Mapping Degree Theory (Graduate Studies in Mathematics 108)

II. Manifolds

58

it is the tangent space to M at x and is denoted by TxM. The vectors in TxM are called tangent to M at x. It is easy to produce a basis for TxM. Just consider the canonical basis ei

= (0, ... ,1, ... ,0) of]Rm and set

Notice that this basis ofTxM consists of the columns ofthe Jacobian matrix of da'P : ]Rm -+ ]RP. The partial derivative notion comes from the fact that one can look at any vector u E ]Rn as a directional derivative. We will not pursue this fruitful viewpoint here, but we keep the notation of partial derivatives anyway. In a different way, one can use local equations to describe the tangent space. Namely, if II, ... ,fq are local equations of M in a neighborhood U of x (11.2.3, p. 55), then u E TxM if and only if

(this is generalized later in II.3.2, p. 62). For one single equation that is, when M is a hypersurface in ]Rm+1, we have

0= dxf(u)

f = II,

af af ~ = (gradx(f), u), gradx(f) = ( aX} (x), ... , axp (x);;

that is, TxM is the linear hyperplane perpendicular to the gradient grad x(f). Finally, note that for a product M x N of two manifolds we have

Example 2.9. Let us look again at spheres §m. One can use the stereographic projection 7rN to obtain a basis Tx§m at the south pole x = as:

Similarly, 7rS gives the same basis of Tx§m at the north pole x = aN. We see that Tx§m ..L x in both cases. This is in fact true for all x E §m, and we invite the reader to pursue the computations at an arbitrary x. However, it is much easier to use the equation f(x) = I:i xT - 1 = 0: TxM is the hyperplane perpendicular to the gradient gradx(f) = 2x. 0

2. Differentiable manifolds

59

Now we can define the following: (2.10) Derivative of a differentiable mapping. Let f : M ~ N be a differentiable mapping of differentiable manifolds, and consider a point x E M. We can extend f to a differentiable mapping f from some open neighborhood of x in RP into some open neighborhood of y = f(x) in Rq, and this extension has a derivative dx ! : RP ~ Rq. Then, it is easy to check that by restriction we get a well-defined linear mapping

which does not depend on the choice of f and is called the derivative of f at x. Alternatively, we can localize, that is, pick coordinate domains U at x and V at y such that f (U) c V, and build the following commutative diagrams:

U

~

V

TxM da'P

r~

Rm

~ TyN

~

1

(db'I/J)-l

~ Rn

(where

We insist that the consideration of manifolds is necessary to have derivatives in any reasonable sense: the notion of differentiable mappings was given from the very beginning with no restriction on the sets involved, but for derivatives we need tangent spaces, available only for manifolds. Then, once derivatives are defined, calculus follows: chain rule, etc. The only caution concerns the boundary sometimes. For instance, it is easy to see that if a differentiable function f : M ~ R has a local extreme at a point x E M \ aM, then the derivative dxf : TxM ~ R is identically zero. In the same vein, the Inverse Mapping Theorem must be modified to read: a differentiable mapping f : M ~ N is a local diffeomorphism at x if and only if dx f is a linear isomorphism and f preserves boundaries locally at x. Of course, if M is boundaryless, the latter condition is meaningless.

II. Manifolds

60

(2.11) Tangent vectors and curve germs. We can describe tangencies in a very geometrical way: a tangent vector to M at x is always tangent to some curve germ in M. This means that for every u E TxM there are differentiable mappings, : [0, c) -+ M c jRP such that ,(0) = x and

u = ,'(0) =

(ri (0), ... , ,~(o»

=

do,(l)

E TxM

Furthermore, if f : M -+ N is differentiable, dxf(u)

c

jRP.

= (f 0 ,)'(0).

Example 2.12. Here there is a simple application of tangent vectors: if a curve and a manifold meet transversally at a point, then the point is isolated in their intersection. Let r be a curve and M a manifold, both in jRP, and let a Ern M. The hypothesis says that no tangent vector to r at a is tangent to M at a. For the proof we can suppose that a is the origin in M = jRm X {o}. Then choose a parametrization, : t H (ri(t» of r with ,(0) = a, and we know that ,'(0) ~ TaM = jRm X {o}. Hence ,~(o) =1= 0 for some i > m, so that ,i(t) is strictly monotonous near 0 and 0 is an isolated zero. This means that ,(t) ~ M = jRm X {o} for small enough t =1= 0, as claimed. D

Exercises and problems Number 1. Prove that a subset M C ]RP is a differentiable manifold if and only if for every point x E M there are an open set A in some half-space IHlm and a differentiable mapping t/J = (t/Jl,'" ,t/Jp) : A -+ ]RP such that (i) t/J is a homeomorphism onto some open neighborhood of x in M and (ii) the Jacobian matrix (~) has rank m at t/J-l(X). J

Number 2. Show the following: (1) The cusp in ]R2 given by the equation x 2 = y3 is not a smooth curve, but it is homeomorphic to ]R. (2) The semi-cone in ]R3 defined by x 2 + y2 = Z2, Z ~ 0, is not a smooth surface, but it is homeomorphic to ]R2. Number 3. (1) Show that the solid torus M C ]R3 generated by the disk y = 0, (x - 2)2 + z2 ::; 1 around the z-axis is a manifold with boundary the torus generated by the circle y = 0, (x - 2)2 + Z2 = 1. (2) Show that the subsets Ml : xi + x~ ::; ~ and M2 : x~ + x~ ::; ~ of the unit sphere §3 C ]R4 : xi + x~ + x~ + x~ = 1 are diffeomorphic to M. (3) Deduce that §3 is the union of two solid tori along their boundaries, so that the meridians of each one are the parallels of the other. Number 4. Let M be a Hausdorff topological space M with a countable basis of open sets. A chart on M is a homeomorphism x : U f-t ]Rm from an open set U of M onto an open set x(U) C ]Rm. Two charts x and y with domains U and V are called cr

61

2. Differentiable manifolds

compatible (we fix the differentiability class r henceforth) when the mapping (of open sets in Euclidean spaces) yo x-I: x(U n V) ---t y(U

n V)

is differentiable. A differentiable atlas in M is a collection of Cr compatible charts whose domains cover M; when such an atlas exists, we say that M is an abstract (boundaryless) manifold. (1) Check that the concrete manifolds M C ]RP defined in the text are also abstract manifolds. Check that dimension is defined well and is the same in the abstract as in the concrete setting. (2) Show that real and complex projective spaces lRlP'm and cpm are abstract smooth manifolds, using homogeneous coordinates (xo : ... : x m ) to define charts with domains Xi i= O. Compute dimensions. Number 5. A continuous mapping f : M ---t N of abstract manifolds is Cr if for every two charts x, y on M, N, with domains U, V such that f(U) C V, the localization yo f 0 X-I: X(U) ---t y(V) is cr. Then we have the notion of diffeomorphism, and the loop closes, because every abstract manifold M is diffeomorphic to some concrete manifold. (1) Check that differentiable mappings between concrete manifolds as defined in this section are exactly differentiable mappings in the abstract sense. (2) Prove that ]Rpm is diffeomorphic to a compact manifold in some ]RP, using the mapping

How can this be adapted to cpm? (3) Show that the real projective line is diffeomorphic to tive line is diffeomorphic to §2.

§l

and the complex projec-

Number 6. Let p : ]Rn+l \ {O} ---t §n be the radial retraction given by x t-+ xlllxll. For every point a E ]Rn+l, a i= 0, let Sa be the sphere centered at the origin with radius lIall, and denote Ha = TaSa. Show, without explicit computations, that daP(a) = 0 and daPIHa is the homothecy of ratio 1/11all. Deduce from this an expression for the derivative daP(u). Number 7. Let k 2: 1 be an integer. Show that the set Mk C y2k + Z2k = 1 is a smooth surface and that the radial retraction X diffeomorphism from Mk onto the unit sphere §2.

]R3 t-+

given by X 2k + xlllxll induces a

Number 8. Let M C ]RP be a differentiable Cr (r 2: 2) manifold of dimension m. Prove that the tangent bundle

TM = {(x,u) E M is a differentiable

VM

x]RP : U E

cr- 1 manifold of dimension 2m.

= {(X,Ul,'" ,Urn) EM

is a non-compact differentiable Cr -

x]RP I

x···

TxM}

C]RP x]RP

Show also that the Stiefel bundle

x]RP: Ul, ...

,Urn is a basis of TxM}

manifold of dimension m(m + 1).

II. Manifolds

62

3. Regular values In this section we come to the key notion behind most constructions in differential topology. We refer to transversality. However, we do not need it in full generality, and for this reason we restrict our discussion to regular values, which is the first instance of transversality. (3.1) Regular values. Let f : M -t N be a differentiable mapping. A critical point of f is a point x E M whose derivative dxf : TxM -t Tf(x)N is not surjective. Critical points form a closed set, because the condition on their derivatives can be expressed by some equations involving the Jacobian determinant of a localization of f. The non-critical points are called regular points, and they form an open set. Now we look at a E N: we say that a is a critical value of f if there is some critical point x E M with f(x) = a. Hence, the set of critical values of f is the image of that of its critical points but need not be closed (unless f itself is closed, e.g., when M is compact). If a E N is not a critical value, then we call it a regular value of f. We denote by Cf C M the set of critical points of f and by Rf C N the set of regular values of f. Hence Rf = N \ f(Cf). Notice that if dim(M) < dim(N), then Cf

= M and

Rf

=N

\ f(M).

Now we recognize that when we defined equations II = ... = fq = 0 of a manifold Me ]RP on an open set U C ]RP (11.2.3, p. 55), we just said that owas a regular value of the mapping f = (II, ... , fq) : U -t ]Rq. In fact, the main feature of regular values is that their inverse images are manifolds. This is an easy consequence of the Inverse Mapping Theorem but must be stated with some care to keep track of boundaries: Theorem 3.2. Let f : M -t N be a differentiable Cr mapping, and let a E N \ 8N be a regular value of both f and flaM. Then (1) f-l(a) is a manifold with boundary f-l(a)

n 8M,

whose dimension

is dim(M) - dim(N). (2) Txf-l(a) = ker(dxf) for every x E f-l(a). Concerning the proof of this result we will at least mention the following. For every x E f-l(a) there are coordinate systems (Xl, ... , xm) of M at x and (Yl, ... , Yn) of N at a such that Yi 0 f = Xm-n+i for i = 1, ... , n (and

3. Regular values

Xl

63

2:: 0 where the boundary is involved). In other words, the localization of

f near x is a linear projection {

In fact,

dxf(a X m a- n +

a~j

(Xl, ... ,

Ix' j

xm -

n)

(Xl, ... , Xm)

.1)x = 1..1

1

VI'" a

= 1, ... , m - n,

H

(X m- n+1,"" xm). Then

for i = 1, ... , n, and generate Txf-l(a).

is a coordinate system of our inverse image at x.

We will use this result mainly when dim(M) - dim(N) = 1, so that f-l(a) is a curve. Then it is important to know that curves always have global parametrizations: Theorem 3.3 (Classification of compact differentiable curves). Let C be a compact Cl curve. Then C is Cl diffeomorphic to either the unit circle §l C JR2 or the closed interval [0,1] c JR.

Of course, to use these theorems we need to find regular values, but this is the famous Sard-Brown Theorem: Theorem 3.4 (Sard-Brown Theorem). Let f : M -+ N be a differentiable Cr mapping, with r > dim(M) - dim(N). Then the set of regular values of f is residual (a countable intersection of dense open sets), hence dense, in N. In particular, if dim(M) < dim(N), then N \ f(M) is residual, hence dense, in N.

