is a unit vector. A corresponding

l : X ∪ϕ1eλ → X ∪ϕ0

eλ

is defined similarly but reversed. We can verify that kl = lk is the identity and kis a homotopy equivalence. �

Lemma 3.15. Let ϕ : eλ → X be an attaching map. Any homotopy equivalencef : X → Y extends to a homotopy equivalence

F : X ∪ϕ eλ → Y ∪f◦ϕ eλ

Proof Sketch of Lemma. Define F such that{F |X = fF |eλ = id

Let g : Y → X be the homotopy inverse of f , and define G : Y ∪fϕ eλ → X ∪gfϕ eλsuch that G|Y = g and G|eλ = id. Since gfϕ is homotopic to ϕ, by previouslemma there is a homotopy equivalence k : X ∪gfϕ eλ → X ∪ϕ eλ. Show that kGF :X∪ϕeλ → X∪ϕeλ is homotopic to the identity. Then F has a left homotopy inverse.We then prove F is a homotopy equivalence by prove the following assertion:Assertion. If a map F has a left homotopy inverse L and a right homotopy inverseR, then F is a homotopy equivalence; and R or L is a 2-sided homotopy inverse.The proof now proceeds as follows: the relation kGF ∼ id asserts that F has a lefthomotopy inverse, and the idea of proving this result can show that G also has aleft homotopy inverse. We then complete the proof by the following 3 steps:

Step 1: Since k(GF ) ∼ id, and k is known to have left inverse, it follows that(GF )k ∼ id.

Step 2: Since G(Fk) ∼ id, and G is known to have a left inverse, it followsthat (Fk)G ∼ id.

Step 3: Since F (kG) ∼ id, and F has kG as left inverse also, it follows thatF is a homotopy equivalence. This completes the proof.

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Proof of Theorem. Let c1 < c2 < ... be critical values of f : M → R. Sincecritical points of f are non-degenerate, by the corollary of Morse lemma they areisolated. Therefore, for any a ∈ R, the compact Ma can only include finitely manycritical points, and thus for every a there are only finitely many ci ≤ a. Thus{ci} has no accumulation point. The set Ma is vacuous for a < c1. Supposea 6= c1, c2, c3, ..., and that Ma is of the homotopy type of a CW-complex. This ispossible if we consider that Ma can be formed by iteratively attaching cells. Letc be the smallest ci > a. By 3.9, 3.10, and 3.11, M c+ε has the same homotopytype of M c−ε ∪ϕ1 e

λ1 ∪ ... ∪ϕj(c)eλj(c) for certain maps ϕ1, ..., ϕj(c) when ε is small

AN INVITATION TO MORSE THEORY 11

enough, and there is a homotopy equivalence h : M c−ε → Ma. We have assumedthat there is a homotopy equivalence h′ : Ma → K, where K is a CW-complex.Then each h′ ◦ h ◦ φj is a homotopy by cellular approximation to a map

ψj : eλj → (λj − 1)− skeleton of K

Then K ∪ψ1 eλ1 ∪ ... ∪ψj(c)

eλj(c) is a CW-complex, and has the same homotopy

type as M c+ε. By induction (on the critical values) it follows that each Ma′ hasthe homotopy type of a CW-complex. If M is compact we are done. If M is notcompact itself but all critical points lie in a compact set, then a similar argumentwill again finish the proof.

If there are infinitely many critical points then the above argument gives us aninfinite sequence of homotopy equivalence

each extending the previous one. Let K denote the union of Ki in the final topologywith respect to the homotopy equivalence maps, and let g : M → K be the limitmap. Then g induces induces isomorphism of homotopy groups in all dimensions.We now applying the following theorem to conclude the proof.

Theorem 3.16 (Whitehead). The map f : X → Y is a homotopy equivalence if,and only if, fn : πn(X) → πn(Y ) (where πn is the projection map) is an isomor-phism onto for every n such that 1 ≤ n < N + 1.

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4. Applications

We can apply the knowledge we have learned to prove the following beautifultheorem:

Theorem 4.1 (Reeb). If M is a compact manifold and f is a differentiable functionon M with only two critical points, both of which are non-degenerate, then M ishomeomorphic to a sphere.

Figure 4. A Manifold of Such Kind

12 ZHENGJUN LIANG

Proof. The two critical points must be the minimum and maximum points. Saythat f(p) = 0 is the minimum and f(q) = 1 is the maximum. If ε > 0 is smallenough then the sets M ε = f−1[0, ε] and f−1[1 − ε, 1] are closed n-cells. But M ε

is homeomorphic to M1−ε since there is no critical point in between. Thus M is aunion of two closed n-cells matched along their boundary. Then we can constructa homeomorphism between M and Sn. �

5. Manifolds in Euclidean Space

Our principal topic of investigation in the previous sections has been functionson manifolds with no degenerate critical points, so we shall give some examples ofsuch functions in this section.

Example 5.1. For a fixed p ∈ Rn consider Lp : M → R defined by Lp(q) = |p−q|2.It will turn out that for a.e. p the function Lp has only non-degeneratre criticalpoints.

Definition 5.2. Let M ⊂ Rn be a manifold of dimension k < n, differentiablyembedded in Rn. Let the total space of normal vector bundle N ⊂ M × Rn bedefined by

N = {(q, v) : q ∈M,v ⊥M at q}It can be verified that N is an n-dimensional manifold differentiably embedded inR2n. Let E : N → Rn be defined by E(q, v) = q + v.

Figure 5. An Intuition of the Mapping E

Definition 5.3. e ∈ Rn is a focal point of (M, q) with multiplicity µ if e = q + vwhere (q, v) ∈ N and the Jacobian of E at (q, v) has nullity µ > 0. The point ewill be called a focal point of M if e is a focal point of (M, q) for some q ∈M .

