MORSE THEORY AND IT’S APPLICATION TO HOMOTOPY …petersen/Bott_book.pdfMORSE THEORY AND IT’S APPLICATION TO HOMOTOPY THEORY 5 Lemma 2.2. Let M be a compact smooth manifold and

MORSE THEORY AND IT’S APPLICATION TO HOMOTOPY

THEORY

RAOUL BOTT

Abstract. The following is a LaTeX version of Bott’s 1960 book on Morse

theory which is no longer in print. Some proofs are supplemented and someambiguous notations are fixed.

Contents

1. Introduction 12. Morse Theory of Smooth Functions on a Manifold 43. The Morse Inequalities 94. Manifolds Embedded in an Euclidean Space 115. Topology of Flag Manifolds 166. The Structure of the Space Ωp,q(M) 197. The Index Theorem 218. Critical Manifolds 299. The Stable Homotopy Groups of the Unitary Group 31Acknowledgement 3510. References 36

1. Introduction

To get a first idea about Morse Theory, we consider a simple example. Let Tbe a 2-dimensional torus, resting on its tangent plane V as indicated in fig.1. Thedistance of the points of T from the tangent place V is a real analytic, hence smooth(C∞) function f on T . We set

T a = {x ∈ T | f(x) ≤ a}

T a is empty for a < 0, {p} for a = 0, homeomorphic with a 2-cell for 0 < a < f(q),homeomorphic with he product of a circle and a line segment for f(q) < a < f(p),is homeomorphic with the figure indicated in fig.2 for f(q) < a < f(s), and thewhole torus for a ≥ f(s).

As f grows from 0 to f(s), T a grows in successive steps from a point to thewhole torus. From the topological point of view, something new comes at the levelof p, q, r, and s. These points are the critical points of f , the points where df = 0.

Here, we touch on a first essential idea of Morse Theory. There is a close relationbetween the behavior of f as a smooth function, especially with respect to its criticalpoints, and the topological structure of T . Let us make this more precise. For that

1

2 RAOUL BOTT

Figure 1.

Figure 2.

purpose, we introduce the concept of attaching an r-cell to a topological spaceX. More generally, let Y be a second topological space, Z a subspace of Y andf : Z → X a continuous map. Let X̃ be the topological space obtained as follows:in the disjoint union of X and Y , we identify the points s of Z with their imagef(s) and provide the resulting space with the quotient topology. We say that X̃ isobtained from X by attaching Y to X according to the pair (Z, f). In particular,if y is an r-cell er and Z its boundary ėr, we simply say that er is attached to Xaccording to the map f and write X̃ = X

⋃f er. Furthermore, we introduce the

index of a non-degenerate critical point of a smooth function f defined on a smoothmanifold Mn.

As we said above, a critical point of f is a point x ∈ Mn, such that at x,df = 0, or expressed in local coordinates (x1, ..., xn):

∂f

∂xi= 0, i = 1, ..., n.

A critical point x of f is called non-degenerate if

Hf =

(∂2f

∂xi∂xj

)is a nonsingular n× n matrix. It is easily proved that the number of positive andnegative eigenvalues of Hf is independent of the local coordinates. The last num-ber is naturally called the index of x as a critical point of f . In the case of ourexample, the index of p, q, r, and s is 0, 1, 1, 2 respectively. From the homotopicpoint of view, the building up of t in successive steps can be described as follows:If there are no critical points between a and b then T a is of the same homotopictype as T b. If there is a single critical point of index k in this range, then T b isobtained by attaching a k-cell.

MORSE THEORY AND IT’S APPLICATION TO HOMOTOPY THEORY 3

Figure 3. The building up of T by successive attaching of cellscorresponding to the critical points of f

The main result of the simplest part of Morse theory states that the situation asdescribed for the special function f on the special smooth manifold T is a generalone. This is expressed by the following theorem.

Theorem 1.1. Let f be a smooth function on the compact smooth manifold M,such that f has only non-degenerate critical points. Set Ma = {x ∈ M,f(x) ≤ a}.Then, if f−1(a ≤ s ≤ b) does not contain any critical point of f, then Ma and M bare homeomorphic. If f−1(a ≤ s ≤ b) contains only one critical point x of indexλ, a < f(x) < b, then M b ∼ Ma

⋃eλ, eλ being attached to M

a by a convenientlychosen map.

As always, “smooth” means “C∞” and “manifold” means “connected manifold”.Let the situation be as in the theorem, and let ai ∈ R, i = 1, ...,m be the critical

values of f , which we suppose to correspond 1-to-1 to the critical points of f and

Ma1 ⊂Ma2 ⊂ ...Mam .

It can be shown, precisely in the same way as it can be shown that πk(Sn) = 0 for

0 ≤ k < n, that πk(Mai ,Mai−1) = 0 for 0 ≤ k ≤ λi−2, where λi denotes the indexof the critical point corresponding to ai, also that the homomorphism

πλi−1(Mai−1)→ πλi−1(Mai)

is onto. Thus from the exact homotopy sequence we can conclude:

πλi−1(Mai−1) = πλi−1(M

ai) 0 ≤ k ≤ λi − 1.

Further, it is easy to see that

Hr(Mai ,Mai−1 ;Z) = 0 0 ≤ r ≤ λiand

Hr(Mai ,Mai−1 ;Z) = Z r = 0, λi.

The second part of Morse theory is analogous to the first one, but deals with adifferent, more complicated situation.

Let M be a compact Riemannian manifold. Let P , Q be a couple of points onM . We denote by µP,Q(M) the set of sectionally smooth curves, joining P and Q,and parametrized proportionally to the length of arc by a parameter t, 0 ≤ t ≤ 1.

4 RAOUL BOTT

Let L(u) be the length of the curve u ∈ µP,Q(M). If we set for every pair of curvesu, v ∈ µP,Q(M)

L(u, v) = max distt∈[0,1]

(u(t), v(t)) + |L(u)− L(v)|

then it is easily seen that L(u, v) is a distance function on µP,Q(M). Now the resultof Morse Theory in the previous theorem has an analogue in this case. The role off by L and the geodesics in µP,Q(M) take the place of the critical points. Moreprecisely, the following theorem holds:

Theorem 1.2. Let M be a compact Riemannian manifold, P and Q a pair ofpoints on M, µP,Q(M) the metric space of sectionally smooth curves from P to Q,parametrized proportionally to length of arc. Set

ΩcP,Q(M) = {u ∈ µP,Q(M),L)(u) ≤ c},

where L(u) denotes the length of u.If there is no geodesic of length l, a ≤ l ≤ b, then

ΩbP,Q(M) ∼ ΩaP,Q(M).If there is just one such geodesic g of length l, a < l < b, then

ΩbP,Q(M) ∼ ΩaP,Q(M)⋃eλ,

where λ denoting the number of conjugate points of P along g which lie between Pand Q.

Since πk(ΩP,Q(M)) = πk+1(M) ([1], p.55), this theorem yields information onthe homotopy structure of M , provided that the “geodesic structure” of M is suf-ficiently known. This is the case for special kinds of manifolds, in particular sym-metric spaces.

2. Morse Theory of Smooth Functions on a Manifold

We start with the first part of Theorem 1.1.

Theorem 2.1. Let M be a compact smooth manifold and f : M → R a smoothfunction on M. Set Ma = {x ∈ M,f(x) ≤ a}. if df(p) 6= 0 for all points p ∈ Mwith a ≤ f(p) ≤ b, then Ma and M b are homeomorphic.

We recall that a local 1-parameter group of diffeomorphisms of M is a (smooth)map

Φ : M × R→M,such that

(1) Φ|M×{t} is a diffeomorphism of M ;(2) Φ(m, t1 + t2) = Φ(Φ(m, t1), t2) for |t1|, |t2|, and |t1|+ |t2| < �.

If p ∈ M is not a fixed point of Φ, then there passes just one orbit curve of Φthrough p, Φ determines in a natural way a vector field Φ̇ on M by setting

DΦ̇(p)g = (Φ̇(g))(p) := limt→0

g(Φ(p, t))− g(p)t

,

p ∈ M , g a smooth function on M . Φ̇ is zero at the fixed points of φ and tangentto the orbits of φ at the other points of M .


Lemma 2.2. Let M be a compact smooth manifold and V a vector field on M. Thenthere exists a (uniquely determined) local 1-parameter group Φ of diffeomorphisms

of M such that Φ̇ = V

For the proof of Lemma 2.2, see [2, p.5]. We now give the proof of Theorem 2.1.

Proof. We introduce a Riemannian metric on M . This can be done for example inthe following way. Let {U1, ..., Um} be a finite open covering of M with coordinatesystems, and {ϕ1, ..., ϕm} a partition of unity corresponding to {U1, ..., Um}. Let

xi1, ..., xin

be coordinates in Ui and x ∈ Ui. for every pair (X,Y ) of tangent vectors to M atx, we define ϕi(X,Y ) be setting(

∂

∂xik,∂

∂xil

)= δkl

and extending by linearity. Then

(X,Y ) =

m∑i=1

ϕi(X,Y )

defines a global Riemannian metric on M . This metric enables us to define a specialvector field ∇f on M , the gradient of f , by

(∇f(p), Y (p)) = df |p(Y (p))

Obviously, ∇f(p) = 0 is equivalent to df |p = 0. Now let φ be the local 1-parametergroup of diffeomorphisms with the property that φ̇ = −∇f . Since by assumption,df |p 6= 0 for all x ∈M with a ≤ f(x) ≤ b, by Lemma 2.2, there passes through everypoint of this subset of M a unique integral curve of −∇f . This is also true if wereplace a by a conveniently chosen a′ < a. Since f(m)− f(φ(m, 12�)) is continuous,this function has a positive minimum on the compact set {x ∈ M,a′ ≤ f(x) ≤b}. Therefore, every integral curve of −∇f which meets M b meets Ma′ and alsoconversely.