The last assertion of the theorem is sometimes called the Easy Sard Theorem, because it can be proved directly by an easier argument. We remark that the proof of this theorem for finite class r < +00 is technically very demanding, as it requires the K neser- Glaeser Theorem to restore the differentiability class lost by partial derivation. Of course, this is no problem in the smooth case r = +00, and the proof is much easier in this case. Furthermore, this is enough for us, as we will reduce the differentiable case to the smooth one. In fact, as mentioned before (11.2.7, p. 57), every differentiable manifold has a smooth model, and this deep result roughly says that our limitation to smooth manifolds is only technical. After all these considerations, we include here the following proof: Proof of the Sard-Brown Theorem for smooth mappings. First of all notice that since M c JRP is locally compact, we can apply the Baire Theorem,

II. Manifolds

64

so that any countable intersection of dense open sets of M is again dense, which is the last remark in the statement. Since manifolds have countable bases of open sets, one easily sees that the statement is of a local nature; hence we can assume that M, N are open sets of Euclidean spaces or half-spaces and that f is the restriction of a smooth mapping g : U -+ IRq defined on an open subset U of IRP. Clearly, the result for f follows from the result for g; hence we can simply suppose f = g (this means we can disregard boundaries). We will prove the following statement, which in fact belongs to measure theory:

(*) The image of the set C, of critical points of f has measure zero. Recall that a set has measure zero when it is contained in a countable union of cubes whose total volume can be taken arbitrarily small. It is easy to see that a countable union of measure zero sets also has measure zero, so that being measure zero is a local question. It is also immediate that a measure zero set has empty interior. In our case, this implies that R, is residual. Indeed, C, is closed, hence a countable union of compact sets, so that f(C,) = IRq \ R, is also a countable union of compact (hence closed) sets, each one contained in f(C,), hence each one with empty interior. We now prove the above assertion (*). The proof is Milnor's by induction on p. Assume the result for smooth mappings with domain in Euclidean spaces of dimension < p. Denote Ci

= {x

E U:

all partial derivatives of order :S i vanish at x}

and consider the sequence of closed sets

We split the argument into several steps.

(a) The set f(C, \

Cd

has measure zero.

c, \

We will see that every point a E Cl has a neighborhood V such that f(C, n V) has measure zero. As we can cover C, \ C l by count ably many such neighborhoods, this implies (a). Fix a and choose some partial derivative of f that does not vanish at a, say ~(a) i= o. Then h(x) = (h (x), X2, . •. , xp ) is a local diffeomorphism at a, and we can replace f by f 0 h- l on a neighborhood V of a. In other words, we can suppose h(x) = Xl on V, so that f preserves every

3. Regular values

65

hyperplane Xl = t; we denote by 9 the restriction of Notice that for X = (t, x') E V it is

Df(X)=(!

0

Dg(X')

f

to that hyperplane.

)

so that if D f(x) is not surjective, Dg(x' ) is not either. This means that if x E Cf n V, then x' E Cg • Thus we have

f(Cf n V) n {Xl = t} C f(Cf n {Xl = t}) C g(Cg ), and the latter set has measure zero by induction. Summing, f(Cf n V) cuts each hyperplane Xl = t on a measure zero set, and the Fubini Theorem (which has an elementary proof for measure zero sets) guarantees that f(Cf n V) has measure zero, as desired.

(b) Each set f (Ci

\

Ci +I) has measure zero.

As before, we see that every point a E Ci \ CHI has a neighborhood V such that f (Ci n V) has measure zero. By definition, there is a partial derivative w of f of order i that vanishes at a and such that 8w(a)/8xj =1= 0; for ease of notation, we suppose j = 1. Then h(x) = (w(x), X2, ... , xp) is a local diffeomorphism at a, and on some neighborhood V of a the inverse mapping h- l : W -+ V is well defined and smooth. We have h(Ci n V) c {Xl = O}, and every point in h(Ci n V) is a critical point of the restriction 9 = f 0 h-IIW n {Xl = O}. Hence, f(Ci n V) = g(h(Ci n V)) c g(Cg ), and the latter set has measure zero by induction. We have proved (b).

(c) Ifi > p/q -1, the set f(Ci) has measure zero. Consider a compact cube K C U of side 8 > 0, and let us see that f (Ci n K) has measure zero. Since Ci can be covered by countably many such K's, the conclusion follows. Now, by Taylor expansion and the definition of Ci, we have f(y) = f(x) + R(x, y), IIR(x, y)1I ~ clly - xW+ I for any x, y E Ci n K. The constant c depends only on f and K. Now, subdivide K into n P cubes L with side 8/n and diagonal JP8/n. Choose X E C i n L, so that for every y E L,

. ~ P = c(JP8)HI Ilf(y) - f(x)1I ~ IIR(x, y)11 ~ clly - xll~+l ----:;;.

66

II. Manifolds

Thus, f(L) is contained in the cube L* with center f(x) and side 2p. Consequently, f(Ci n K) is contained in the union of n P cubes with side 2p, whose total volume is

c'nPSince i

q (i+1) .

> p/q - 1, this volume goes to zero when n -+ 00. This proves (c).

It is clear that the three facts (a), (b), and (c) imply (*), and this completes the proof of the Sard-Brown Theorem for smooth mappings. 0

Exercises and problems Number 1. Let P be a homogeneous polynomial of degree d in n + 1 indeterminates. (1) Prove that n+1

oP

LXiOX i=l

= d·P.

1.

(2) Show that every equation P = e:, e: t= 0, defines a smooth hypersurface Me in jRn+l (except if there are no zeros). (3) Prove that Me is diffeomorphk to Ml for e: > 0 and to M-l for e: < O. (4) Also, must Ml and M-l be diffeomorphic? How does d affect this? (5) What about the equation P = O? Number 2. Let M C jRn be a non-empty real quadric given by

L

aijXiXj = 1,

l$:i,i$n

with det(aij) t= o. Show that M is a smooth hypersurface diffeomorphic to a product §k x jRt, for suitable k and f. Number 3. Prove that the set M C jRrnxn of all matrices of rank k is a smooth manifold of codimension (m - k)(n - k). Number 4. Let E C jRnxn be the linear space of all symmetric matrices of order n, and consider the smooth mapping f : jRnxn -+ E : A I-t At A. (1) Show that dAf(B) = At B + Bt A, A, B E jRnxn, and deduce that the orthogonal group O(n) = f-l(I) C jRnxn is a compact manifold of dimension n(n - 1)/2. (2) Prove that O(n) has two diffeomorphic connected components, one, SO(n), defined by the condition det = +1. (3) Compute the tangent space to O(n) at A = I. At which other A E O(n) is the tangent space the same? Number 5. Let M C jRP be a differentiable cr (r ::::: 2) manifold of dimension m. Prove that the (orthonormal) Stiefel bundle OM

= {(x, Ul, ... ,Urn) E M xjRP x· .. xjRP : Ul, ... ,Urn

is an orthonormal basis of TxM}

is a differentiable Cr - 1 manifold of dimension ~m(m + 1), compact if and only if Mis. (Compare with Problem 8 in 11.2.)

4. Thbular neighborhoods

67

f : U -+ IRn be a smooth mapping, m ~ n. Suppose that 0 is a critical point and f(O) = O. Consider the matrices m x n as elements of IRmxn and consider

Number 6. Let U C IRm be an open set, and let

+ Ax. (1) Check that the origin is a regular value of F, hence that M = F- 1 (0) is a smooth F: (U \ {O}) x IRmxn -+ IRn

:

(x, A)

f-t

f(x)

manifold. (2) Show that if A E IRfflxn is a regular value of the projection the mapping FA : U -+ IRn : x f-t f(x) + Ax

7r :

M -+ IRmxn , then

has no critical point x =1= 0 with value FA(X) = O. (3) Conclude that there are A's with entries arbitrarily small such that the origin is the only critical point with value o. Number 7. A real projective hypersurface of degree k is a subset Me lU'm defined by a homogeneous equation P(xo, .. . ,xm ) = 0 of degree k.

(1) Under what conditions on P is M a smooth manifold? (2) Show that if k = 2£ is even,

f

: lU'

m

(

-+ IR: Xo:···:

) Xm

P(xo, ... ,xm ) Xo + ... +xm2 )l

f-t ( 2

is a well-defined smooth mapping. When is 0 a regular value of f? (3) What about real projective hyperplanes H C lU'm, which are defined by linear homogeneous equations?

4. Tubular neighborhoods Here we recall the constructions of tubular neighborhoods. These are a key ingredient in the next section for the approximation results that reduce homotopy to smooth homotopy, but, also, we will need them in a crucial way to prove the Poincare-Hopf Index Theorem at the very end this book (V.7.2, p. 219). The essential fact is that every differentiable boundaryless manifold in a Euclidean space ]RP is the differentiable retraction of some neighborhood of it in ]RP. We first take a purely topological approach that preserves differentiability class well: Proposition 4.1. Let M c ]RP be a boundaryless differentiable Cr manifold. Then there is an open neighborhood U of M in]RP and a Cr retraction p: U -+ M (that is, a Cr mapping with p(x) = x for x EM).

Proof. Since the pair M c ]RP is locally Cr diffeomorphic to the pair ]Rm c ]RP (II.2.3, p. 55) and linear projections ]RP -+ ]Rm are smooth retractions, each point x E M has an open neighborhood U in ]RP equipped with a

68

II. Manifolds

cr

retraction p : U -+ U n M. Henceforth the proof consists of a careful glueing of such local retractions. We split the argument into several steps. First we note that since M is locally closed in jRm, it is closed in some open neighborhood n in jRP; for the rest of the proof, all closures are considered inside this open set n. Also, all mappings are cr.

Step I: Clueing two local retractions. Let a : V -+ V n M and r : W -+ W n M be two retractions defined on open sets V, W of n. Then there is a retraction p : U -+ un M = (V u W) n M. If T = V n W n M = 0, there are disjoint open sets VI, WI c n such that V n M C VI C V and W n M C WI C W. Indeed, in the metric space V U W, the two disjoint closed sets V \ w, W \ V can be separated by disjoint open sets VI C V, WI C W. Now, we take U = VI U WI, and p is the trivial glueing of two mappings defined on disjoint open sets. After this remark we assume T =1= 0.

The idea is to glue a and r through a homotopy that deforms one retraction into the other over T. To that end, consider D = a-I(T) n r-I(T). This set is open in n, hence in jRP, and the set

E = {x ED: D:J [a(x), r(x)]} :J T is open too. Indeed, suppose D contains a segment [a(x),r(x)]. Since D is open in jRP, it contains a convex open neighborhood Q of that segment, and then a-I(Q) n r-I(Q) C E is an open neighborhood of x. Then we can define mappings H t , 0 :S t :S 1, on E by

Next, notice that the disjoint sets V n M \ Wand W n M \ V are closed in V U W. For instance V n M is closed in V, and hence V n M \ W is closed in V \ W, which is closed in V U W. Consequently, we can find a smooth bump function 8, which is == 0 on a neighborhood Uu C V of V n M \ W and == 1 on a neighborhood UT C W of W n M \ V. Finally, we define the retraction p on the open set U = Uu U E U UT by

p(x) = {

a(x)

on Uu ,

H(J(x) (x)

on E, on UT •

r(x)

Some straightforward computations show this construction is consistent.

69

4. Tubular neighborhoods

Step II: Exhaustion of M by local retractions. There is a sequence of retractions Pk : Uk --+ Mk = Uk n M such that

M = UMk

and

Mk C Mk+l·

k

As mentioned before, we can cover M by local retractions T : V --+ V n M, and then, since M has a countable basis of open sets, we get a sequence Tk : W k --+ Wk n M, with M c Uk Wk. Now, by Step I, for each k we can glue Tk and Tk-l, and Tk-2, and so on, to obtain a retraction Uk : Vk --+ Vk n M, with

Next, set

Mk = {x EM: dist(x,M\ Vk) >

i}.