Remark 5.4. Intuitively, a focal point of M is a point in Rn whereby normalsintersect.

Theorem 5.5 (Sard). If M1 and M2 are differentiable manifolds having a countablebasis, of the same dimension, and f : M1 → M2 is of class C1, then the image ofthe set of critical points has measure 0 in M2.

Proof. A proof can be seen on, for example, Topology from the Differential View-point, Milnor. �

Corollary 5.6. For almost all x ∈ Rn, the point x is not a focal point of M .

AN INVITATION TO MORSE THEORY 13

Proof. N is an n-dimensional manifold. The point x is a focal point iff x is in theimage of the set of critical points of E : N → Rn. Therefore the set of focal pointshas measure 0. �

In the following setting, let (u1, ..., uk) be a coordinate for a region of the manifoldM ⊂ Rn. The the inclusion map from M to Rn determines n smooth functions

x1(u1, ..., uk), ..., xn(u1, ..., uk)

These functions will be written briefly as x(u1, ..., uk) where x = (x1, ..., xn).

Definition 5.7 (First Fundamental Form). The first fundamenral form associatedwith the coordinate system is defined to be the symmetric matrix of real valuedfunctions

(gij) = (∂x

∂ui· ∂x∂uj

)

Definition 5.8 (Second Fundamental Form). The vector∂2x

∂ui∂ujat a point of M

can be expressed as the sum of a vector tangent to M and a vector normal to M .

Define lij to be the normal component of∂2x

∂ui∂uj. Given any unit vector v which

is normal to M at q the matrix

(v · ∂2x

∂ui∂uj) = (v · lij)

can be called the second fundamental form of M at q in the direction v.

By change of coordinates we may assumr that (gij) at q is the identity.

Definition 5.9. In the setting stated above, the eigenvalues of the matrix (v · lij)is called the principal curvatures K1, ...,Kk of M at q in the normal direction v.The reciprocals of principal curvatures are called pricipal radii of curvature.

Remark 5.10. The concepts above are well-defined iff the matrix is not singular.

Lemma 5.11. Consider the normal line l consisting of all q+tv, where v is the fixedunit vector orthogonal to N at q. The focal points of (M, q) along l are preciselythe points q+k−11 v, where 1 ≤ i ≤ k, Ki 6= 0. Then there are at most k focal pointsof (M, q), each counted with its proper multiplicity.

The proof of the lemma is technical and involves heavy use of linear algebra, forwhich reason we omit it here.

Now based on previous lemma, we have a result that validates the statement ofexapmle 5.1:

Lemma 5.12. The point q ∈ M is a degenerate critical point of f = Lp iff p is afocal point of (M, q). The nullity of q as a critical point is equal to the multiplicityof p as focal point.

Combining this result with the corollary to Sard theorem, we obtain the follow-ing:

Theorem 5.13. For almost all p ∈ Rn (all but a set of measure 0) the function

Lp : M → Rhas no degenerate critical points.

14 ZHENGJUN LIANG

Corollary 5.14. On any manifold M there exists a differentiable function, withno denegrate critical points, for which each Ma is compact.

Proof. This follows from the previous theorem and the fact that an n-dimensionalmanifold M can be embedded differentiably as a closed subset of R2n+1. �

In many areas of study, an appximation theorem can greatly reduce the difficultyof the problems we want to deal with. The Weierstrass approximation theorem,the cellular appximation theorem introduced above, and the approximation of L1

functions by simple functions, etc, are theorems of such kind. Here we introduce an-other theorem, which approximates a bounded smooth function by another smoothfunction with no degenerate critical points. This is important since many of thetheorems above requires the function to have no degenerate critical point.

Corollary 5.15. Any bounded smooth function f : M → R can be uniformlyapproximated by a smooth function g which has no degenerate critical points. Fur-thermore g can be chosen so that the i-th derivatives of g on the compact set Kuniformly approximate the corresponding derivatives of f , for i ≤ k.

Proof Sketch. Choose some imbedding h : M → Rn of M as a bounded subset ofsome Euclidean space so that the first coordinate h1 is precisely the given functionf . Let c be large, and choose p = (−c + ε1, ε2, ..., εn) close to (−c, 0, 0, ..., 0) ∈ Rnso that the function Lp : M → R is non-degenerate; and set

g(x) =Lp(x)− c2

2c

then show that g is the function desired. �

Lemma 5.16 (Index theorem for Lp). The index of Lp at a non-degenerate criticalpoint q ∈M is equal to the number of focal points of (M, q) which lie on the segmentfrom q to p; each focal point being counted with its multiplicity.

Proof. The index of the matrix

(∂2Lp∂ui∂uj

) = 2(gij − t~v ·~lij)

is equal to the number of negative eigenvalue. Assuming that (gij) is the identity,

this is equal to the number of eigenvalues of (~v ·~lij) which are greater than or equalto 1

t . Comparing this with lemma 5.11 we finish the proof. �

6. Reflections

The book is indeed a good book, and gives me a glimpse of Morse theory. How-ever, since I have not formally studied algebraic topology, Riemannian geometry,and algebraic geometry, I am unable to go deep into certain chapters of the book,which I am sure are interesting and elegant. Therefore, I would recommend at-tempting this book after an initial exposure to the areas of study I enumeratedabove.

Acknowledgments. I would like to express my gratitude to professor Kefeng Liufor his willingness of mentorship and excellent recommendation of textbook. Hisgenerous support makes this reading course possible.

AN INVITATION TO MORSE THEORY 15

References

[1] Allen Hatcher. Algebraic Topology. Cambridge University Press. 2002.[2] John Milnor. Morse Theory. Princeton University Press. 1973.

[3] John Milnor. Topology from the Differential Viewpoint. Princeton University Press. 1997.