A homeomorphism h : M b →Ma is constructed as follows. For y ∈Ma′ , we seth(y) = y. Now let y ∈ M b −Ma′ . Suppose the integral curve of −∇f through yintersects Ma, Ma

′, and M b in ya, ya′ , and yb respectively. We determine h(y) on

this curve C by setting

l(h(y)ya′) =l(yaya′)

l(ybya′)l(yya′)

where l denotes the length of arc along C. �

For the proof of the second part of Theorem 1.1, we need some lemmas.

Lemma 2.3 (Morse). Let f : Rn → R be a smooth function on Rn, such thatf(0) = 0, fxi(0) = 0, i = 1, ..., n, and det(fxixj (0)) 6= 0. Then in a neighborhood Uof 0, coordinates (y1, ..., yn) can be introduced such that in U

f = −y21 − y22 − ...− y2λ + y2λ+1 + ...+ y2n,

where λ is the index of f at 0.

6 RAOUL BOTT

Figure 4.

Proof. We start with the construction of smooth functions aij(x), i, j = 1, ..., n,such that

f =

n∑i,j=1

aij(x)xixj ,

where aij(x) = aji(x). Using f(0) = df(0) = 0, we find in an elementary way:

f(x) =

∫ 10

∂

∂t(f(xt))dt =

n∑i=1

xi

∫ 10

∂i(f(xt))dt

=

n∑i=1

xi [∂if(xt)(t− 1)]10 −n∑i=1

xi

∫ 10

∂

∂t(∂if(xt))(t− 1)dt

=

n∑j=1

n∑i=1

(−∫ 1

0

∂j(∂if(xt)(t− 1))dt)xixj

=

n∑i,j=1

aij(x)xixj .

We may assume: a11(0) 6= 0(at least one element aij 6= 0, in the case this elementis aii with i 6= 1, we simply permute, in the case this element is Aij with i 6= j, weintroduce new coordinates x̄1, ..., x̄n by setting x̄i = xi + xj , x̄j = xi − xj , x̄k = xkfork 6= i, j).

Firstly, let a11(0) > 0. In a neighborhood of 0, we can set

ỹ1 =√a11

(x1 +

∑a1αxαa11

)α = 2, ..., n

ỹk = xk k = 2, ..., n

where√a11 denotes the positive root of a11. It follows:

∑aijxixj = ỹ1

2 +

n∑α,β=2

bαβ ỹαṽβ .

dỹ1 = (d√a11)

(x1 +

∑a1αxαa11

)+√a11

(dx1 + d

(∑a1αxαa11

))


dỹ1(0) =√a11dx1 +

∑cαdxα

dỹα(0) = dxα, α = 2, ..., n

dỹ1 ∧ ... ∧ dỹn =√a11dx1 ∧ ... ∧ dxn 6= 0

Hence we have det(∂ỹ∂x (0)

)6= 0.

Since det(bαβ(0)) = det(aij(0))(

det ∂ỹ∂x (0))2

and det(aij(0)) =1

2n detfxixj (0) 6=0, we also have det(bαβ) 6= 0

In the case that a11 < 0, we set∑aijxixj = −y21 +

∑α,β

yαyβ .

By induction, we get functions y1, ..., yn such that

f =

n∑i=1

�iy2i , �i = ±1,

and dy1 ∧ ... ∧ dyn(0) 6= 0. Since (y1, ..., yn) can be used as coordinates in a neigh-borhood of 0, the number of negative �i’s equals the index of f at 0. This completesthe proof. �

Lemma 2.4. Let X be a topological space, er an r-cell, Is an s-cell, given by

Is = {(t1, ..., ts) ∈ Rs, 0 ≤ ti ≤ 1}and let er×Is be attached to X by a map f : ėr×Is → X. If we set g = f |ėr×(0,...,0),then X

⋃f (er

⋃Is) ∼ X

⋃g er.

Proof. As illustrated in fig. 5, X⋃f (er

⋃Is) can be deformed into X

⋃g er by a

standard deformation. �

Figure 5.

Theorem 2.5. Let M be a compact smooth manifold, f : M → R a smooth functionon M. Set Ma = {x ∈ M,f(x) ≤ a}. If f−1(a ≤ x ≤ b) conatins exactly one non-degenerate critical point p of index λ, a < f(p) < b, then M b ∼Ma

⋃eλ, where eλ

is attached to Ma by a conveniently chosen map.

Proof. We may assume that f(p) = 0. From Theorem 2.1, it follows that it issufficient to prove the existence of a number �, 0 < � ≤ b, such that M � ∼Ma

⋃eλ.

By Lemma 2.3, there is a neighborhood U of p and local coordinates such that inU , f is given by

f = −y21 − y22 − ...− y2λ + y2λ+1 + ...+ y2n,

8 RAOUL BOTT

Furthermore, we may assume M to be provided with a Riemannian metric ds2

which is Euclidean in U . If we set:

A� = {y ∈M �⋂U, y21 + ...+ y

2λ ≤ ρ}

M �∗ = M� −A�

then M � = A�⋃M �∗. For conveniently chosen � > 0, ρ > 0, this just mean that M

�

is obtained from M �∗ by attaching a product eλ × In−λ to M∗ in the way describedin Lemma 2.4. By this Lemma, we find

M � ∼M �∗⋃eλ.

It can be shown in the same way as in the proof of Theorem 2.1 that M �∗ and Ma

are homeomorphic. It only has to be shown that the gradient of f is not zero andthat any integral curve starting from the boundary of Ma meets the boundary ofM �∗. Using the fact that ds

2 is Euclidean in U , this is easily proved. �

Figure 6. The case of n = 2, λ = 1

Figure 7. The case of the torus, considered in the Introduction

A smooth function f on a smooth manifold M is called non-degenerate if allcritical points of f are non-degenerate; and such a function is called strictly non-degenerate if f(p) 6= f(q) for every pair of critical points p and q. In Theorem 2.1and Theorem 2.5, we only considered strictly non-degenerate functions.

It is natural to question whether there always exist non-degenerate and strictlynon-degenerate functions on a given manifold. The answer is given by the following.

Theorem 2.6 (Morse-Thom). let M be a compact smooth manifold, Z the space ofsmooth functions on M, provided with the compact open topology. Then the subsetof non-degenerate functions is everywhere dense in Z.


For the proof of this theorem, see [3], p.155.An immediate consequence of Theorem 2.6 is the following theorem.

Theorem 2.7. Let the situation be as in Theorem 2.6. Then the subset of strictlynon-degenerate functions is also dense in Z.

Proof. It is sufficient to prove the following local result:Let f be a function on Rn, (x1, ..., xn) with only one non-degenerate critical

point at (0, ..., 0). Then there exists a function f ′ on Rn such that:(1) |f − f ′| < � everywhere on Rn and f = f ′ outside a neighborhood of

(0, ..., 0);(2) f ′ also has only one (non-degenerate) critical point, and this is again the

point (0, ..., 0);(3) f ′(0) = f(0) + �.

Here � denotes an arbitrarily but sufficiently small positive number. Set

A = {(x1, ..., xn) ∈ Rn,n∑i=1

x2i ≤ 1}

B = {(x1, ..., xn) ∈ Rn, 1 ≤n∑i=1

x2i ≤ 2}

C = {(x1, ..., xn) ∈ Rn,n∑i=1

x2i ≥ 2}

Let g be a function on Rn with g(x) = 0 if x ∈ C, 0 ≤ g(x) ≤ 1 if x ∈ B, andg(x) = 1 if x ∈ A. Taking for f ′(x) the function f(x) + �g(x), � > 0, we can seethat f ′ fulfills the conditions of (1), (2), (3) provided that � is sufficiently small.The only thing to be verified is that f ′ has no critical points on B. But on B, wehave uniform bounds α and β, α > 0, such that ‖df‖ ≥ α and ‖dg‖ ≤ β. From

‖df ′‖ ≥ ‖df‖ − �‖dg‖ ≥ α− �β

it follows that f ′ has no critical points on B provided that � is sufficiently small.Hence f ′ has only one (non-degenerate) critical point and that is the point 0. �

3. The Morse Inequalities

Let M be a compact manifold, which can be built up by successively attachingcells, int he way described in section 2. Then it can be shown (see [8]) that there isa CW-complex K such that its cells are in dimension preserving 1-1 correspondencewith the attaching cells, and the homology of K is the homology of M (with respectto any group of coefficients). Accepting this result, the well known Morse inequal-ities follow immediately from the results of section 2 if we take R as a domain ofcoefficients. In fact, let

C =

n∑i=0

Ci

the (naturally graded) vector space of chains of K,

Z =

n∑i=1

Zi

10 RAOUL BOTT

the space of cycles,

B =

n∑i=1

Bi

the space of boundaries and

H =

n∑i=1

Hi

the real homology group of K. By definition, we have the exact sequences:

(1) 0→ Z → C δ→ B → 0(2) 0→ B → Z → H → 0

where δ reduces the degree by 1. Set dimCi = ci, dimZi = zi, dimBi = bi,dimHi = hi (the i-th Betti number of K), i = 0, 1, 2, .... Combining (1) and (2),we find:

zi + bi−1 = ci, bi + hi = zi, ci − hi = bi + bi−1 i = 0, 1, ... (b−1 = 0)

Since bi ≥ 0, i = 0, 1, ..., these relations lead to the following sequence of inequali-ties:

c0 ≥ h0c1 − c0 ≥ h1 − h0

c2 + c1 − c0 ≥ h2 − h1 + h0......