This is an open subset of Vk n M, and

because Vk C Vk+l. On the other hand, for every x E M, say x E Wk C Vk, we have {

dist(X, M \ Vi) ~ dist(x, M \ Vk) dist(x, M \ Vk) >

i

for.e ~ k (as Vi n M ~ Vk n M), for .e large (as dist(x, M \ Vk) > 0),

and thus x E Me for large.e. Consequently, M = Uk = u;l(Mk) and Pk = ukluk.

Uk Mk.

Finally, take

Step III: Splitting into two sequences of disjoint retractions and conclusion.

First, we shrink the domains of the retractions Pk of Step II as follows. Take targets MlI

= Ml, M2I = M2, MkI = Mk \M k-2 for

k ~3

and domains U~ = p;l(M~). The virtue of this is that the M~'s still cover M, since M k C Mk+l, but besides, we have M~

(and k

n Mi

=

0 if k ==.e mod 2

=f: .e, of course). Indeed, suppose .e =

k

+ 2n.

{ M~ C Mk c Me-2' Mi = Me \ Me-2,

Then k

~

.e - 2, and

II. Manifolds

70

hence our assertion. Now put

V'

= UU~e-l' W' = UU~e' e

e

These two sets are open, and M c V' U W'. Let us look at V' first. The sets M~e-l = U~e-l n M c V' n M are closed in V' and form a discrete family. Indeed, suppose that some x E M~eo-l is the limit of a sequence of points Xe E M~e-l c M. Then Xe E U~eo-l for e large enough; hence Xe E U~eo-l n M = M~eo-l' a contradiction. Consequently, the sets M~e-l can be separated by disjoint open sets P2e-l restricts to a retraction U~e-l n A 2e- 1 -+ M~e-l' All these restrictions glue trivially, because their domains are disjoint, to give another retraction

A 2e- 1 C V', and

e

e

Similarly, starting with W', we get a second retraction Peven :

U(U~e n B 2e ) -+ UM~e' e

e

and we are done, because by Step I we can glue these two retractions to obtain P : U -+ M, the one we sought. D The preceding proof shows that being a retract of an open set in ~p is a local property. Hence, the differentiable content of the last theorem is the basic fact that manifolds are locally retracts. Actually, this local property characterizes manifolds: Proposition 4.2. Let M c ~p be a subset such that every point x E M has an open neighborhood U in ~p on which there is a differentiable cr retraction P : U -+ U n M. Then M is a boundary less cr manifold. Proof. Fix x E M and a Cr retraction P : U -+ un M, with x = p(x) E unM. After a translation, we can assume x is the origin. Denote 7r = dop, and define h = (ld]Rp -7r) 0 (ld]Rp -p) + 7r 0 p.

Since p = pop, we have 7r

= dop = do(p

0

p) =

7r 0 7r,

4. Tubular neighborhoods

71

which readily gives doh = IdlR.p. Hence h is a Cr diffeomorphism on an open neighborhood V C U of the origin. Further computation shows that hop = 7f 0 h; hence h(W n M)

= h(p(W)) = 7f(h(W))

for W = V n p-l(V). But 7f(h(W)) is open in L = 7f(~P), because linear maps are open. Thus W n M is Cr diffeomorphic to an open set of a linear D space; that is, W n M is a Cr manifold. There is a second and more popular method to construct retractions. We present this method now for smooth manifolds. As usual, we denote by -L perpendicularity in Euclidean spaces. Proposition and Definition 4.3. Let M C manifold of dimension m. Then the set vM

=

~p

be a boundaryless, smooth

{(x,u) E ~p x ~p: u -L TxM, x E M}

is a boundaryless, smooth manifold of dimension p, and the smooth mapping {} : vM --+

~p :

(x, u)

H

x

+ u.

induces a diffeomorphism from an open set fl :J M x {O} onto another U :J M. Consequently, we have the smooth retraction

p: U --+ M : y = {}(x, u)

H

X

and it follows that (maybe for a smaller U): (1) M is closed in U, (2) y - p(y) -L Tp(y)M for all y E U, and

(3) dist(y, M) = Ily - p(y) II for all y E U. In fact, (3) characterizes p. We summarize the situation by saying that p : U --+ M is a tubular retraction of M in ~p. Proof. To start with, we prove that

(a) vM is a boundaryless, smooth manifold of dimension p. Consider any local equations for M, that is, differentiable functions II, ... , fq, q = p - m, defined on an open set V C ~P, such that:

II. Manifolds

72

(i) VnM={xEV: JI(x)= .. ·=fq(x)=O} and

(ii) the rank of the Jacobian matrix

(;~i.)

is q at every point x E VnM.

J

Condition (i) means that the gradients

are perpendicular to TxM, and condition (ii) means that they are independent. Hence, they generate TxM.1. Consequently we can define a bijection (V n M) x IRq -+ (V x JRP) n vM: (x,y)

M

(x,u)

by q

u

= LYi gradx(Ji). i=l

This is in fact a diffeomorphism: its inverse is obtained by solving the linear system (*) in the unknowns Yi. Since we can cover M by open sets like V, we see that v M is locally diffeomorphic to M x IRq, which is a manifold of dimension m + q = p. Next we see that

(b) iJ is a local diffeomorphism at every point (x, 0). Indeed, it is enough to check that d(x,o)iJ is a linear isomorphism. But iJ induces by restriction the identifications

iJIMx{O} : M x {O} == M

and

iJl{x}xT",Ml.: {x} x TxM.1 == TxM.1,

so that the image of d(x,o)iJ contains TxM + TxM.1 = IRP. Hence, that derivative is surjective, and since dim(vM) = dim(IRP), we are done. Once we know that iJ is a local diffeomorphism around M x {O}, what remains to show is that (c) iJ is injective on some neighborhood of M x {O}. Clearly, iJ is injective on the set E = {(x, u) E vM: where

T

is defined as follows:

Ilull < T(X)},

4. Tubular neighborhoods

73

T(X) is the infimum of all s E JR. such that there are (x, u), (y, v) E • vM with Ilvll :s; Ilull = s, x

=1=

y, 19(x, u) = 19(y, v).

Thus, we must check that E is a neighborhood of M x {o}. Fix z E M. We already know that 19 is injective in a neighborhood V of (z, 0), say V : IIx - zll + lIull < E for a suitable E > O. We claim that

T(X) ~ 1E if Ilx - zll < ~E. For, suppose IIx-zll < ~E and let (x, u), (y, v) in the definition of T(X) verify Ilvll :s; Ilull < E• Then (x, u) E V and (y, v) ¢: V; moreover, IIx - yll = Ilu - vii :s; 211ull· Consequently:

1

iE > 211ull ~ Ilx - yll ~ Ily - zll - Ilx - zll ~ (E -lIvll) - ~E > (E -1E) - ~E = ie, a contradiction. Now, the claim implies

E:J {(z,u)

liz - xii

E vM:

< ~e, lIull < 1e},

and the latter set is an open neighborhood of (x, 0). Consequently, there is some open neighborhood 19 is an injective local diffeomorphism; hence

(d) 19 is a diffeomorphism from

n of M x {O} on which

n onto an open neighborhood U

of M.

Now, of the three conditions on p. 71, (1) and (2) are immediate by construction; hence it only remains to show that, perhaps after shrinking U, condition (3) also holds true. But this is a local question: it suffices to prove that (e) every point x E M has a neighborhood U X C U on which (3) holds. Indeed, then one replaces U by

Ux U

X•

Thus, fix x E M to prove (e).

Choose a compact neighborhood L of x in M. Let W be the interior of Lin M, and set e = ~ dist(x, M \ W). Now, let V = W n {liz - xII < e}, and choose 8 < ~e such that L x {liull :s; 8} c n. We claim that UX = 19(V x {liull < 8}) does the job. Indeed, pick y E U X , that is, p(y) E V and Ily - p(y)11 < 8. Let us estimate the distance dist(y, M). For z E M \ L, the various choices above give the following sequence of bounds: liz - yll 2: liz - xii - Ilx - yll ~ 2e - (11x - p(y)11 + IIp(y) - yll) > 2e - (E + 8) = e - 8> 8 > Ily - p(y)lI,

74

II. Manifolds

which show that {

IIY-ZII > Ily-p(y)11 for Z E M\L and 8> Ily - p(y)11 ~ dist(y, M) = dist(y, L).

But L is compact; hence dist(y,L) = Ily - zil for some z E L. This means that the function h : t H Ily - tl1 2 has a global minimum on M at z, and consequently, its derivative vanishes on TzM; equivalently, the gradient gradAh)

= 2(y - z)

is perpendicular to TzM, and v = y - z E TzMl.. We have Ilvll = Ily - zll = dist(y, L) < 8, so that (z, v) E L x {liull :=s: 8} c n. Since {} is injective on nand z + v = p(y) + (y - p(y)), we conclude that z = p(y) and dist(y,M)

= dist(y,L) = Ily - zll = Ily - p(y)ll·

All of this also shows that p(y) is the unique point z E M such that lIy- z II = lIy - p(y)lI; hence (3) characterizes p. 0

If one copies the above proof for differentiable manifolds of finite class C the set liM is a Cr - 1 manifold (the gradients of the local Cr equations are of class Cr - 1 ), and we end up with a Cr - 1 retraction. But notice the way in which we apply the Inverse Function Theorem on liM, which requires r - 1 ~ 1, that is, r ~ 2. These restrictions were avoided in 11.4.1, p.67, by purely topological means, but there no information concerning perpendicularity and distances was obtained. That information will be essential for the Poincare-Hopf Index Theorem. r,

Exercises and problems Number 1. Let M C lR m + 1 be a differentiable hypersurface and consider a point c off M. Prove that a E M is a critical point ofthe Euclidean distance M -+ lR : x H dist(x, c)

if and only if a - c.l TxM.

1 : lRP -+ lRn be a smooth map such that the origin is a regular value of hence M = 1-1(0) is a boundaryless smooth manifold of dimension m = p - n. Prove that there is a diffeomorphism V' : Al x lR n -+ vM such that {) 0 V'(x, u) = x. Number 2. Let

I;

Number 3. Let M C lRP be a boundaryless smooth manifold of dimension m. Show

that N

= {(x,u,v)

E lRP x lR P x lR P

:

x E M,u.l TxM, v E TxM}

is a boundary less smooth manifold diffeomorphic to M x lRP . Number 4. Let M C lRP be a compact boundaryless smooth manifold of dimension m. For c > 0 we denote veM = {(x,u) E vM: lIuli < c}.

5. Approximation and homotopy

75

Show that: (1) For c small enough, iJ restricts to a diffeomorphism from veAf onto an open neighborhood of A! in lRP . (2) There is a diffeomorphism h from vA! onto an open neighborhood of A! in lRP such that h(x,O) = x for every x E A!. Number 5. Let A! C lR P be a boundaryless smooth manifold of dimension m. For every continuous mapping c : A! -+ lR everywhere > 0 we denote veAl

= {(x, u)

E vA!:

Ilull < c(x)}.

Show that: (1) There is an c such that iJ restricts to a diffeomorphism from veAl onto an open neighborhood of A! in lR P • (2) There is a diffeomorphism h from vA! onto an open neighborhood of A! in lRP such that h(x,O) = x for every x E A!.