Let f be a non-degenerate smooth function on the compact smooth manifold M .By Theorem 2.7, there is a strictly non-degenerate function f ′ on M such that f andf ′ have the same number of critical points of index i, i = 0, 1, 2, .... Combinationof Theorem 2.5, the result stated at the beginning of this section, and the aboveset of inequalities leads to

Theorem 3.1. Let f be a non-degenerate smooth function on the compact smoothmanifold M. Let ci be the number of critical points of index i of f and hi, i =0, 1, 2, ... the i-th Betti number of M. Then there exists a sequence of non-negativeintegers b−1, b0, b1, ... such that

ci − hi = bi + bi−1 i = 0, 1, 2, ...

Therefore, there is a sequence of inequalities:

c0 ≥ h0c1 − c0 ≥ h1 − h0

c2 + c1 − c0 ≥ h2 − h1 + h0......

These inequalities are known as the Morse inequalities. They imply that a (non-degenerate) smooth function on a compact smooth manifold necessarily has a cer-tain number of critical points of index i, i = 0, 1, ... .


4. Manifolds Embedded in an Euclidean Space

In this section, the results of section 2 are applied to the special case that Mn isembedded smoothly in a Euclidean space Rn+k, (x1, ..., xn+k) and f is the distancefunction from the points of M to a fixed point p, p ∈ Rn+k. We denote this functionby lp = lp(x), x ∈Mn.

Let q ∈ Mn be a critical point of lp. This is equivalent to saying that pq isperpendicular to the tangent space TMq of M at q. Therefore, q is also a criticalpoint of the function lr, where r denotes any point of the line pq, r /∈Mn.

We choose a convenient coordinate system as follows. For q, we choose q = 0 =(0, ..., 0) and TM0 = {xn+1 = xn+2 = ... = xn+k = 0}. In a neighborhood of 0,x1, ..., xn can be used as a system of local coordinates on M , and in Rn+k, Mn canbe given locally by

xn+l = gl(x1, ..., xn), l = 1, ..., k

Since→p0 is perpendicular toMn, the coordinate of p can be taken as (0, ..., 0, p1, ..., pk).

Finally, we set (0, ..., 0, tp1, ..., tpk) = tp for 0 < t.Let, as usual, Hltp(0) denote the Hessian of ltp at 0. A straightforward calcula-

tion gives:

Hltp(0) =1

t‖p‖I −

(k∑l=1

pl‖p‖

∂2gl∂xi∂xj

(0)

),

where I denotes the n × n identity matrix and ‖p‖ = (∑kl+1 p

2l )

1/2. Using a wellknown result on quadratic forms (see [4], p. 158), we conclude that it is possible tochoose a base in M0, such that

1

‖p‖I and −

k∑l=1

pl‖p‖

∂2gl∂xi∂xj

(0)

are simultaneously reduced to diagonal form, in such a way that

1

‖p‖I

is reduced to the identity itself. With respect to this base, Hltp(0) is given by amatrix

Hltp(0) =

a11 +1t 0

. . .

0 ann +1t

.We see

(1) the coefficients in the diagonal are strictly decreasing functions of t;(2) only for a finite number of values of t, t1, ..., tm, Hltp(0) is non-singular,

and t is a degenerate critical point of ltp;(3) for 0 < t� 1, Hltp(0) is positive definite.

It follows that the index of Hltp(0) is a decreasing (integer valued) function of1t , which only jumps at the points t1p, ..., tmp, and at these points this jump justequals ν(Hltip(0)) = the dimension if the nullity of Hltip(0), i + 1, ...m. Since by(3), the index of Hltp(0) is zero in a neighborhood of 0, we can state

Theorem 4.1. Let the smooth manifold M be embedded smoothly in the Euclideanspace R. For the point p ∈ R, p /∈M , let lp(x) denote the distance from the points

12 RAOUL BOTT

x ∈ M to p. Let q be a critical point of lp(x) (degenerate or not). Then the indexof the Hessian Hlp(q) is given by

index Hlp(q) =∑

0


A variation of g induces in a natural way a vector field J(t) along g:

J(t) =dpi(s)

ds

∣∣∣∣s=0

+dqi(s)

ds

∣∣∣∣s=0

Such a field, induced by a variation of g will be called a Jacobi field along g .

Jacobi fields are also characterized by the property d2Jdt2 = 0 all along g. It follows,

that the Jacobi fields along g form a vector space of dimension 2n, which will bedenoted by Jg.

Let Mm be a proper smooth submanifold of Rn, and let g be perpendicular toM at g(0). A variation of g relative to M is a variation V (s, t) of g with thefollowing properties:

(1) V (s, 0) ∈M(2) all lines V (s, t) are perpendicular to M at V (s, 0).

Figure 10.

Let Jg(M) be the set of Jacobi fields along g which are induced by variations of grelative to M . In the sequel to this section, it will become obvious that Jg(M) is alinear subspace of Jg. So far, at every point g(t0) of g, there is a natural restrictionhomomorphism

rt0 : Jg(M)→ Tg(t0)R.

Definition 4.3. g(t0), t0 6= 0, is called a focal segment of M of multiplicity ν,ν > 0, if and only if dim Ker rt0 = ν.

The fundamental result of this section is the following

Theorem 4.4. Let the notations be as introduced before. Then g(t0) is a focalsegment of M of multiplicity ν if and only if g(0) is a degenerate critical point ofLg(t0)(x) of nullity ν.

Proof. Let ν be the nullity of Lg(t0)(x). We may assume g(t0) to be the point(0, ..., 0). If (u1, ..., um) is a system of local coordinates on M in a neighborhood ofg(0), and as usual

L =

n∑i=1

x2i (uα),

then the number ν equals m minus the rank of the quadratic form

HL|g(0) =m∑

α,β=1

∂2L

∂uα∂uβ

∣∣∣∣g(0)

UαV β ,

14 RAOUL BOTT

where U = (Uα) and V = (V β) are tangent vectors to M at g(0). Setting

∂xi∂uα

= Λiα and∂

∂uβ(Λiα) = Λ

iα,β , i = 1, ..., n; α, β = 1, ...,m,

we find:

∂L

∂uα= 2

n∑i=1

xi(uα)∂xi∂uα

= 2

n∑i=1

xiΛiα

∂2L

∂uα∂uβ= 2

n∑i=1

ΛiαΛiβ + 2

n∑i=1

xiΛiα,β

If we denote by (U, V ) the inner product, induced by the embedding of M in Rn,and set

n(U, V ) =

m∑α,β=1

n∑i=1

xil

Λiα,βUαV β ,

where l denotes the distance from g(0) to 0, we find

HL|g(0)(U, V ) = 2{(U, V ) + l · n(U, V )}.Let V (s, t) = p(s) + tq(s) be a variation of g relative to M . From the definition

of Jg(M), it follows immediately

(1) p(s) ∈M ;

(2)

n∑i=1

qi(s)Λiα(p(s)) = 0, α = 1, ...,m for all values of s.

Differentiation of (2) with respect to s gives

(3)

n∑i=1

dqids

Λiα(p(s)) +

m∑β=1

n∑i=1

qi(s)Λiα,β(p(s))

∂uβ∂s

(p(s)) = 0, α = 1, ...,m

where uβ = uβ(s) is the curve on M , described by xi = pi(s). Setting in particulars = 0, we get

(4)

n∑i=1

dqids

(0)Λiα(g(0)) +

m∑β=1

n∑i=1

pi(0)

lΛiα,β(g(0))

∂uβ∂s

(g(0)) = 0, α = 1, ...,m.

The Jacobi field along g induced by V (s, t) can be written as

ηi(t) = ηi(0) + tη̇i(0), i = 1, ..., n

with

dpids

∣∣∣∣s=0

= ηi(0)

dqids

∣∣∣∣s=0

= η̇i(0), i = 1, ..., n.

Obviously, η(0) ∈ Tg(0)M and ηβ =∂uβ∂s , β = 1, ...,m. Now let U = (U

α) be anytangent vector to Mm in g(0). Multiplying (4) with Uα, we get by summation:

( ˙η(0), U)− n(η(0), U) = 0.Conversely, let η(t) be a Jacobi field along g, such that

(a) η(0) ∈ Tg(0)M ;


(b) (η̇(0), U)− n(η(0), U) = 0 for all U ∈ Tg(0)M .Let pi(s) be a curve on M with pi(0) = pi, and

dpids

∣∣∣∣s=0

= ηi(0)

Consider (3) as a system of differential equations for q1, ..., qn with s as independentvariable. There exists a solution with

qi(0) = ηi(0) anddqids

∣∣∣∣s=0

= η̇i(0), i = 1, ..., n,

since by (b), (4) is satisfied for these values of qi(0) anddqids

∣∣∣s=0

. Let qi(s) be such

a solution. For this solution, we have:

n∑i=1

qi(s)Λiα(p(s)) = constant.

Sincen∑i=1

qi(0)Λiα(p(0)) = 0,

this constant is zero. It follows that there exists an element of Jg(M) inducing η(t)along g. Therefore, the properties (a) and (b) are characteristic for the elements ofJg(M).

The elements η ∈ Jg(M), for which rt0(η)(0) = 0 are characterized as thoseelements of Jg, which satisfy the following conditions:

(a) η(0) ∈ Tg(0)M ;(b) (η(0), U) + l · n(η(0), U) = 0 for all U ∈ Tg(0)M ;(c) η(0) + l · η̇(0) = 0.

From this, it is immediate that

dim Ker rt0 = ν(HLg(0)(p)) = ν.

This proves Theorem 4.4. �

Remark 4.5. If we define a linear transformation

T∗ : Tg(0)M → Tg(0)M

by setting (T∗U, V ) = n(U, V ), the conditions (a) and (b) for the elements of Jg(M)can be replaced by the following ones:

(a’) η(0) ∈ Tg(0)M ;(b’) η̇(0)−T∗(η(0)) ∈ Tg(0)M⊥ (the orthogonal complement of Tg(0)M in Tg(0)R).

This leads to m + (n −m) = n linearly independent conditions for an element ofJg to be in Jg(M). Therefore:

dim Jg(M) = n.

From this it follows easily that dim Ker rt0 = ν is equivalent to

dim(TRg(t0)/rt0(Jg(M))) = ν.

Combination of Corollary 4.2 and Theorem 4.4 gives

16 RAOUL BOTT

Theorem 4.6. Let M be a proper smooth submanifold of Rn. Let a ∈ Rn, a /∈M ,and let b ∈M be a critical point of the function La(x). Let ν(t) be the multiplicityof the focal segment, and zero otherwise. Then the index of b equals∑

0


line AB is a focal segment of multiplicity ν of LB(M) if and only if dimMA =dimMC = ν

This lemma is a very special case of a more general theorem, which states,for example, the same results for the case that U(n) is replaced by an arbitraryconnected Lie group.

The proof of Lemma 5.2 is very analogous to the considerations in section 9 andwill therefore be omitted.

From Lemma 5.2, Theorem 4.4, and the fact that the dimension of a complexflag manifold is always even, we conclude that the index of every critical point ofa function LP (M) is always even. Applying the results of section 3 to a generalpoint P , we obtain the statement of Theorem 5.1 (The existence of such a point Pcan be proved easily in this case; it follows also from the following considerations).

As a second example, we sketch how elementary Morse theory can be used toobtain the Betti numbers of a complex flag manifold.

It follows from the definition of m that the Jacobi bracket [X,Y ] = XY = Y X,X,Y ∈ R has the property

(1) ([X,Y ], Z) = (X, [Y,Z])

Lemma 5.3. The tangent space TX to MX at X is given by

(a) TX = {Z = [Y,X], Y ∈ R}

and the normal space to MX at X by

(b) NX = {Y ∈ R, [Y,X] = 0}.

Proof. (a) For every element Y ∈ R, etY is a 1-parameter family subgroup of U(n),to which Y is the tangent vector in I. We get all vectors Z of TX by taking

(2) Z = limt→0

AdetYX −Xt

for all elements Y ∈ R. But instead of (2), we can write

Z = limt→0

(1 + tY + ...)X(1− tY + ...)−Xt

= limt→0

t(Y X −XY ) + t2(...)t

= Y X −XY = [Y,X].

(b) [X,Y ] = 0 is equivalent to (A, [X,Y ]) = 0 for all A ∈ R. This on its turnis by (1) equivalent to ([A,X], Y ) = 0 for all A ∈ R and, by (a), just means thatY ∈ NX . �

Lemma 5.4. If at one of its points, a line is perpendicular to an orbit, then thatline is perpendicular to all orbits which it intersects.

Proof. Let the line B + tA be perpendicular to MB at B. By Lemma 5.3, thismeans that ([X,B], A) = 0 for all X ∈ R. Since ([X,A], A) = (X, [A,A]) = 0 forall X ∈ R, we find: ([X,B + tA], A) = ([X,B], A) + t([X,A], A) = 0 for all X ∈ R,and this is precisely the formal expression of our assertion. �

18 RAOUL BOTT

With respect to a suitable basis in V , every element of R can be written asiθ1 a12 · · · a1n

−ā12 iθ2...

.... . .

...−ā1n · · · · · · iθn

then

R = h⊕ e12 ⊕ ...⊕ e(n+1)n = h⊕∑

ekl

where

h =

iθ1 0. . .

0 iθn

, ekl =

0 · · · · · · 0...

. . . akl...

... −ākl. . .

...0 · · · · · · 0

, k 6= 1

Let h∗ denote the subset of h for which the real number θ1, ...θn are all different.In geometric language, h∗ consists of those points in the vector space h, which donot lie on any of the hyperplanes θi − θj = 0, i, j = 1, ..., n, i < j. Obviously, h∗consists of “almost all” points of h.

Lemma 5.5. If X ∈ h∗, then NX = h.

Proof. It is readily verified that in this case, [X,∑ekl] =

∑ekl. From R = h ⊕∑

ekl,we deduce [X,R] = [X,h]⊕ [X,∑ekl] =

∑ekl, since [X,h] = 0 by definition

of the bracket. By lemma 5.3, [X,R] = TX , NX = h. �

Lemma 5.6. (a) Let M be any orbit of AdU(n) in R and P ∈ h∗. Then the criticalpoints of the function LP (M) are precisely the points of M ∩ h. (b) In particular,these points are independent of the choice of P in h∗.

Proof. (a) Let L be a critical point of LP (M). The line PL is perpendicular to Mat L. From Lemma 5.4, it follows that PL is perpendicular to MP at P . SinceP ∈ h∗, and , by Lemma 5.5, we also have the direction of PL lies in h, so we seethat PL lies in h, and in particular that L ∈ h.

(b) For every point X ∈ h, we have [X,h] = 0. If L ∈ M ∩ h, then every linethrough L in h is perpendicular to M at L (Lemma 5.3), in particular PL. SinceM is always compact, LP (M) certainly has critical points. Therefore, every orbitintersects h. �

Let P ∈ h∗ and L a (non-degenerate) critical point of LP (M). By Theorem 4.4and the index λ(L) of L is given by

λ(L) =∑i

ν(Fi),

where the points Fi are the focal points on the segment PL and ν(Fi) are theirmultiplicities as focal points. From Lemma 5.2, we see that the focal points Fiare the points where the segment PL intersects orbits of lower dimension, andfurthermore that, if F is such a point, then

ν(F ) = dimMP − dimMF .


But since PL lies in h, these points Fi are precisely the points of intersection withthe hyperplanes θi− θj = 0, i < j. It is obvious, that dimMP − dimMF equals twotimes the number of θi − θj = 0, i < j, which vanish at F .

The problem to determine the Betti numbers of a flag manifold is now reducedto a problem of Euclidean geometry. In fact, let M be any orbit, that is some typeof flag manifold. The intersection points L1, ...,Ls of M with h can be determinedexplicitly by an algebraic procedure. Taking a general point P in h∗, the intersec-tions of the segments PL1, ..., PLs with the hyperplanes θi − θj = 0, i < j, can bedetermined also, and therefore the index of every critical point of LP (M). Sincethe homology of M vanishes in odd dimensions, the i-th Betti number of M is justthe number of critical points Lj of index i.

Needless to say that similar methods apply to many other situations.

6. The Structure of the Space Ωp,q(M)

Let M be a compact Riemannian manifold and p, q a couple of points on M . Wedenote Ωp,q(M) the set of sectionally smooth curves joining p with q, i.e. the set ofpiecewise smooth mappings f : [0, 1] → M , with f(0) = p, f(1) = q, parametrizedproportionally to the length of arc (constant speed); by L(c), c ∈ Ωp,q(M) thelength of c; by Ωap,q(M) the subset of Ωp,q(M) determined by

Ωap,q(M) = {c ∈ Ωp,q(M)|L(c) ≤ a};

by ρ(x, y) the distance of pair of points x, y on M , i.e. the infimum of L(d),d ∈ Ωx,y(M). A simple argument (see [1], pg. 45) shows that

ρ(c, c′) = max0≤t≤1

ρ(c(t), c′(t)) + |L(c)− L(c′)|

is a distance function on Ωp,q(M). With respect to the topology induced by thismetric, L is a continuous function on Ωp,q(M). We consider Ωp,q(M) provided withthis topology.

Figure 11.

ρ(x, y) is a continuous, but in general not a differentiable function on M ×M .However, there exists a number ρ > 0 such that for ρ(x, y) < ρ, there is just onegeodesic arc from x to y, of length ρ(x, y). For this ρ, ρ2(x, y), restricted to thepart of M ×M determined by ρ(x, y) < ρ, is a differentiable function of (x, y). ([1],pg. 49).

In the sequel, Ωp,q(M) will be considered for a fixed pair of points p, q on M .So we can write Ω instead of Ωp,q(M) and Ω

a instead of Ωp,q(M)a. Let n be such

that b = a2

n+1 < ρ2. If we set M × ...×M(n times) = M∗ and

ρ2(p, x1) + ...+ ρ2(xn, q) = φ(x1, ..., xn) = φ(x),

20 RAOUL BOTT

then φ is a differentiable function on M b∗ , where Mb∗ is determined by

M b∗ = {x ∈M∗, φ(x) ≤ b}.

The following theorem relates Ωa very strongly with M b∗ .

Theorem 6.1. Ωa and M b∗ are of the same homotopy type, i.e. there exist mapsα : Ωa → M b∗ and β : M b∗ → Ωa, such that β ◦ α and α ◦ β are homotopic with theidentity map of Ωa and M b∗ respectively.

Proof. The proof is carried out in four steps.

(a) Definition of α.For c ∈ Ωa, we define

α(c) = {c(t1), ..., c(tn)}, ti =i

n+ 1, i = 1, ..., n.

We have to verify: α(c) ⊂ M b∗ . The length of the arc c(ti)c(ti+1) of c equalsL(c)n+1 ,

hence

ρ(c(ti), c(ti+1)) ≤L(c)

n+ 1.

From the definition of φ, it follows that

φ(α(c)) ≤ (n+ 1) L2(c)

(n+ 1)2≤ a

2

n+ 1= b.

So for every

c ∈ Ωa, α(c) ∈M b∗ or α(Ωa) ⊂M b∗ .