5. Approximation and homotopy In this section we show that homotopies can always be made differentiable,

which is essential for studying homotopies of continuous mappings using differentiable methods. As is well known, a continuous homotopy is a continuous mapping H : [0,1] x M --+ N (of course, one can use any closed interval), and as customary, we usually denote Ht(x) instead of H(t,x). Thus a homotopy is a continuous uniparametric family of mappings H t : M --+ N. Two mappings f, g : M --+ N are homotopic if there is a continuous homotopy Ht with Ho = f and HI = g. As is well known, this is an equivalence relation, whose set of classes is denoted [M, N]. Here there is other standard notation: the k-th homotopy group is 7rk(N) = [§k, N] for N connected, and the k-th cohomotopy group is 7rk(M) = [M, §k] (this is indeed a group, because M is a manifold). This notion is enough for studying continuous mappings on compact manifolds, but here we will transfer the compactness assumption from manifolds to maps, which requires some additional care. (5.1) Proper mappings. A continuous mapping f : M --+ N is called proper when it is closed and the inverse image of every point of Y is compact. In the context of manifolds, these two conditions are equivalent to the single condition that the inverse image of every compact set is compact. In our setting, we can also formulate properness in terms of convergency: f is proper if and only if every sequence Xk whose image f(Xk) converges has some convergent subsequence.

Note also that if M is compact, then all continuous mappings are proper.

II. Manifolds

76

As mentioned before, for proper mappings plain homotopy is not convenient. Instead, we consider the notion of a proper homotopy, which simply is a homotopy H : [0, 1] x M ---t N that is a proper mapping. Then, two proper mappings f, 9 : M ---t N are properly homotopic if there is a proper homotopy H t with Ho = f and HI = g. The preceding definitions can be used in the differentiable setting, just by requiring all mappings involved to be differentiable. But then, we need two facts: (i) that every proper mapping is homotopic to a proper differentiable one and (ii) that two differentiable mappings that are homotopic are homotopic by a differentiable homotopy. To prove this, we must use approximation techniques and these techniques require the use of retractions (described in the preceding section). Theorem 5.2. Let X c R.P be a locally closed set, N c R.q a boundaryless differentiable CT manifold, and f : X ---t N a continuous (proper) mapping. For every positive continuous function c : X ---t R. there is a (proper) CT mapping 9 : X ---t N such that

Ilf(x) - g(x)11 < c(x)

for every x E X.

Proof. Since X is closed in some open neighborhood W C R.P, f extends to a continuous mapping, which we still denote f : W ---t R.q, but the target need not be N (Tietze Extension Theorem). In the same way, c extends to a continuous function c : W ---t R., and substituting {c > O} for W, we can suppose c is positive on W. We proceed now in several steps. Step I: Approximation of f : W ---t R.q. Since f is continuous, every point x E W has a neighborhood W X C W such that Ilf(z) - f(x)11 < c(z) for all z E W X. Let {Ox} be a smooth partition of unity for the covering {WX} of W, and define x

Clearly, h is smooth, and for every z E W we have

Ilh(z) - f(z)11

= IIL:f(x)Ox(z) - (L:Ox(z))f(z)11 x

~

L: Ox(z)llf(x) -

x

f(z)1I < c(z)

x

(notice that if Ilf(x) - f(z)11 2 c(z), then z ¢ W X , and so Ox(z)

= 0).

Step II: Retraction of the approximation to N. Let p : U ---t N be a CT retraction onto N from an open set U C R.q (H.4.l, p. 67). The function

77

5. Approximation and homotopy

c'(x) = dist (f (x) , Rq \ U) is continuous and positive on X. Thus, we can replace W by {c' > O}, or merely assume f(W) c U. Using min{c,c'} instead of c in Step I, we get h(W) cU. For each x E X, let Kx be a compact neighborhood of x in Wand let Cx

> 0 be the minimum of con Kx. The set Vx = {y

E

U: IIp(y) - f(x)11 < ~cx}

is an open neighborhood of f(x) = p(f(x)) in Rq and contains a ball centered at f(x) with radius c5x < CX. Let Bx be the ball centered at f(x) with radius ~c5x. Now, by continuity, in an open neighborhood AX c f-I(Bx) n Kx of x in W it follows that Ilf - f(x)1I < ~c5x. By replacing W with the union of the sets A x, we can assume they cover W, and using a smooth partition of unity {(x}x for the cover {AX}x, we define

c"

=

L ~c5x(x : W --+ R, x

which is positive and continuous. Next, we substitute min{c,c',c"} for c in the preceding step and claim that the composite mapping 9 = po h, which is well defined and Cr because h(W) c U, is the approximation we sought. Indeed, let us estimate IIg - fll on X. Pick z E X, and we will have (x(z) =I 0 exactly for finitely many points x = Xl, ... , Xs E X. Then s

c"(z)

=

L ~c5Xi(Xi(Z) ::; ~c5Xll i=l

c5XI being, to simplify notation, the biggest c5Xi . As z E AX I , f(z) E B XI ; hence Ilh(z) - f(xI)1I ::; Ilh(z) - f(z) II

+ IIf(z) -

f(xI)1I < c"(z) + ~c5XI ::; c5Xll

so that h(z) E VXI and IIp(h(z)) - f(XI)11 < ~CXI. On the other hand, z E AXI also implies Ilf(z) - f(XI)11 < ~CXI' so that

IIg(z) - f(z)11 ::; IIp(h(z)) - f(XI)11

+ Ilf(z) -

f(XI)11

< cXI·

To conclude, notice that cXI ::; c(z), because z E AXI C K XI . Step III: Properness. Suppose f proper. The concern then is to choose c small enough. Pick two locally finite open coverings {Vi} and {Ui} of N such that Vi C Ui and Vi is compact. Then the compact sets Ki = f-I(Vi) form

II. Manifolds

78

a locally finite covering of X. Hence, each point x E X has a neighborhood Wx that meets finitely many Ki's, and we denote

Ex = m~ndist(Vi' N \ t

Ud

for the indices i with Wx n Ki =I 0. There is a partition of unity {T/x} for the covering {Wx }, and define the positive continuous function

E*(Z) =

! L T/x(Z)Ex. x

We claim that if the approximation 9 is constructed using E ::; E*, then 9 is proper. The key fact is that for such a 9 we have g(Ki) C Ui for all i's. Indeed, suppose Z E Ki = f- 1 (V i ). Since T/x(z) =I 0 implies Z E W x , we see that i is one of the indices in the definition of Ex, so that Ex ::; dist(V i ,N \ Ui ). It follows that

Ilf(z) - g(z)11 < E*(Z) ::;

! (L T/x(z)) dist(V i ,N \ Ui) = ! dist(Vi ,N \ Ui). x

As f(z) E Vi, necessarily g(z) E Ui , as desired. Finally we deduce from this that 9 is proper. Let LeN be compact. As the family {Ud is locally finite, L meets finitely many Ui's. But then, by the property just proved, C = g-1(L) meets only finitely many Ki'S. As the Ki'S cover X, the closed set C is contained in the union of finitely many of them. Such a union is compact; hence C is compact too. D Approximation settles homotopy matters, because:

Proposition 5.3. Let X C RP be a locally closed set, N C Rq a boundaryless differentiable CT manifold, and f : X ~ N a continuous (proper) mapping. There is a positive continuous function E : X ~ R such that every continuous mapping g : X ~ N with IIf(x) - g(x)1I < E(X) for all x E X is (properly) homotopic to f. Proof. To start with, note that any continuous mapping 9 : X ~ N C Rq is homotopic to f in Rq, by Ht = (1 - t)f + tg. But this homotopy can map points off N. Then consider a tubular retraction p : U ~ N, and set c(x) = dist(f(x), Rq \ U) > o. If IIf(x) - g(x) II < E(X) for all x E X, we deduce

cr

IIf(x) - Ht(x) I = Ilf(x) - (tf(x) + (1- t)g(x)) II = 11 - tlllf(x) - g(x) I < E(X) = dist(f(x), R n

\

U),

5. Approximation and homotopy

79

so that Ht(x) E U. Consequently, we can define a homotopy in N by po H t . That po H t is proper when f is follows by applying the method of Step III in the proof of the preceding result 11.5.2 to the proper mapping [0,1] x X ---+ N: (t,x) t-+ f(x). 0 In the last proof, if f and g are Cr , so is the homotopy obtained. But in fact, we have the following more general result, essential in the differentiable setting: Proposition 5.4. Let X c jRP be a locally closed set and let N c jRq be a boundaryless differentiable Cr manifold. If two Cr mappings f, g : X ---+ N are (properly) homotopic, then they are homotopic by a (proper) Cr homotopy. Proof. Let H : [0,1] x X ---+ N be a (proper) homotopy with Ho = f, HI = g. By approximation (11.5.2, p. 76), there is a (proper) Cr mapping H' : [0,1] x X ---+ N with IIH - H'li arbitrarily small. We thus have a but we only know that Ilf - Hbll and Ilg - H~II (proper) Cr homotopy are small. Anyway, 11.5.3, p. 78, provides (proper) Cr homotopies F t from f to Hb and G t from H~ to g. Clearly, we can paste Ft to H: and then with G t to get another homotopy from f to g. The only difficulty is for the pasted homotopy to be Cr at the junctions. But this can be arranged easily: replace Ft , H:' and G t by F8(t) , H~(t), G 8(t), for any bump function [0, 1] ---+ [0, 1] with

H:,

o:

°

{ O(t) = O(t) = 1

for t ::;

i

and

for t ~ ~.

These modifications flatten the homotopies near t the pasting cr.

°

= and t = 1 and make 0

Exercises and problems Number 1. Let M and N be two differentiable Cr manifolds. Two Cr mappings f, 9 : M -+ N are cr homotopic if there is a Cr homotopy H : [0,1] x M -+ N such that Ho = f and Hl = g. Show that this is an equivalence relation for cr mappings. State and prove the analogous result for proper Cr mappings. Number 2. Let f, 9 : M -+ lR be two continuous functions such that f (x) < g( x) for every x E M. Prove that there is a cr function h : M -+ lR such that f(x) < h(x) < g(x) for every x E AI.

80

II. Manifolds

Number 3. Let M C RP be a differentiable cr manifold and C c M a closed subset. Let f : M -+ R be a continuous function whose restriction fie : C -+ R is cr differentiable. Show that for every real number c > 0, there is a CT· mapping 9 : !vI -+ R such that glc = fie and IIg(x) - f(x) II < c. Use this result to prove that if !vI is connected, each two points in M can be connected by a differentiable cr curve.

Number 4. Let H : [a,lJ x M -+ R be a Cr homotopy, where M is a compact differentiable Cr manifold. Show that if a E R is a regular value of H Q , then it is a regular value of H t for every t > a small enough. Number 5. Let M be a differentiable cr manifold and let h : M -+ R be a Cr function, whose zero set we denote by X. Let f : M -+ R be a continuous function that vanishes on X, and let c : M -+ R be a continuous strictly positive function. Show that there is a cr function 9 : M -+ R such that IIf(x) - g(x)h(x)1I < c(x) for every x E M.

6. Diffeotopies Here we discuss homogeneity, which is the property that points in manifolds can be moved around on demand. To start with, we introduce the key definition:

Definition 6.1. Let M be a differentiable Cr manifold. A Cr diffeotopyof M is a differentiable Cr homotopy F : [0, 1] x M -+ M such that Fo = IdJl,[ and all mappings Ft , ~ t ~ 1, are Cr diffeomorphisms of M. We say that F joins each two points x E M and Fl (x) . We say that F is the identity off a set A C M if Ft(x) = x for x tJ. A and ~ t ~ 1.

°

°

Remarks 6.2. (1) Every diffeotopy is a proper homotopy. Indeed, the mapping P : (t, x) I--t (t, Ft(x)) is a diffeomorphism of [0,1] x M (use the Inverse Mapping Theorem to check that p- 1 is differentiable) ,'\ hence a proper mapping. Now the projection 7r : [0,1] x M -+ M is proper,) because [0,1] is compact, and we conclude that F = 7r 0 F is proper too. / (2) Since every point x E M is connected to Ft(x) by the path t I--t Ft(x), they lie in the same connected component. Hence, the Ft's preserve connected components. This allows many reductions to the case where M is connected. (3) As for homotopies, the unit interval can be replaced by any other closed interval. 0

81

6. Diffeotopies

Example 6.3. The mapping (t, x) M tx gives a diffeotopy of JRm on any closed interval [1, AJ, and it can be transported to the sphere §m c JRm +1 via the stereographic projection from the north pole to obtain a diffeotopy with equations:

It is clear that Ft moves every parallel Xm+1 = h > -1 to another Xm+1 = ht, with limt-too h t ---+ 1. Thus we can move the set Xm+1 2:: h into Xm+1 2:: h' for any given heights h, h'. We say that Ft collapses the sphere 0 towards the north pole.