(b) Definition of β.If we set p = x0 and q = xn+1, from the definition of φ, we derive immediately

for x = (x1, ..., xn) ∈M b∗ :n∑i=0

ρ2(xi, xi+1) ≤ b, or, for i = 0, ..., n : ρ(xi + xi+1) ≤√b =

a√n+ 1

< ρ.

Hence for i = 0, ..., n, xi and xi+1 can be joined by a unique geodesic arc. Theunion of these arcs determines an element of Ω, which, by definition, is β(x). Weonly have to verify that β(x) ∈ Ωa, i.e. that L(β(x)) ≤ a. This is an immediateconsequence of the Schwarz inequality:

L(β(x)) =

n∑i=0

ρ(xi + xi+1) ≤√n+ 1

√φ(x) ≤

√n+ 1

a√n+ 1

= a.

(c) Construction of a deformation Dτ : Ωa → Ωa, 0 ≤ τ ≤ 1, with D0 = identity

and D1 = β ◦ α.Let c ∈ Ωa and let for i = 0, ..., n + 1, xi = c(ti), ti = in+1 . Set x

τi =

c((1− τ)ti + τti+1) and define Dτ (c) as the (conveniently parametrized) element ofΩa, consisting of the (unique determined) geodesic arc pxτ0 , the arc x

τ0x1 of c, the

geodesic arc x1xτ1 , ..., the arc x

τnq of c. Obviously, D0 is the identity and D1 = β◦α.

It has to be verified that Dτ is actually a deformation. This is straightforward butnot completely trivial. We refer to [1], pg. 51.


Figure 12.

(d) Construction of a deformation ∆τ : Mb∗ → M b∗ , with ∆0 = identity and

∆1 = α ◦ β.For x ∈ M b∗ let si ∈ [0, 1] be such that β(x)(si) = xi, i = 0, ..., n + 1. Set

β(x)(ti) = yi, β(x)((1 − τ)si + τti) = xτi , xτ = (xτ1 , ..., xτn) and ∆τ (x) = xτ . Wehave to verify xτ ∈M b∗ .

Since the distance from xτi to xτi+1 at most equal the distance from x

τi to x

τi+1

along β(x), we find

ρ(xτi , xτi+1) ≤ L(β(x))[(1− τ)si+1 + τti+1 − (1− τ)si − τti]

= L(β(x))[(1− τ)(si+1 − si) + τ(ti+! − ti)]

If we set the distance from xτi to xτi+1 along β(x) as δi, this inequality becomes

ρ(xτi , xτi+1) ≤ (1− τ)δi + τ

L(β(x))

n+ 1.

From this follows, by the definition of φ:

φ(xτ ) =

n∑i=0

ρ2(xτi , xτi+1)

≤ (1− τ)2(

n∑i=0

δ2i

)+ τ2

L2(β(x))

n+ 1+ 2τ(1− τ)L(β(x))

n+ 1

(n∑i=0

δi

)

= (1− τ)2φ(x) + τ2L2(β(x))

n+ 1+ 2τ(1− τ)L

2(β(x))

n+ 1

= φ(x) +

(φ(x)− L

2(β(x))

n+ 1

)(1− τ)2 −

(φ(x)− L

2(β(x))

n+ 1

).

Since, by Schwarz inequality, φ(x)− L2(β(x))n+1 ≥ 0, we find for 0 ≤ τ ≤ 1:

φ(xτ ) ≤ φ(x) ≤ b.

This means that xτ ∈M b∗ .∆0 is the identity and ∆1 = α ◦ β by definition. For the straightforward proof

that ∆τ is actually a deformation, we again refer to [1].�

7. The Index Theorem

We use the notations of Section 6. On M b∗ , we shall study the function, whichsome may refer as the energy function:

φ(x) =

n∑i=0

ρ2(xi, xi+1)

22 RAOUL BOTT

Theorem 7.1. x = (x1, ..., xn) ∈M b∗ is a critical point of φ if and only if(i) px1x2..xnq is a geodesic from p to q;

(ii) ρ(p, x1) = ρ(x1, x2) = ... = ρ(xnq).

Figure 13.

Proof. We assume the First Variation Formula, which can be stated in the followingway (see [3]). Let a, b ∈ M , a 6= b, with ρ(a, b) < ρ (see section 6), let g be theunique geodesic segment from a to b. X0 and X1 are the unit tangent vector to g ata and b, respectively. U0 and U1 are tangent vectors to M at a and b, respectively.The function ρ(x, y) is differentiable in a neighborhood of (a, b) on M ×M . Sowe can speak of the directional derivative of ρ(x, y) in the direction of the tangentvector (U0, U1) at (a, b) as

〈X1, U1〉 − 〈X0, U0〉.Furthermore, if we fix a and consider the function ρ(a, x), x 6= a, on M , thedirectional derivative in direction U1 is simply 〈X1, U1〉.

We denote by si the length of the geodesic segment gi from xi to xi+1. On gi,denote the unit tangent vectors to gi at xi, xi+1 by s

−i , s

+i , respectively, i = 1, ..., n.

Then the directional derivative of φ in direction (U1, ..., Un) is:

2s0〈s+0 , U1〉 − 2s1〈s−1 , U1〉+ 2s1〈s

+1 , U2〉 − ...− 2sn〈s−n , Un〉.

For this expression, we can write:

2[〈(s0s+0 − s1s−1 ), U1〉+ 〈(s1s

+1 − s2s

−2 ), U2〉+ ...+ 〈(sn−1s

+n−1 − sns−n ), Un〉]

x is a critical point of φ, if and only if this expression vanishes for all tangentvectors (U1, ...Un). This means precisely that x is a critical point of φ, if and onlyif px1x2...xn is a geodesic from p to q and ρ(p, x1) = ρ(x1, x2) = ... = ρ(xn, q). �

Remark 7.2. If we consider the continuous function L =∑ni=0 ρ(xi, xi+1), then L

is differentiable in a neighborhood of all points x = (x1, ..., xn) ∈M b∗ with xi 6= xjfor all i 6= j. In the same way as in the proof of Theorem 7.1, we see that such apoint x is a critical point of L, if and only if px1x2...xnq is a geodesic from p to q.

The index theorem gives an expression for the index of a critical point of φ. Toderive this expression, we need several preliminaries.

Lemma 7.3. Let A be a quadratic form on the real vectors space Y. then the indexof A can be characterized as the maximal dimension of a subspace of V on whichthe restriction of A is positive.


Proof. Let d de this dimension, λ the index of A, and �1, ..., �n a coordinate systemon V , such that A is given by

λ∑i=1

�2i +

n∑j=λ+1

�2j .

Since A is negative definite on the subspace of V spanned by x1, ..., xλ, d ≥ λ.Conversely, suppose A to be negative definite on a subspace W of V of dimensiond′ > λ. W has a subspace of dimension at least 1 in common with the subspacespanned by xλ+1, ..., xn on which A is semi-positive definite. This is impossible,thus λ ≥ d. Combination gives λ = d. �

Lemma 7.4. Let A(t), 0 ≤ t ≤ 1, be a continuous family of quadratic forms onthe n-dimensional real vector space V n, with the following properties:

(i) A(t1) ≤ A(t2) for t1 ≤ t2;(ii) A(t) is degenerate for a finite number of t-values, 0 < t < 1: α1, ..., αr and

possibly for t = 0, but not for t = 1.

Then: λ(A(0))− λ(A(1)) =∑rk=1 ν(A(αk)).

Proof. Let us look at the point αi. From condition (i) and Lemma 7.2, we deduce

(2) λ(A(t)) ≤ λ(A(αi)) for t ≥ αi.On the other hand, if A(αi) is negative definite on the subspace W of V

n, thenA(t) is negative definite on W for |t− αi| < �. From Lemma 7.2, it follows

(3) λ(A(t)) ≥ λ(A(αi)) for |t− αi| < �.Combining (2) and (3) we find:

(4) λ(A(t)) = λ(A(αi)) for αi ≤ t ≤ αi + �.Denoting by µ(A(t)) the number of positive terms in a reduction of A(t) to diagonalform, we prove in the same way for αi − � < t < αi:

(5) µ(A(t′)) = µ(A(αi)).

Combination of (4) and (5) gives:

λ(A(t′)− λ(A(t)) = n− µ(A(t′))− λ(A(t)) = n− µ(A(αi))λ(A(αi)) = ν(A(αi))the nullity of A(αi). Application of this last result to the points α1, ..., αr clearlygives the statement of the lemma. �

Let g(t) be a constant-speed geodesic on M . A family of geodesics gα(t),−∞ < α < ∞, all parametrized proportionally to arc-length from gα(0), is calleda variation of g(t) if

(i) gα(t) depends differentiably on α;(ii) g0(t) = g(t).

A variation of g(t) induces a vector field U(t) along g by

U(t) =∂V (α, t)

∂α

∣∣∣∣α=0

.

A vector field along g, induced by a variation of g is called a Jacobi field along g.Clearly, this definition generalizes the concept of Jacobi field along a straight linein a Euclidean space, as defined in Section 4. A vector field along the segment s ofg is called a Jacobi field along s, if and only if it is the restriction to s of a Jacobi

24 RAOUL BOTT

field along g. If a and b are the endpoints of s, the vector space of Jacobi fieldsalong g which vanishes at both a and b is denoted by

∧sab.

We admit the following properties of Jacobi fields which are either trivial or canbe proved by standard methods.