The main goal of this section is to move points around using diffeotopies. We start in Euclidean spaces.

°

Proposition 6.4. There is a positive radius E > such that in JRm the origin can be joined to every point in the open ball Ilxll < E by a smooth diffeotopy that is the identity off Ilxll 2:: 2. Proof. Let {} : JR ---+ [0,1] be a bump function such that

{

=1 (}(t) == {}(o)

°

and on

It I 2::

1.

Let a 2:: 1 be such that I{}I(t)1 ~ a for all t E JR, and take E = 1/a i; J. Let c be a point with a = Ilcll < E. After a rotation we can assume c = (a,O, ... ,O) E JR x JRm-l, and we consider coordinates x = (y,z) in JR x JRm-l. We will produce a diffeotopy F joining the origin to c. Let T : JRm- I ---+ [0,1] be another bump function which is == 1 on and == on Ilzll 2:: v3. Then define

°

Ft(x)

= Ft(y, z) =

(y

Ilzll

~

1

+ t{}(y)T(z)a, z).

It is clear that Fo is the identity and FI (0) = c. Furthermore, if Ilxll = Vy2 + IIzll2 2:: 2, it must be either y2 2:: 1 or IIzl12 2:: 3; hence either (}(y) = or T(Z) = 0, and Ft(x) = x. Now, we are to prove that each Ft is a diffeomorphism. To that end, we claim:

°

For every fixed t, z, the function Ft,z : JR ---+ JR : y M Y + t{}(Y)T{z)a is bijective.

82

II. Manifolds

Indeed, the function Ft,z is strictly increasing, because its derivative is 1 + to'(y)r(z)a 2: 1 - aa > 0, i

and F t ,z is not bounded, because

IFt,z(Y) I 2: when

Iyl ~ 00,

1

Iyl- ItO(y)r(z)al 2: Iyl - -a

~

00

This proves the claim.

It immediately follows from the claim that Ft is bijective; hence to prove it is a diffeomorphism, we can apply the Inverse Mapping Theorem and check that the derivative dx (Ft) : jRm ~ jRm is a linear isomorphism, which can be readily seen by computing the Jacobian determinant:

1 + to'(y)r(z)a

* 1

o

o

1

o The proof is finished.

= 1 + to'(y)r(z)a > O.

o

Using this Euclidean construction, the result follows easily for manifolds: Theorem 6.5. Let M be a boundaryless differentiable cr manifold. Let a, b be two points of M and let A be a connected open set containing both. Then there is a cr diffeotopy joining a and b, which is the identity off a compact set K c A, which is a neighborhood of a and b. Proof. Choose for each point x E A a coordinate domain Ux c A, diffeomorphic to jRm. By II.6.4, p. 81, we find two open relatively compact neighborhoods Wx C Vx of x such that Kx = V x C Ux and each point of Wx can be joined to x by a diffeotopy of Ux which is the identity off Kx; notice that this latter condition means that the diffeotopy can be extended to the whole manifold M by the identity. Now, since A is connected and covered by the Wx's, we find a finite chain W X1 , ... , W Xp with a E W X1 and b E W Xr ' Then pick points Yk E W Xk n WXk+l' k = 1, ... ,p - 1, and set Yo = a, YP = b. By construction, for each k = 0, ... ,p - 1, we can join Yk and Yk+l by a diffeotopy F(k) of M which is the identity off the compact set KXk+l cA. Clearly, if we paste those diffeotopies, we get one which joins a = Yo and b = yp and which is the identity off the compact set

6. Diifeotopies

K =

U KXk+l

C

83

A. The only problem, that the pasting be Cr at the junc-

tions, is arranged by a bump function () as in 11.5.4, p. 79. The composite G * F of two diffeotopies F and G is (G

*

F)(

for t ~ {F(()(2t), x) t,x = FI(G(()(2t _ 1),x)) for t ~

)

!, !.

D

Corollary 6.6. Let M be a boundaryless differentiable Cr manifold of dimension m ~ 2. Let al, ... ,ap and bI, ... ,bp be two collections of points of M, and let A be a connected open set containing both. Then there is a Cr diffeotopy joining ak and bk for all k = 1, ... ,p, which is the identity off a compact set K c A. Proof. By induction on p. Assume the result for fewer that p points, and let N = M \ {a p , bp }. By assumption, there is a diffeotopy of N joining ak and bk for k = 1, ... ,p-l, which is the identity off a compact set LeA \ { ap , bp } (which is connected because dim (A) ~ 2); such a diffeotopy extends to M. Then, by 11.6.5 above, there is a diffeotopy of M joining a p and bp that is the identity off a compact set K contained in the connected set A \ {ak' bk : 1 ~ k < p}. One concludes by pasting these two diffeotopies. D

Remarks 6.7. (1) On M = li every diffeomorphism is either decreasing or increasing, which immediately gives a restriction to joining series of points: they must be numbered in exactly the same (or the reverse) order. Then it is easy to produce by hand the diffeotopy that connects the two series (or to do the pasting as in the proof above). This is the essential restriction for manifolds of dimension 1. (2) An easy consequence of the preceding result is that connected manifolds are n-homogeneous (in particular homogeneous): any two collections of n points can be transformed to each other by a diffeomorphism (in dimension 1 with the restriction explained in the above remark). (3) Another interesting consequence that often makes life easier is that any finite collection of points of a connected manifold has an open neighborhood diffeomorphic to lim. Indeed, given Xl, ... , X p , pick an open set U diffeomorphic to lim, and choose any YI, ... , Yp E U. By remark (2), there is a diffeomorphism f, mapping Xk t--t Yk; hence f-I(U) is the open neighborhood we sought. D

84

II. Manifolds

All these results obviously work for bordered manifolds as long as the points involved are not in the boundary. We will not need an elaborate discussion.

Exercises and problems Number 1. Let f : IRm -+ IRm be a linear isomorphism with positive determinant. Show that there is a smooth diffeotopy F : [0,1] x IRm -+ IRm such that Ft = f. Number 2. Let h : IR -+ [0,1] be a smooth function with h(t) = 1 for It I ~ 1 and h(t) = 0 for It I ~ 4. Prove that there exists a positive real number E > 0 such that for every a E IRm with Iiall < E and every linear mapping u : IRm -+ IRm with II IdRm -ull < E, the map Ft(x) = x + th(lIxIl2)(U(X) - X + a) is a smooth diffeotopy of IRm verifying (i) Ft(x) = x for 0 ~ t ~ 1 and Ilxll ~ 2 and (ii) Fl(X) = u(x) + a for Ilxll ~ 1. (Check that each Ft is a proper local diffeomorphism whose fibers all have the same number of points.) Number 3. Let f : IRm -+ IR m be a smooth diffeomorphism with f(O) = 0 and dof = IdRm. Show that there is a smooth diffeotopy F: [0,1] x IRm -+ IRm such that Ft = f. Number 4. Let U be an open neighborhood of the origin in IRm, and let f : U -+ IRm be a differentiable Cr mapping with f(O) = 0, dof = IdRm. Consider the function h in Problem Number 2, and prove that there is a positive real number E > 0 such that

Ft(x)

= x + th( lI~f )(f(x) -

is a well-defined Cr diffeotopy of IR m verifying (i) Ft(x) and (ii) Fl(X) = f(x) for IIxll ~ E.

x)

= x for 0 ~ t ~ 1 and Ilxll

~ 2E

Number 5. Let sm C IRm+1 be a sphere of odd dimension m. Construct a diffeotopy F ofSm such that F1 (x) = -x for all x E sm. Number 6. Let F: [0,1] x sm -+ sm be a diffeotopy. (1) Show that tP: [0,1] x U -+ U

= IRm +1 \

{O} : (t, x)

I-t

IlxllFt (II~II)

is a diffeotopy. (2) Pick any point a E sm. Show that the Jacobian determinant at a of tPt is positive. Use this to prove that if m is even, there is no isotopy of sm such that Fl(X) = -x for all x E sm.

7. Orientation In this final section of the chapter we quickly survey the notion of orientation and some related constructions.

85

7. Orientation

(7.1) Orientation of differentiable manifolds. An orientation ( on a linear space E of finite dimension m 2: 1 is a choice of a basis B = {UI, ... , urn}, two choices being equivalent if the determinant of the base change matrix is positive; we denote ( = [UI, ... , urn] and say that B is a positive basis of (. Clearly, there are only two orientations, which are denoted by ( and -( and are called opposite. In E = jRm there is a canonical orientation (rn, corresponding to the canonical basis {el' ... ,ern}. We know the two first examples of this very well: in jR2, the counterclockwise orientation; in jR3, the screwdriver orientation. Now, let M be a differentiable manifold of dimension m. An orientation on M is a family (M = {(x : x E M} of orientations (x in each tangent space TxM, such that each point a E M has a coordinate system x on whose domain U

[a~llx'···' a:rn Ix] = (x

for every x E U.

When this condition holds, the coordinate system (and the corresponding parametrization) is said to be compatible with the orientation. Notice here that the base change matrix for two bases of partial derivatives is the Jacobian matrix of the corresponding change of coordinates; hence the Jacobian determinant of the change of coordinates of a pair of compatible parametrizations is positive (we just say that the change is positive). By definition, if M has an orientation, it has a covering by coordinate domains with positive changes, which is called a positive atlas. Conversely, if there is a positive atlas, we can use (*) above to define an orientation ( onM. Suppose M is connected and oriented by (. Pick a coordinate domain U, with coordinate system x. To check condition (*) for x, we must use a coordinate system compatible with ( and check whether the change is positive. By continuity, this holds (or not) locally; hence, (*) holds (or not) locally. If the coordinate domain is connected, (*) holds (or not) on the whole domain. Combining this with the fact that any two points of Mean be joined by a chain of connected coordinate systems, we conclude that once we know ( at a point, then we know it everywhere. In other words, M has exactly two different orientations, ( and its opposite: -( = {-(x : x E M}. Finally, let us mention here that the product M x N of two oriented manifolds is immediately oriented at (x, y) E M x N by

II. Manifolds

86

given that

= [UI,"" urn] and we write (MxN = ((M, (N). (x

In short,

(y

= [VI, ... , vn].