(i) Let U ′(t) be the covariant derivative of U(t) along g. Then there is one andonly one Jacobi field along g with given initial values U(t0) and U

′(t0);

Proof sketch. Jacobi fields are determined by the Jacobi equation, which isa second-order differential equation, so (i) is trivial. �

(ii) If the segment g(t), 0 ≤ t ≤ 1, of g lies completely in a neighborhood inwhich every two points can be joined by a unique geodesic arc, then thereis one and only one Jacobi field along g(t) with prescribed boundary valuesat g(0) and g(1);

Proof sketch. g is the unique geodesic from g(0) to g(1), so g(0) and g(1)are in the injective radius of each other. In particular, they are not conju-gate. Then for U(g(0)) = 0, the linear transformation Tg(0)M → Tg(1)M ,U̇(g(0)) 7→ U(g(1)) is non-singular. So combining with (i), there existsunique Jacobi field with U(g(0)) = 0, U(g(1)) = b. Similarly, there existsunique Jacobi field with U(g(0)) = a, U(g(1)) = 0. Summing up these twoJacobi fields yields the desired Jacobi field. �

(iii) If the Jacobi field U(t) is perpendicular to g(t) for t = t1 and for t = t2,t1 6= t2, then U(t) is perpendicular to g(t) for all values of t.

Proof sketch. By g̈ = 0 and Ü +R(U, ġ)ġ = 0, we have

d

dt〈U̇ , ġ〉 = 〈Ü , ġ〉 = 〈R(U, ġ)ġ, ġ〉 = 0.

Thus 〈U̇ , ġ〉 = ddt 〈U, ġ〉 is constant, then 〈U, ġ〉 is linear. Since 〈U, ġ〉 is 0for t = t1 and for t = t2, t1 6= t2, it must be constant. �

Figure 14.

Let U ⊂M be an open set in which every two points can be joined by a uniquegeodesic arc. Let a, b ∈ U , a 6= b, and let g(t), 0 ≤ t ≤ 1, be the constant-speedgeodesic from a to b. The function l = ρ(x, y) is a differentiable function on U ×Uin a neighborhood of (a, b). Let Ca and Cb be differentiable cells of codimension1, passing through a and b, respectively, and perpendicular to g(t) at a and b,respectively. C = Ca ×Cb is a submanifold of U × U . Let l∗ be the restriction of l


to C. It follows from (4), that (a, b) is a critical point of l∗, and the Hessian Hl∗a,bis well-defined.

We admit the following expression for Hl∗a,b (this is the so-called Second Vari-

ation Formula, see [3]):

(6) XYHl∗a,b = (Xb, Y′b )− (Xa, Y ′a) + nb(Xb, Yb)− na(Xa, Ya),

where the symbols have the following meaning: Xa and Ya are tangent vectors toCa at a, Xb and Yb are tangent vectors to Cb at b. (Xa, Xb) and (Ya, Yb) can beconsidered as tangent vectors of C at (a, b). By property (ii) of Jacobi fields, thereis a unique Jacobi field Y (t) along g(t) with Y (0) = Ya and Y (1) = Yb. Y

′(t)denotes the covariant derivative of Y along g(t). Finally, na(Xa, Ya) and nb(Xb, Yb)denote the second fundamental forms at a and b with respect to g(t), evaluated atXa, Ya and Xb, Yb respectively. For a definition of this for which plays no furtherrole in these notes, we refer to [5], pg. 257, and the reference given there.

Furthermore, if we fix a and consider the function l = ρ(a, x), x 6= a, and alsoits restriction l∗ to Cb, then b is a critical point of l

∗, and

(7) XbYbHl∗ = (Xb, Y

′b ) + nb(Xb, Yb),

where Y ′b is the covariant derivative of the Jacobi field which is 0 at a and Yb at b.

Figure 15. Caption

Again, we use the notations of section 6, but replace p, q by c, d. Supposecx1x2...xnd is a geodesic from c to d such that xi 6= xj for i 6= j. From the remarkafter Theorem 7.1, it follows that x = (x1, ..., xn) ∈ M b∗ is a critical point of thefunction

L =

n∑i=0

ρ(xi, xi+1).

Let Ci, i = 1, ..., n be differentiable cells of codimension 1, passing through xi andorthogonal to g(t) at xi. We set C1× ...×Cn = N and consider N as a submanifoldof M b∗ . Clearly, x ∈ N . Let L∗ be the restriction of L to N . Obviously, x is acritical point of L∗. From (6) and (7), we deduce the following expression for theHessian HL∗x:

(8) UVHL∗x =

n∑i=1

Ui(V+i − V

−i ).

where U = (U1, ..., Un), V = (V1, ..., Vn), Ui and Vi being tangent vectors to Ci inxi, i = 1, ..., n. V

−i and V

+i denote the values at xi of the covariant derivative along

26 RAOUL BOTT

xi−1xi and xixi+1, respectively, of the Jacobi field determined by Vi−1, Vi and Vi,Vi+1, respectively, with V0 = Vn+1 = 0.

If V is in the null space of HL∗, then it follows from (8) and property (i) that theJacobi fields determined by V0 and V1 along rx1, by V1 and V2 along x1x2,... forma global Jacobi field along g(t), 0 ≤ t ≤ 1, which vanishes at r and s. Conversely,by (8) and property (iii), such a global field determines a vector V in the null spaceof HL∗.

Proposition 7.5. Let g(t), 0 ≤ t ≤ 1, be a geodesic on M , g(0) = r, g(1) = s. Letx = (x1, ..., xn) ∈ M b∗ be a subdivision of g(t), x0 = r, xn+1 = s and xi 6= xj fori 6= j. Let L∗x be defined as above. Then ν(HL∗x) = dim

∧g(t)rs .

Again we use the notations of Section 6. Let g(t), 0 ≤ t ≤ 1, be a constant-speedgeodesic, with g(0) = p, g(1) = q. Let g(t) be subdivided by x = (x1, .., xn) ∈M b∗ ,set p = x0, q = xn+1 and suppose xi 6= xj for i 6= j for i, j = 1, ..., n+ 1.

We claim that there are only a finite number of t-values, 0 < t < 1, which wecall α1, ...., αr, such that dim

∧gpg(αi)

6= 0. (see [3] for the proof)Let g(s1) be a point of g(t) between xn and q, s1 6= αi, i = 1, ..., r. For s1 ≤ t ≤ 1,

we define the functions L(t) and L∗(t) in the same way with respect to p and g(t)as the functions L and L∗ are defined with respect to p and q.

From the remark after Theorem 7.1, it follows that x is a critical point of L(t)and therefore of L∗(t) for all t, s1 ≤ t ≤ 1. Using the triangle inequality, we findfor t1 ≤ t2:

L(t2)(x′) ≤ L(t1)(x′) + ρ(g(t1), g(t2))

for x′ in a neighborhood of x. Since

L(t2)(x) = Lx(t1)(x) + ρ(g(t1), g(t2))

we findHL(t1)|x ≥ HL(t2)|x.

By restriction (Lemma 7.2):

HL∗(t1)|x ≥ HL∗(t2)|x.Applying Lemma 7.3 we find:

λ(HL∗|x) = λ(HL∗(s1)|x) +∑

s1≤t≤1

ν(HL∗(t)(x)),

and by Proposition 7.4:

(9) λ(HL∗|x) = λ(HL∗(s1)|x) +∑

s1≤t≤1

dim∧gpg(t) .

Now we replace the subdivision x of g(t) by a subdivision y = (y1, ..., yn) ∈ M b∗of the segment pg(s1) of g(t), in such a way that the t-value of yn is smaller thanthat the t-value of xn.

On M b∗ there is a path Y from x to y such that L∗(s) non-degenerate at all of

its points. Since HL∗(s) depends on s continuously and is non-degenerate for allthe points of Y , we find

λ(HL∗y(s)) = λ(HL∗x(s)).

We choose a point g(s2) 6= α1, ..., αr between yn and g(s1), and have the samereasoning for the segment g(s1)g(s2) as we did above for the segment g(s1)q. Wecan go this way, such that after a finite number of steps we arrive at a point sl on


g(t), lying close enough to p that the function L(sk) has an absolute, non-degenerateminimum at the subdivision z = (z1, ..., zn) ∈M b∗ , obtained by repeated applicationof our procedure. It follows that HLz(sk) and therefore HL

∗z(sk) is positive definite;

i.e., λ(HL∗z(sk)) = 0. Repeated application of (9) finally gives

(10) λ(HL∗(x)) =∑

0

28 RAOUL BOTT

The first relation follows from the fact that dτ |Np is an automorphism of Np; thatis, from condition (ii). The second we already used. The third is a consequenceof condition (iii). Indeed, let F (x) = f(x) − f(τ(x)), x ∈ U . Then F ≥ 0, withF (p) = dF (p) = 0. Hence HFp is non-negative. But this Hessian is preciselyHf − τ∗(Hf), whence Hf − τ∗(Hf), and this clearly implies λ(Hf) ≤ λ(τ∗(Hf)).On the other hand, we also have

(13) λ(τ∗(Hf∗)) = λ(τ∗(Hf)).

To see this remark that τ∗(Hf∗) is just the restriction of τ∗(Hf ) to Np. Hence

(14) λ(τ∗(Hf∗)) ≤ λ(τ∗(Hf)).

Suppose A is a subspace on which τ∗(Hf) is negative definite. Then A does notintersect the kernel of dτ , therefore B = dτ(A) ⊂ Np has the same dimension as A.By definition, τ∗(Hf∗) is negative definite on B and since dimB = dimA, we find:

(15) λ(τ∗(Hf∗)) ≥ λ(τ∗(Hf)).

Combining (14) and (15), we get (13).From (13), it follows that the inequalities in (12) must in fact be equalities. In

particular, λ(Hf) = λ(Hf∗). This completes the proof of Lemma 7.5, and now wehave all the necessary preparations for the proof of (11). �

Proof of (11). Let F = φ− 1n+1L2. By the Cauchy inequality F ≥ 0. Further, near

x, F is differentiable and F (x) = 0. Hence dF (x) = 0, and the Hessian of F is notnegative at x. Thus Hφx ≥ 1n+1HL

2, whence

(16) λ(Hφx) ≤ λ(HLx).