(7.2) Orientation and differentiable mappings. Consider a differentiable mapping of oriented manifolds of the same dimension, f : M -+ N. Let x E M be a regular point of f, so that dxf : TxM -+ Tf(x)N is a linear isomorphism. Then we say that f preserves (resp., reverses) orientation at x E M if dxf maps a positive basis of (M,x onto a positive basis of (N,/(x) (resp., -(N,/(x»' We denote signxU)

= +1 (resp., -1)

and call this the sign of f at x. The sign can be easily computed by localization through coordinates compatible with both orientations (M and (N: as the Jacobian matrix of such a localization is the matrix of dxf with respect to the bases of partial derivatives, one only has to check whether that Jacobian has positive determinant. In particular, we see that if f : M -+ N preserves (resp., reverses) orientation at x, it does the same on a whole neighborhood of x. In particular, if a diffeomorphism of connected manifolds preserves (resp., reverses) orientation at some point, it does so at all of them. The following is an application of this:

Proposition 7.3. Let Ft be a diffeotopy of an oriented manifold M. Then all Ft's preserve orientation. Proof. Since diffeotopies preserve connected components, we can suppose !vI connected, and then it is enough to check that the Ft's preserve the orientation at a given point a E M. To see that, let t E [0,1] and choose a local parametrization 'I/J : V -+ M compatible with the orientation of M, such that 'I/J(O) = Ft(a). Then F-I('I/J(V» is a neighborhood of (t, a), and there are (i) a local parametrization rp : U -+ M compatible with the orientation of M, with a = rp(O), and (ii) a neighborhood J c I of t, such that F(J x rp(U» c 'I/J(V). In this situation, the mapping (J' :

J -+ lR : s

f--t

det do ('I/J -1

0

Fs

0

rp)

is well defined and continuous. Since 'I/J, rp, and the Fs's are diffeomorphisms, (J' never vanishes and has constant sign on a neighborhood of t. In conclusion, for s close enough to t, Fs preserves or reverses orientation as Ft

7. Orientation

87

does. It follows that all Ft preserve or reverse orientation simultaneously. But Fa = IdM preserves; hence all Ft preserve also. 0 (7.4) Orientation of hypersurfaces. Let M c ]R.m+1 be a differentiable hypersurface, that is, dim(M) = m. We look for a normal vector field v : M -+ ]R.m+1, which is a mapping such that 0 =lv(x) 1.. TxM for every xEM.

To start with, fix a coordinate domain U corresponding to a parametrization Then for every x E U, v(x) must be perpendicular to the partial derivatives

cp.

a

~I UXi x

=

(aCP1 ~(cp -1 (x)), ... , acpm+1 a (cp -1) (x)) ,1 ~ z.~ m. UXi

Xi

This perpendicular vector is easily obtained as a vector product. Take new variables u = (U1,"" Um +1) and let

A(u)

=

Then, let the i-th component of v(x) be the signed adjoint to Ui in A(u). Computing detA(u) for u = (1 ~ i ~ m) through its first column, it follows immediately that this v(x) is a normal vector field on U, which can be assumed unitary after dividing by its norm.

a'!i Ix

Now, notice that at each point x E M there are only two unitary vectors perpendicular to TxM. This makes it reasonable to paste together the v's obtained from different coordinate systems. For this, we must understand when v(x) = v'(x) for two different parametrizations. But by construction

(

m+ 1 _

-

[a a ] _ [, a a] v, aX1 ' ... 'aXm - v, ax~ , ... , ax'm

is the canonical orientation in ]R.m+1, and it follows readily that v(x) = v'(x) if and only if the change of coordinates is positive. From this we see that M has a global normal vector field if and only if M is orientable. Furthermore, in that case, there are exactly two different unitary normal vector fields, opposite each other. We can distinguish them by chosing an orientation (: v is compatible with ( if the coordinate systems in (*) above are compatible with (.

II. Manifolds

88

Examples 7.5. (1) Let Me jRm+l be a closed differentiable hypersurface, which has a global equation M : f = 0, that is, a differentiable function f : jRm+l ~ jR such that is a regular value of f and M = f- 1 (0). Then, M disconnects jRm+l: otherwise, f would have constant sign; hence every point x E M would be extremal, hence a critical point of f!

°

Once f is given, we know (11.2.8, p. 57) that each tangent space TxM is perpendicular to the gradient gradx(f) = (/t(x), ... , axa.!+1 (x)), which is always 1= 0, and consequently is a global normal vector field (and 11 = grad(f)/II grad (f) II is unitary). Thus, Mis orientable.

In fact, this is another formulation of the Jordan Separation Theorem for differentiable hypersurfaces: every closed differentiable hypersurface of a Euclidean space disconnects it, has a global equation, and is orientable.

We will prove this using degree theory (111.6.2, p. 124, 111.6.4, p. 129)

(2) The first case to which we apply the above construction is the unit

sphere §m C jRm+l, whose equation is I:i x~ = 1. Then the gradient 1I(x) = x is already unitary and gives the canonical orientation ( of §m. We always equip §m with this orientation; 11 is called the outward normal vector field. Let us now look at two antipodal points a and -a in §m. The two tangent spaces Ta§m and T _a§m coincide: both are perpendicular to the line through a and -a. Thus we can compare the orientations (a and (-a:

and (-a =

Thus we can take (-a = -(a.

[VI, ... , vmJ V1

=

-UI, V2

if (m+l = [-a, V1,···, vmJ. =

U2, .•. , Vm

=

Um

and conclude that

We have just seen that the sphere has opposite orientations at antipodal points.

(3) Now suppose a = (0, ... ,0,1) and -a = (0, ... ,0, -1). Let 7fN be the stereographic projection from a and let 'Irs be that from -a (11.2.4, p. 55). We have (a = [e1, ... , emJ

if and only if m is even.

After this remark, from 11.2.9, p. 58, we see that 'Irs preserves orientation if and only if m is even. Since (-a = -(a, we similarly see that 'lrN preserves orientation if and only if m is odd. Hence, 'Irs and 'lrN do not form a

7. Orientation

89

positive atlas. To get one, change the sign of one coordinate in one of the projections. 0

(7.6) Orientation of inverse images. Let f

: M -+ N be a differentiable

cr mapping of oriented manifolds, and let a E N \ aN be a regular value of both f and flaM. As we know, f-l(a) is a manifold (II.3.2, p. 62), and we can use the orientations of M and N to define an orientation on f- l (a). We do it as follows. Let x E f-l(a). By II.3.2, p.62, L = Txf-l(a) is the kernel of the derivative dxf; hence this restricts to a linear isomorphism E -+ TaN for any linear supplement E C TxM of L. Choose any such E with a basis VI, ... , Vn that dxf maps onto a positive basis WI, ... , Wn of TaN. Then we define an orientation ~x on L by declaring positive any basis UI, ... , U m - n of L such that VI, ... ,Vn , UI, ... ,Um - n is a positive basis of TxM. We must see that this does not depend on the choice of data, so assume other v~, uj are given. Consider the matrix A = ( ~l of the base

112 )

change from {vLuj} to {Vi,Uj} and the matrix C = (CliO) of dxf with respect to the bases {Vi, Uj} and {Wi}. In view of the way orientations determine the choices, we have the following: (i) The matrix C 1 is the base change from {d xf (Vi)} to {wd, so 0 det(CI ).

<

(ii) Since CA = (CIAIIO), the matrix CIA I is the base change from {dxf(v~)} to {Wi}; hence 0 < det(CI ) det(A I ), and by (i), 0 < det(Ad. (iii) The matrix A is the base change from {v~, uj} to {Vi, Uj }; hence 0 < det(A) = det(AI) det(A 2 ), and by (ii), 0 < det(A 2 ). But A2 is the base change from {uj} to {Uj}, and consequently {uj} defines the same orientation as {Uj}. Since we know that the definition does not depend on choices, let us see why ~ = {~x : x E f-l(a)} is indeed a well-defined orientation on our inverse image. To do this, pick coordinates (XI, ... , xm) of M and (YI, ... ,Yn) of N as in the proof of II.3.2, p. 63, so that (Xl' ... ' X m- n ) are coordinates of f- l (a) and

90

II. Manifolds

.1,

.

Thus we can use the vectors Vi = ox m-n+. 0 ~.I x x Wi = ayOI • a ' and Uj = VA] It is quite clear that depending on how x, y respect orientations, either (Xl, ... ,Xm - n ) or (Xl, ... , -x m - n ) defines ~ consistently at all points of f-l(a) close to x. We are done.

(7.7) Orientation of boundaries. The definition of orientation applies of course to manifolds with boundary, as no boundaryless assumption has been made as far. If M has boundary 8M, all tangent spaces TxM carry an orientation, despite whether or not x belongs to 8M, In fact, it is easy to see that M is orientable if and only if the boundaryless manifold M \ 8M is; in other words, the orientation of M at boundary points is determined by the nearby interior points. A more interesting matter is that when M is oriented, its boundary can be oriented as a manifold itself. We describe this here. Let M be a manifold with boundary 8M, and fix an orientation (on M. Choose any point a E 8M. The tangent space Ta (8M) is a hyperplane in TaM, and we pick a coordinate system at a, x : U -+ lliIm with Xl 2 O. Then the linear isomorphism dax : U f--t t = (tl' t2,'" ,tm ) maps Ta (8M) c TaM onto {O} x IR m- 1 c IRm. Thus we find two types of vectors that are tangent to M but not to 8M: the inward vectors, with tl > 0, and the outward vectors, with tl < O. As usual, via changes of coordinates, one sees that these conditions do not depend on the choice of the coordinates, and we have a consistent notion of outward tangency. Curve germs provide a more geometric way to determine inward and outward vectors:

(1) If we have "( : [0, c) -+ M with "((0) = a TaM \ Ta(8M), then

U

E 8M

U

=

U

E

U

E

is inward.

(2) If we have "( : (1 - c, 1] -+ M with "((1) = a TaM \ Ta(8M), then

and "('(0)

is outward.

-

X

E 8M

and "('(I)

=

91

7. Orientation

In (1), Xl 0, : [0, c) -t [0,00) must be increasing, and tl = (Xl 0,)'(0) is positive; hence U is inward. In (2) we get an outward vector, because Xl 0 , : (1 - E, 1] -t [0,00) is decreasing. Finally, we define an orientation on Tx(8M) by

8(= [u] (= [v,u]

Again, some straightforward computations confirm that this indeed defines an orientation 8( on 8M.

Examples 7.S. (1) The orientation of ~m = ~m X {O} as the boundary of lHIm+l : Xl 2: 0 is the "wrong" one! Indeed, lHIm+1 carries the canonical orientation (m+l, but since el is inward tangent, we get 8(m+l = _(m.

-el = outward

(2) Let us look at the unit sphere §m as the boundary of the ball L:i x~ ~ 1. Then the vector v(x) = X is outward tangent to the ball, and we see that the orientation of the sphere as boundary of the ball is the canonical orientation defined in 11.7.5(2), p. 88. (3) A case of relevance for later discussion is that of a cylinder. Let M be an oriented boundaryless manifold and consider the cylinder M' = [0,1] xM oriented as described in 11.7.1, p. 85: if UI, ••. , U m give the orientation (x at X EM, at (t, x) E M' take the orientation ~(t,x)

=

[(1,0), (0, UI),

.•• ,

(0, um)].

II. Manifolds

92

This ~ induces an orientation copies of M:

8~

on 8M', which is a disjoint union of two

8M'=MoUM1 , 'Po :Mo

= {O} xM == M: (O,x) == X,

'PI :Ml

= {l}xM == M: (l,x) == x.

Then 'PI preserves orientations and 'Po reverses them. In other words, the orientation 8~ induced on 8M' is the right one on Ml and the wrong one on Mo. The picture below describes the situation.

•

•

t

~

• [0,1]

Indeed, it is enough to remark that (1,0) is inward at (0, x) and outward at (1, x). Moreover, this is immediate, looking at the curve germs {

M:

1'0 ~ [0,1) -+ ~ t r-t (t, x), 1'0(0) 1'1 . (0,1] -+ M . t r-t (t, x), 1'1 (1)

= (0, x), 1'~(0) = (1,0), = (1, x), 1'1 (1) = (1,0).

0

Exercises and problems Number 1. Let M and N be diferentiable Cr manifolds of dimensions m and n, respectively, M boundaryless. Let (M and (N be orientations on M and N, respectively. (1) Equip M x Nand N x M with the orientations (MXN and (NXM, and determine when the canonical diffeomorphism T : M x N -7 N x M : (x, y) >--+ (y, x) preserves orientation. (2) Note that M x aN = o(M x N) and compare the orientations (Mx8N and O(MXN. Number 2. Let f : jRm -7 jRm be a diffeomorphism. Show that there is a diffeotopy F: [0,1] x ]Rm -7 ]Rm with F[ = f if and only if f preserves orientation. Number 3. Show the following orient ability criterion. A manifold M is orientable if and only iffor any orientations (u, (von two connected open sets U, V C M and for any two points x, y E Un V, (u,x = (v,x if and only if (u,y = (v,y.

7. Orientation

93

Number 4. Use the criterion of the preceding problem to show that: (1) The Mobius band M C R3 parametrized by

x

= cosO + t cos 0 cos !,

y

= sinO + tsinOcos!,

z

= tsin!

is not orientable. (2) The real projective plane is not orientable.

Number 5. Show that real projective spaces RJlDTn are orientable exactly for odd m. What about complex projective spaces? Number 6. Let M C RP be a boundaryless smooth manifold of codimension n with a smooth equation 1 : RP -t Rn (that is, M = 1-1(0) and 0 is a regular value of f). Prove that M is orientable. Find another sufficient condition for orientability involving 11M. Number 7. Find the orientation of the cilinder M C R3 : x 2 + y2 = 1, 0 ::; z::; 1 such that the local diffeomorphism R2 -t M : (t, x) t-+ (cos t, sin t, z) preserves orientation. Then determine the orientation induced on each connected component Co, C 1 of the boundary aM:

and study whether the mapping Co -t C 1 not.

:

(x, y, 0)

t-+

(x, y, 1) preserves orientation or

Number 8. Let M be a boundaryless differentiable Cr manifold of dimension m. Prove the following: (1) Let a, bE M be two points in a connected open set A c M and let.>. : TaM -t nM be a linear isomorphism, preserving orientation il M is oriented. Then there is a C diffeotopy joining a and b, which is the identity off a compact neighborhood K C A of both a and b and such that da F1 = .>.. (2) If M is oriented and connected and 'P, t/J : B -t Ml are two parametrizations compatible with the given orientation, both defined on the unit open ball BeRm, then there is a Cr diffeotopy F : [0,1] x M -t M such that F1 o'P = t/J on some smaller open ball B' c B.

Chapter III

The Brouwer-Kronecker degree Here we present the beautiful Brouwer-Kronecker degree theory for arbitrary proper mappings from the differentiable viewpoint. To start with, §§1 and 3 contain the construction of the Brouwer-Kronecker degree. Also, we use it to prove the Fundamental Theorem of Algebra. In §2 we take a small diversion to describe the degree through integration by means of the de Rham cohomology. Next, we discuss in §4 the extension of the theory to arbitrary differentiable manifolds, which is based on the computation of the degree of a C1 mapping. In §5 we define the so-called Hopf invariant, for mappings between spheres of unequal dimension, which is based upon the notion of link coefficient. We devote §6 to a beautiful application of the theory: the Jordan Separation Theorem. We show that every closed differentiable hypersurface disconnects the Euclidean space, and from this we deduce that such a hypersurface has a global equation and is orientable. We conclude the chapter by proving in §7 the famous Brouwer Fixed Point Theorem, jointly with some interesting consequences, and the equally important theorem by Brouwer concerning the existence on spheres of vector fields without zeros.

1. The degree of a smooth mapping Here we present the notion of degree for smooth data. The first step towards the definition of the Brouwer-Kronecker degree is as follows: Proposition 1.1. Let f : M --+ N be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension m. Then each regular value a E RJ C N of f has an open neighborhood V C RJ such that for every b E V, the inverse image f-l(b) is a finite set and the integer

d(f, b) =

L

signx(f)

xEJ-1(b)

only depends on a. Proof. First note that for any regular value b the compact set f-l(b) is discrete. This follows because f is a local diffeomorphism at each x E f-l(b) 95

-

96

III. The Brouwer-Kronecker degree

by the Inverse Mapping Theorem. We must now show that b t---+ d(f, b) is locally constant in Rf (which is open because f is proper). To that end, we need an open neighborhood V of a such that f-l(V) is a disjoint union of finitely many connected open sets Uk such that each restriction fJUk is a diffeomorphism onto V. To obtain V, let f- l (a) = {Xl, ... , x r }. By the Inverse Mapping Theorem, there are disjoint open neighborhoods Uk of Xk, such that fJu~ is a diffeomorphism onto some open neighborhood V' of a, which we can take common for all the k's. Then since f is closed, the set f(M \ Uk Uk) is closed. As that set does not contain a, we can find a connected open neighborhood V C V' of a such that

V

n f(M \

UUk) = 0. k

Hence, f-l(V) C Uk Uk·

Uk Uk'

and we take Uk = f-l(V)

n Uk' so that

f-l(V)

=

Note also that, Uk being connected, signx(f) is constant on Uk, say == (Tk. Thus, for every bE V we have

d(f, a)

= L signxk (f) = L (Tk = L signYk (f) = d(f, b). k

k

k

D

Proposition 1.2 (Boundary Theorem). Let X be an oriented smooth manifold of dimension m + 1 with boundary aX = Y and let N be an oriented, boundaryless, smooth manifold of dimension m. Let H : X -t N be a proper smooth mapping and let a be a regular value of Hand HJy. Then d(HJy,a) =

o.

Proof. Denote f = HJy, which is a proper smooth mapping, because Y is closed in X. By II.3.2, p. 62, the inverse image C = H-l(a) is a compact smooth curve with boundary

aC = C n Y =

f-l(a).

1. The degree of a smooth mapping

97

By the classification theorem for compact curves (II.3.3, p. 63), the finitely many connected components rk of e are either circles (without boundary points) or arcs with two boundary points Pk, qk E 1-1 (a). Thus, the boundary points of Be come in pairs, and we have d(f,a)

= Lsignpk(f) + Lsignqk(f). k

k

We claim that signpk (f) = - signqk (f), which concludes the argument. Henceforth we omit the indices k in all notation. The situation is depicted in the figure below.

')" (1) -')" (0)

We consider in the component under consideration, r, the orientation it has as an inverse image (II.7.6, p.89). Let t t--+ ')'(t) E r be a parametrization such that ')'(0) = p, ')'(1) = q. After renaming p,q as q,p and reparametrizing by t t--+ ')'(1- t), we can suppose that the inverse image orientation in T'Y(t)r is in fact given by the tangent vector ')"(t). We next discuss orientations at the end points of r. (i) Orientations at p. Let Ul, ... , Urn be a positive basis of TpY. As Y is oriented as the boundary of X, v, UI, . .. , Urn is a positive basis of TpX for any outward tangent vector v. But ')"(0) is inward (II.7.7, p. 90); hence we can take v = -1"(0). Thus

-,),'(0), UI, ... , Urn is a positive basis ofTpX. On the other hand, we know that 1"(0) gives the orientation of r as an inverse image; hence

III. The Brouwer-Kronecker degree

98

is a positive basis of TpX if and only if dpf maps Ul, ... , U m onto a positive basis of TaN, that is, if and only if signp(f) = +1. But the determinant of the change from the first base to the second is (-1) m+l, and so signp(f)

= +1 if and only if m is odd.

(ii) Orientations at q. The argument is the same, but at q the tangent vector 'Y'(I) is outward; hence we do not need -'Y'(I), and the sign of the base change is (_1)m. Thus: signq (f)

= +1 if and only if m is even.

From these two equivalences, we see that the signs signp(f) and signq(f) are opposite, as desired. This concludes the proof of the proposition.

D

Next we consider smooth homotopies: Proposition 1.3 (Homotopy invariance). Let H : [0,1] x M -+ N be a proper smooth homotopy of oriented, boundaryless, smooth manifolds, and let a be a regular value of both HfJ and HI. Then

d(Ho, a) = d(Hl' a). Proof. By the Sard-Brown Theorem, II.3.4, p. 63, we can choose regular values b of H arbitrarily close to a, and then d(Ho, a)

= d(Ho, b), d(Hl' a) = d(Hl' b)

by IILl.I, p. 95. We apply the preceding proposition with X = [0,1] x M, so that Y = {O} x M U {I} x M, and get d(Hly, b) = O. But, since Y is a union of two disjoint components,

d(Hly, b) = d(HI{o}XM, b)+d(HI{l}xM, b). We can identify {O} x M {I} x M

== M and HI{o}XM == Ho, == M and HI{l}xM == HI

and then see what the degrees are. But we must care for the orientations induced on Y as a boundary. Indeed, as we know (II.7.8, p. 91), in a cylinder like X the orientations induced in {O} x M and {I} x M are reversed; hence if, say, the equivalence {O} x M == M preserves orientations, then {I} x M == M does not, and we have

d(HI{o}xM, b) = d(Ho, b)

and

d(HI{I}XM, b) = -d(HI' b).

1. The degree of a smooth mapping

99

Substituting in (*), we see that d(Ho, b) = d(Hl' b). Clearly, if {O} xM reverses orientations, the conclusion is the same.

== M 0

Proposition and Definition 1.4. Let f : M -t N be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension m; furthermore, N is connected. Then the integer d(f, a) does not depend on the choice of the regular value a E RI' It is called the degree of f and is denoted by deg(f). Proof. Let b be another regular value of f. Since N is connected, by 11.6.5, p. 82, there is a smooth diffeotopy Ft of N with Fl(b) = a. Then F t 0 f is a proper smooth homotopy of f and Fl 0 f. Since a is a regular value of these two mappings, from III.1.3, p. 98, we get d(f, a)

= d(Fl 0

f, a).

But Fl is a diffeomorphism that preserves orientation everywhere (II.7.3, p. 86); hence by the chain rule: d(Flof,a)

=

o Remarks 1.5. (1) It is clear that the above definition depends on the chosen orientations. The essential thing here is that if the orientation of M is reversed, then the degree changes sign. (2) It is also clear from the definition that every non-surjective smooth mapping has degree zero. 0 Examples 1.6. Let

§m C ~m+l

denote the standard unit sphere.

(1) Of course the identity has degree 1. Now let us look at the antipodal map f : x t--+ -x. It is a diffeomorphism. Hence its degree is ±1 according to the behavior of orientations, which can be checked at any point. Pick the north pole a = (0, ... ,0,1). Then f(a) = -a = (0, ... ,0, -1) is the south pole, and the two tangent hyperplanes Ta§m and T _a§m coincide as linear spaces, but they carry opposite orientations. Now, daf is the symmetry: u t--+ -u, which maps any basis {Ul, ... ,um } to {-Ul, ... ,-um }. The determinant of the latter with respect to the former is (-l)m, that is:

{ +1, and so the bases define equal orientations, if m

is even, -1, and so the bases define opposite orientations, if m is odd.

III. The Brouwer-Kronecker degree

100

Hence, a positive basis of Ta§m is mapped to {

a negative basis of T _a§m if m is even, a positive basis of T_a§m if m is odd,

and we conclude that daf { All in all, deg(f)

reverses orientation if m is even, preserves orientation if m is odd.

= -1 if m is even, and deg(f) = +1 if m is odd.

(2) To produce a mapping with degree -1 in any dimension, we turn to symmetries, like h : x t---+ (Xl, ... , X m , -X m +1)' This h maps again the north to the south pole, but here dah is the identity, because the tangent vectors have their last component zero. Consequently, a discussion as above shows that dah : Ta§m --t T_a§m reverses orientations, and deg(h) = -1. (3) We can identify the unit circle §l C R2 with the group of complex numbers of module 1. Then the mapping §l --t §l : z --t zd (complex multiplication) has degree d. If d = 0, this mapping is constant and its degree is indeed O. Hence, we assume d t= O. In polar coordinates, we can represent the mapping by z

= cosO + isinO t---+ zd = cos(d·O) + isin(d·O).