Our next steps is to show that λ(HLx) = λ(HL∗x) by means of Lemma 7.5. We

have to construct a map τ : U → N of a suitable neighborhood U of x with theproperties (i), (ii), and (iii) of Lemma 7.5. For thsi purpose, let �, 0 < � < L(x)2(n+1) ,

be a real number and let C�i be the parallel translates of Ci along g(t) by a distance�, in the direction from p to q. Similarly, let C−�i be the �-translate along g(t) inthe opposite direction. The geodesic g meets C±�i transversely. Hence there is aneighborhood U ′� of x, such that for all y ∈ U ′� the polygon β(y) (Section 6) alsomeets the C±�i transversely, say at points η

±�i (y). These points, taken in the obvious

order p, η−1 , η+1 , η

−2 , ..., defines a polygon η�(y). We now define the components of

τ�(y) to be the intersection of η�(y) with Ci. The transformation τ� is well definedand differentiable on some smaller vicinity U� of x, again because τ�(x) = x, so thatη�(y) will meet the Ci’s transversely for y close enough to x.

Figure 17.


In figure 17, the situation is described graphically. τ� maps U into N , furtherτ�(x) = x and L(τ(y)) ≤ L(y) as follows from the triangle inequality. This allrequirements of Lemma 7.5 are satisfied except possibly (ii). We therefore still

have to show that τ�|N has a non-singular differential for 0 < � < L(x)2(n+1) , andclearly depends continuously on �. However, for � = 0, τ0|N is the identity. Hencefor � sufficiently small, d(τ�|N ) is non-singular. Applying Lemma 7.5, we haveestablished

(17) λ(HLx) = λ(HL∗x).

From (7) we deuce

(18) UVHφ∗x =2L(x)

n+ 1

n∑i=1

Ui(V+i − V

),

i

where the symbols have the same meaning as in (8). From (8) and (18) it follows

(19) λ(HL∗x) = λ(Hφ∗x).

Finally, the first part of Lemma 7.5 gives:

(20) λ(Hφ∗x) ≤ λ(Hφx)Combining (16), (17), (19), (20), we proved (11). �

From (10) and (11), we deduce the Index Theorem.

Theorem 7.7 (The Index Theorem). Let g(t), 0 ≤ t ≤ 1, be a geodesic on M ,g(0) = p, g(1) = q and let x = (x1, ..., xn) ∈M b∗ be the subdivision of g(t) given byρ(p, x1) = ρ(x1, x2) = ... = ρ(xn, q). Then x is a critical point of the function

φ = ρ2(p, x1) + ρ2(x1, x2) + ...+ ρ

2(xn, q),

and

λ(Hφx) =∑

0

30 RAOUL BOTT

Lemma 8.3. Let f be a smooth function on M which assumes its absolute minimumon the non-degenerate critical manifold V ⊂M . If there are no other critical pointsof f on Ma and if Ma is compact, then Ma ∼ V .

Proof. Let p ∈ V , and let f ′ be the restriction of f to the “geodesic plane” per-pendicular to V at p. The gradient of f ′ is transverse to an �-sphere in this planeof � small enough. This follows from lemma 8.2, since Hf ′p is non-degenerate bythe definition of a non-degenerate critical manifold. From the compactness of V ,we deduce the existence of a global tubular neighborhood of N of V in M suchthat ∇f is transverse to the boundary of N . In the same way as in the proof oftheorem 2.1, a deformation of Ma in N can be constructed. Combining this withthe obvious fact that N � ∼ V , we get the statement of the lemma. �

Lemma 8.4. Let f be a smooth function on M such that for a ≤ x ≤ b, there isonly one critical value x = c, a < c < b. Suppose furthermore that f−1(c) is thenon-degenerate critical manifold V . If M b is compact, then:

M b ∼Ma ∪ eλ1 ∪ ... ∪ eλr .with λi ≥ λ(V ).

Proof. Let N �1 and N �2 be closed, sufficient;y small tubular neighborhood of V inM , of radius �1, �2, respectively, �1 < �2. Let φ(x) be a smooth function on M

b

with the following properties:

φ(x) = 1 on N �1

φ(x) = 0 outside N �2

Figure 18.

Let π : N �2 → V be the fibre projection and g a (strongly) non-degeneratefunction on V . We set

f ′ = f + αρ(x)π∗(g), α > 0,

and assert that for α sufficiently small, f ′ has non-degenerate critical points only,all lying on V and precisely the critical points of g. We consider f−1(a < x < b) ⊂M . Outside N �2 , f ′ has no critical point. Between N �1 and N �2 and onto theirboundaries, we have:

|df | > A, d|ρ(x)π∗(g)| < B,and therefore:

|df ′| ≥ |df | − αd|ρ(x)π∗(g)| > A− αB > 0for α sufficiently small. By looking at a restriction to a fibre of N �1 , we see that onN �1 , f ′ has only critical points on V and there only at the critical points of g. In


these points, the null space of Hf is just the tangent space to V (by definition) andthat of π∗(g), the tangent space to the fibre of N �1 through that point (by directverification). It follows that the critical points are non-degenerate and that theirindices are at least λ(f)p = λ(V ).

�

9. The Stable Homotopy Groups of the Unitary Group

In this final section, we shall apply the results of the preceding sections to thecalculation of the stable homotopy group of the unitary groups. This means thefollowing: The exact homotopy sequence of the fibering

U(n)/U(n− 1) = S2n−1

gives usπk(U(n)) = πk(U(n− 1)), 0 ≤ k ≤ 2n− 2,

from which it follows that

πk(U(n)) = πk(U(m)), n,m ≥k + 2

2.

This group is called the k-th stable homotopy group of the unitary group(s). It willbe denoted by πk(U).

We start with a simple result concerning Jacobi fields on a compact connectedLie group, considered as a Riemannian manifold by providing it with a left andright invariant Riemannian structure. Since we need only the result for SU(n), weshall state and prove it for this group, but the general case is no more difficult. Thereason, that we consider SU(n) rather than U(n) is that Ω(SU(n)) is connectedbut Ω(U(n)) is not. The way in which we use this will become clear in the sequel.

The Lie algebra L of SU(n) is provided with a metric m′ in a natural way, namelywith the restriction to L of the metric m, introduced in Section 5.

For Y ∈ L,V (ρ, t) = eρY · etX · e−ρY , −∞ < ρ < +∞

defines a variation of the geodesic g(t) = etX , X ∈ L. On g(t), we consider inparticular the segment s : 0 ≤ t ≤ 1, starting at the identity e of SU(n) and endingat a = g(1). V (ρ, t) induces a Jacobi field along g(t), given by

(1) UY (t) =∂

∂ρ(eρY · g(t) · e−ρY )

∣∣∣∣ρ=0

.

Writing instead of eρY · g(t) · e−ρY , eρY · e−Adg(t)ρY · g(t), we see that(2) UY (t) = (Y −Adg(t)(Y ))g(t).

The result we shall prove is, that all elements of∧gea can be written in the form

(1) for a suitable choice of Y .Let J denote the vector space of Jacobi fields along s, which vanishes at e

(but not necessarily at a). As remarked in Section 7, a Jacobi field along g(t) iscompletely determined by its value at e and the value of its covariant derivativealong g(t) at e. From this, it is obvious that dim J = dimSU(n) = dimL.

Let P ⊂ L be the subspace of L, consisting of all those elements of L whichinduce the zero Jacobi field along s. From (2) and from

[X,Y ] = limt→0

AdetXY − Yt

,

32 RAOUL BOTT

it follows that

(3) P = {Y ∈ L, [X,Y ] = 0}.Consider the sequence

(4) 0 −→ P α−→ L⊕ P β⊕γ−→J −→ 0,where α denotes the injection of P into L, β the homomorphism which attachesto Y ∈ L, UY ∈ J , and where γ is defined as follows: γ(Z), Z ∈ P , will be theJacobi field along g(t) induced by the variation

(5) W (ρ, t) = et(ρZ+X), −∞ < ρ < +∞of g(t). Since W (ρ, 0) = e for all ρ, γ(Z) ∈J for all Z ∈ P .

We shall prove that the sequence (4) is exact. For this, it is sufficient to prove:β(Y ) + γ(Z) = 0, Y ∈ L, Z ∈ P implies: Y ∈ P , Z = 0. Indeed, if we provethis, then β ⊕ γ is automatically surjective, since dim kernel(β ⊕ γ)) = dim P ,dim image(β ⊕ γ)) = dim L + dim P − dim P = dim L = dim J .

Now, (3) allows us to write instead of (5),

W (ρ, t) = etρZ · etX ,

(6)∂

∂ρW (ρ, t)

∣∣∣∣ρ=0

= tZetX .

If β(Y ) + γ(Z) = 0, then, by (2) and (6):

(7) Y −AdetX (Y ) + tZ = 0, 0 ≤ t ≤ 1.Denoting by ( , ) the inner product with respect to m′, we find:

(Y,Z)− (AdeX (Y ), Z) + (Z,Z) = 0.Since (Y,Z) = (AdeX (Y ),AdeX (Z)), this leads to

(8) AdeX (Y ){AdeX (Z)− Z}+ (Z,Z) = 0.Z is in P , therefore by (2), the first term on the left side of (8) vanishes and wefind: (Z,Z) = 0, hence Z = 0, hence, by (7), Y ∈ P . This proves the exactness of(4).

Finally, to prove that the elements of∧gea can be written in the form of (1), it

is sufficient to prove that (β⊕ γ)−1(∧gea) ⊂ L. But the pair (Y,Z) (Y ∈ L, Z ∈ P )

lies in the inverse image if and only if (7) holds for t = 1. This gives again Z = 0.

Proposition 9.1. Let SU(n) be provided with a left and right invariant Riemann-ian structure, let g(t), 0 ≤ t ≤ 1, g(0) = e, g(1) = p be a geodesic segment onSU(n).

Then every element of∧gep is induced by a variation of g(t) of the form

eρY · g(t) · e−ρY , −∞ < ρ < +∞,where Y denotes an appropriate element of the Lie algebra of SU(n).

Our argument gives the same result for any compact Lie group, but we do notneed this fact, in its turn a very special case of the “variational completeness” ofsymmetric spaces (see [6]; also [5] for the proof given above).

We consider the situation of Section 7, with M = SU(2m), p = e and q = −e.According to Theorem 7.1, the critical points of φ on M bn = M

b∗ correspond to the

n-tuple (x1, ..., xn), such that ex1x2...xn(−e) is a geodesic segment on SU(2m),


and such that ρ(e, x1) = ρ(x1, x2) = ... = ρ(xn,−e). Let y = (y1, ..., yn) be such ann-tuple. Then all the transforms of y by the natural operation of SU(2m) on M∗(induced by the adjoint action of SU(2m) on itself) are also critical points of φ. Itfollows that the critical points of φ on N b∗ = M

b∗ minus its boundary consist of a

series of compact submanifolds of N b∗ , on each of which φ is constant.We shall prove that these critical manifolds are non-degenerate in the sense of

Section 8.Let

X =

iα1 0. . .0 iα2m

be an element of the Lie algebra of U(2m), considered as the space of skew-symmetric matrices (see Section 5). X lies in the Lie algebra of SU(2m) if andonly if

2m∑k=1

αk = 0.

In that case, the geodesic

etX =

eiα1t 0

. . .

0 eiα2mt

lies on the maximal torus D, consisting of the diagonal matrices of SU(2m).

Now let C be a critical manifold of φ on N b∗ , let x ∈ C, and let g be a geodesicex1x2...xn(−e). There is an element ω ∈ SU(2m), such that ωx1ω−1 ∈ D. ωgω−1passes through ωx1ω

−1 lying on D, so ωgω−1 ⊂ D. Therefore, every criticalmanifold contains at least one point x = (x1, ..., xn), such that ex1...xn(−e) ⊂ D.

To prove our statement, we have to show, that for a point p on a critical manifoldC of φ, the zero space of Hφp is precisely the tangent space to C at p. A trivialcalculation shows that every tangent vector to C at p certainly lies in the zero space.It therefore suffices to show that ν(Hφp) = dimC. By the same kind of argument asused in Section 7 to prove that λ(Hφp) = λ(Hφ

∗p), it can be proved (in the general

case) that ν(Hφp) = ν(Hφ∗p). By Proposition 7.5, this nullity equals dim

∧se(−e),

where s denotes the geodesic segment ex1...xn(−e). If we denote by G the subgroupof SU(2m), leaving s fixed, we have: dimC = dimSU(2m)− dimG. On the otherhand, if follows from Proposition 9.1, that dim

∧se(−e) = dimL − dimP . Since

dimL = dimSU(2m), and since it follows easily from (3) that dimP = dimG, wehave proved our theorem that C is a non-degenerate critical manifold.

From the proof, it follows also that to determine the indices of the critical mani-fold C, it is sufficient to determine the indices of geodesic segments g(t), 0 ≤ t ≤ 1,from e to −e, lying on D and considered as representing a critical point x of φ onN b∗ . By Theorem 7.7 (the index theorem), this index equals∑

0

34 RAOUL BOTT

provided that dim∧(g)eg(t) differs from 0 only for a finite number of t-values, 0 ≤ t ≤ 1.

The argument, giving us the equality

dim∧(s)e(−e) = dimSU(2m)− dimG,

tells us also, that

dim∧(g)eg(t) = dimGg(t) − dimG,

where Gg(t) denotes the subgroup of SU(2m), leaving the point g(t) fixed.Let the segment s : 0 ≤ t ≤ 1 on g(t) be given by

g(t) =

e2πiα1t 0

. . .

0 e2πiα2mt

Since g(t) ⊂ SU(2m),

2m∑k=1

αk = 0,

and since g(1) = −e,

αk =2βk − 1

2, k = 1, ..., 2m, βk integer.

Conversely, every sequence of rational numbers α1, ..., α2m, fulfilling these con-ditions gives us a geodesic segment from e to −e on D.

If we take α1 = ... = αm =12 , αm+1 = ... = α2m = −

12 , then it can be checked

easily that for all values of t, 0 < t < 1, dimGg(t) = dimG. Therefore, thisα-sequence gives a geodesic ex1...xn(−e), such that λ(x) = λ(C1) = 0, x ∈ C1.Consequently, on this C1, φ assumes its absolute minimum a0.

C1 = SU(2m)/S(U(m)× U(m)) = U(2m)/(U(m)× U(m))

is a Grassmann variety.If we take for example the sequence

α1 =3

2, α2 = ... = αm−1 =

1

2, αm = ... = α2m = −

1

2,

then on the corresponding geodesic, there is for 0 < t < 1 just one t-value t0 =12 ,

such that dim∧(g)eg( 12 )

6= 0 (there is just one value of t, 0 < t < 1, namely t = 12 ,such that 32 t and −

12 t are congruent modulo 1).

dim∧(g)eg( 12 )

= dimS(U(m+ 2)× U(m− 2))− dimS(U(1)× U(m− 2)× U(m+ 1))

= 2m+ 2.

It can be checked that the index of all geodesic g corresponding to α-sequence inwhich not only αk = ± 12 appears, is at least this number. In fact, it is sufficient toremark that in such a case, there is at least one point g(t0) with dim

∧geg(t0)

6= 0,0 < t0 < 1 (and on the other hand only a finite number), and this dimension equals

S(U(n1)× ...× U(nk))− S(U(m1)× ...× U(ml)),

where m1 + ...+ml = 2m is a refinement of n1 + ...+nk = 2m. Then this expressionis at least 2m+ 2. (see Milnor’s Morse Theory, p.131)


Since two geodesic segments on D from e to −e and corresponding to α-sequenceswith all αk = ± 12 clearly lies in the same critical manifold C, we deduce fromTheorem 6.1, Lemma 8.2, Lemma 8.3 and the results above that for every a > a0

Ωae(−e)(SU(2m)) = SU(2m)/S(U(m)× U(m)) ∪ ed1 ∪ ed2 ∪ ... ∪ edi

with d1, d2, ..., di ≥ 2m+ 2, and i depending on a. Therefore

πk(Ωe(−e)(SU(2m))) = πk(SU(2m)/S(U(m)× U(m)))= πk(U(2m)/U(m)× U(m)), 3 ≤ k ≤ 2m,

Since, as stated at the start of Section 1,

πk(Ωe(−e)(SU(2m))) = πk+1(SU(2m)),

we can conclude:

(9) πk+1(SU(2m)) = πk+1(U(2m)) = πk(U(2m)/U(m)×U(m)), 3 ≤ k ≤ 2m.

At the very beginning of this section, we remarked that

πk(U(2m)/U(m)) = 0

Using the exact homotopy sequence of the fibering

(U(2m)/U(m), U(2m)/U(m)× U(m), U(m))

we find:

(10) πk(U(2m)/U(m)× U(m)) = πk−1(U(m)), 1 ≤ k ≤ 2m− 2.

Combining (9) and (10), we get for large m:

πk+1(U(2m)) = πk−1(U(m)), k ≥ 3.

Since π0(U) = π2(U) = 0 and π1(U) = π3(U) = Z ([7], p. 132), we have

πk+2(U) = πk(U), k ≥ 0,

and explicitly

π2k(U) = 0,

π2k+1(U) = Z, k = 0, 1, 2, ...

Acknowledgement. I typed this book as part of my Summer 2020 REU on Morsetheory. I want to thanks my mentor Professor Peter Petersen for recommendingthis book, resolving my confusions, and proofreading this document. I also wantto thanks Zhenyi Chen for our fruitful discussions on Morse Theory over the twomonths.

36 RAOUL BOTT

10. References

[1] H. Seifert and W. Threlfall: Variationsrechnung im Grossen, Teubner, Leipzig1938.

[2] K. Nomizu: Lie Groups and Differential Geometry, Publications of the Math-ematical Society of Japan, vol. 2, Tokyo 1956.

[3] M. Morse: The Calculus of Variations in the Large, A. M. S. ColloquiumPublications, vol. 18, New York 1934.

[4] W. Graeub: Linear Algebra, Die Grundlehren der Mathematichen Wissenschaften,vol. 97, Springer, Berlin 1958.

[5] R. Bott: An Applicaition of the Morse Theory to the Topology of Lie Groups,Bulletin dela Societe Mathematique de France, vol. 84(1956), pp. 251-282.

[6] R. Bott and H. Sameloon: Application of the Theory of Morse to SymmetricSpaces, American Journal of Mathematics, vol. 80(1958), pp. 964-1029.

[7] N. Steenrod: The Topology of Fibre Bundles. Princeton 1951.For the materials of these notes, see also[8] R. Bott: The Stable Homotopy Groups of the Classical Groups, Annuls of

Mathematics (2), vol. 70(1959), pp. 313-337.

1. Introduction2. Morse Theory of Smooth Functions on a Manifold3. The Morse Inequalities4. Manifolds Embedded in an Euclidean Space5. Topology of Flag Manifolds6. The Structure of the Space p,q(M)7. The Index Theorem8. Critical Manifolds9. The Stable Homotopy Groups of the Unitary GroupAcknowledgement

10. References

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