Clearly, there are no critical values, and each value has check orientations, just look at the representation §l --t §l:

(cosO,sinO)

t---+

Idl

preimages. To

(cos(d.O),sin(d.O)),

which preserves orientation for d > 0 and reverses it for d < O. In fact, we have just formalized the idea of winding §l around itself. We will generalize this later (V.1.1, p. 183). This example shows quite clearly why degree is called degree! 0

The degree of a composite mapping is easy to compute:

Proposition 1. 7. Let f : M --t Nand g : N --t P be two proper smooth mappings of oriented, boundaryless, smooth manifolds, with Nand P connected. Then deg(g 0 f) = deg(g) . deg(f).

1. The degree of a smooth mapping

101

Proof. Let a E P be a regular value of 9 0 f. Then every y E g-l(a) is a regular value of f. Indeed, let x E f-l(y). As a is a regular value of 9 0 f, the derivative dx(g 0 f) is an isomorphism. But by the Chain Rule,

and we conclude that dxf is injective, hence an isomorphism, because all spaces have dimension m. Consequently, we have deg(f)

=

L

signx(f)

for every y E g-l(a).

XE/- 1(y)

Thus deg(g 0 f)

=

L signx(g f) = L L L signy(g) L signx(f) = L

xE(gof)-l(a)

=

signy (g) signx(f)

0

yEy-l(a) xE/-1(y)

yEg-l(a)

XE/-1(y)

signy(g) deg(f)

YEy-l(a)

= deg(g) . deg(f).

o

Now we prove two fundamental properties of deg:

.

Proposition 1.8 (Boundary Theorem). Let X be an oriented smooth manifold of dimension m+ 1 with boundary ax = Y, and let N be a connected, oriented, boundaryless, smooth manifold of dimension m. Let H : X -+ N be a proper smooth mapping. Then

deg(Hly)

= o.

Proof. The smooth mapping Hly is proper, because Y is closed in X. Then, by the Sard-Brown Theorem, we can pick a point a E N that is a regular value of both Hand HIY. Hence

deg(Hly)

= d(Hly,a),

and the result follows readily from 111.1.2, p. 96.

o

Proposition 1.9. Let H : [0,1] x M -+ N be a proper smooth homotopy of oriented, boundaryless, smooth manifolds, N connected. Then deg(Ho) = deg(Hl)'

III. The Brouwer-Kronecker degree

102

Proof. By the Sard-Brown Theorem, the two proper smooth mappings Ho and HI have some common regular value a E N, and then deg(Ho) = d(Ho, a),

deg(HI)

From this and III.1.3, p. 98, we get deg(Ho)

= d(HI' a).

= deg(HI)'

D

We end this section with a proof of the Fundamental Theorem of Algebra, which in the first chapter was presented as the origin of degree theory. The proof that follows brings in the preceding notions and makes clear the interest of developing the theory for proper mappings. Later we will give a second proof of this result, closer to Gauss's ideas, appealing to the Euclidean degree (IV.2.7, p. 150) Proposition 1.10 (Fundamental Theorem of Algebra). Every algebraic equation zP + CIZp-1 + ... + Cp = 0 with complex coefficients has some complex solution.

Proof. The mapping p:]R2

== C --+ C: (x, y) == x + iy

= z H zP

+ qzp-I + ... + cp

is smooth and proper (because limlzl-too IP(z)1 = 00). By III.1.5(2), p. 99, it is enough to show that deg(P) i= O. To do that, consider the homotopy Pt(z)

= zP + tCIZp-1 + ... + tcp , 0:::; t :::; 1,

which is proper and smooth. Hence deg(P) = deg(Po). But the degree of the mapping Po : z H zP is exactly p. Indeed, one checks immediately that (i) any a i= 0 is a regular value of Po with p different roots c in C and (ii) at each of them Po preserves orientation: dcPo is multiplication by the complex number pcp-I. D

Exercises and problems Number 1. For each i = 1, ... , r, consider a proper smooth mapping Ii : Ali ~ Ni with dim(Mi) = dim(Ni ), Ni connected. Prove that deg(/1 x ... x Ir) = deg(fI)··· deg(fr). Deduce that 011 the torus T m = §1 X ... X §1 there are smooth mappings T m ~ T m of arbitrary degree. Number 2. Let p, q, ml, ... ,mr be even positive integers, and let n L:i mi . (1) Show that the following smooth mapping has degree 2:

=

p

+ q,

m

=

103

2. The de Rham definition

(2) Define by induction a mapping fr : §ml not null-homotopic.

X ... X §mr

---+

§m

of degree 2r-l, hence

Number 3. Check that the following quadratic mapping is well defined, and compute its degree:

Number 4. Let m = 2n + 1 be an odd positive integer. (1) Compute the degree of the canonical surjection 11": §m

---+

JR]pm :

(xo, ... ,Xm)

1-+

(xo : ... : Xm).

(2) Show that the mapping h of the preceding problem factorizes through obtain a smooth mapping f : JR]pm ---+ §m with deg(f) = l. (3) Take a different view of f via stereographic projections.

11",

and

Number 5. Fix an orientation on C m == R2m. Let U C C m be an open set, and let f : U ---+ C m be a proper holomorphic function. Show that deg(f) is the cardinal of the generic fiber of f: the inverse image f-l(a) of every regular value a of f has exactly deg(f) points. In particular, deg(f) ~ O. Number 6. It can be shown that the degree of a smooth mapping CIP'2 ---+ CIP'2 must be a perfect square. Construct all of them. Number 7. Fix an orientation on the complex projective space ClP'm to compute the degree of the mapping

ClP'm ---+ ClP'm : z where

Zi

= (zo

: ... : Zm)

stands for the complex conjugate of

Zi

1-+ Z

= (zo

: ... : Zm),

E C.

Number 8. Given a proper smooth mapping f : M ---+ N of smooth manifolds with boundary, N connected, such that f(aM) c aN, define deg(f)

=

L

signx(f),

xE!-l (a)

where a is a regular value of f off aN. Develop the corresponding theory, using homotopies H t such that Ht(aM) c aN. Prove also that deg(f) = deg(flp), where P C aM is the inverse image f-l(Q) n aM of any chosen connected component Q of aN.

2. The de Rham definition In this section we describe the Brouwer-Kronecker degree of a smooth mapping in terms of differential forms and integration. We will not fully depict all details behind the constructions, only those directly linked to the notion of degree. (2.1) Forms and de Rham cohomology. Let M be a connected, oriented, boundaryless, smooth manifold of dimension m. We denote by

104

III. The Brouwer-Kronecker degree

r~(M) the linear space of all smooth differential forms of degree k, kforms for short, with compact support. For k = 0, r~(M) consists of all smooth functions f : M -+ JR. with compact support. As is well known, the derivatives of such a function f give its total differential df : x t-+ dxf, and this induces a linear operator d : r~(M) -+ r~(M). Then, this operator extends to d : r~(M) -+ r~+l(M) for all k, so that dod = 0 and

d( a 1\ ,8)

= da 1\ ,8 + (-1 r a 1\ d,8

for every r-form a and s-form ,8; d is called exterior differentiation. A k-form w with compact support is called closed when its exterior differential is zero, Le., dw = 0, and exact when w = da for some (k - 1)form with compact support a. The closed k-forms with compact support are a linear subspace of r~(M) and the exact k-forms with compact support another linear subspace. As dod = 0, exact implies closed, and we can consider the quotient linear space

Hk(M JR.) c'

= {closed k-forms with compact support} {

exact k-forms with compact support} .

This is the so-called k-th de Rham cohomology group of M with compact support. It is a deep theorem that this cohomology coincides with the singular cohomology H~(M, Z) after extension of coefficients (H~(M, JR.) = JR. 0z H~(M,Z)), but we will not discuss this matter here. Of course, if M is compact, everything has compact support and we get the ordinary cohomology groups, but in the non-compact case, it is important to consider compact supports. For instance, H:;n(JR.m, JR.) = JR., but disregarding compact supports in all definitions above, we get Hm(JR.m, JR.) = O. Here we will concentrate on the maximum degree m and look solely at the m-th cohomology group H:;n(M, JR.). This group can be beautifully described by integration. Indeed, since M is boundaryless, Stokes' Theorem says simply that

1M da=O, which means that the linear mapping

is well defined. But in fact, it can be shown that it is an isomorphism. By some standard reduction using partitions of unity and local coordinate systems, this amounts to proving the following assertion:

2. The de RhaIn definition

105

Let h : IRm -? IR be a smooth function with compact support, such that IIRn h = O. Then

for some smooth functions hI' ... ' h m with compact support. In other words, h is the divergence of some vector field with compact support. This is proved in parametric form, by induction on m, using elementary integration.

(2.2) Homomorphisms on cohomology. Let M and N be two connected, oriented, boundaryless, smooth manifolds of dimension m. Every proper smooth mapping f : M -? N induces linear mappings

j* : r~(N)

-?

r~(M) : w t---+ j*w.

These pull-back mappings are compatible with all operations with forms, including exterior differentiation, so that they induce homomorphisms on the cohomology groups, which we still denote f*. Again, we concentrate on the maximum degree group. Then, we have a commutative diagram of linear mappings

H:-(N,IR)

r

~

L 1~

H:-(M,IR)

lL

~ ~

IR

Thus we are ready to prove the main result concerning degree and cohomology. Theorem 2.3. Let f : M -? N be a proper smooth mapping of connected, oriented, boundaryless, smooth manifolds of dimension m. Then there is an integer d such that

fMj*W=d·Lw for every m-form w with compact support on N. Moreover, for any regular value a E N of f we have

d

=

L

signx (f)

(0 if f-I(a) = 0).

xEf-l(a)

Proof. The mapping oX in the preceding commutative diagram is a linear mapping IR -? 1R; hence it must be multiplication by some real number 6,

III. The Brouwer-Kronecker degree

106

and commutativity reads:

for every m-form w with compact support on N. Then, by II.3.4, p. 63, we can pick a regular value a of f, and the theorem follows at once if we show that 6= signxU)

2:

xEf-l(a)

(note that the right-hand side is indeed an integer). In other words, it is enough to see that

1J*w = 2: M

signA!)

1w N

xEf-l(a)

for a suitable w that we construct now. First of all, since a is a regular value, its inverse image f- l (a) is finite, suppose non-empty, say f- l (a) = {Xl, ... , x r }. Actually, as in the proof of IILl.1, p. 95, we find an open neigborhood V of a in Nand r disjoint open neighborhoods Ul , •.. , Ur of Xl, ... , xr in M, such that the restrictions fk = fluk : Uk --+ V are diffeomorphisms and f-l(V) = U1 U ... U Ur . Finally, we shrink V to a domain of coordinates diffeomorphic to jRm. Next, using any diffeomorphism V --+

jRm,

we pull back to V the m-form


where

r w = 1vr w = ± 1~m r
1N

m

=f= 0

(change of variables for the difeomorphism V --+ jRm). On the other hand, the support of the form J*w is contained in f-l(V) = U Uk, and since the Uk'S are disjoint, we have

r J*w = 2:1

1M

k

Uk

J*w.

2. The de Rham definition

107

Here we can compute each summand using the corresponding diffeomorphism fluk : Uk ---t V, and again by the change of variables for integrals, we get

1

f*w

= Uk f

Uk

Jv

W,

where Uk = ±1 according to whether or not fluk preserves or reverses orientation (which holds on the whole Uk because this open set is connected). In other words, since Xk E Uk,

and we conclude that

as desired. Note that this determines the integer fN w 1= O.

L:k sign

xk

(f), because

The reader will easily supply the simplified argument that settles the case f-l(a) = 0. 0 Thus, the de Rham approach gives the following most general form of the change of variables for integrals: Corollary 2.4. Let f : M ---t N be a proper smooth mapping of connected, oriented, boundaryless, smooth manifolds of dimension m. Then

1M f*w = deg

(f)·1

w

for every m-form w on N with compact support. In particular, the integral is zero if f is not surjective

Of course, this statement contains the classical change of variables of calculus, when f is a diffeomorphism. On the other hand, the invariance of homotopy follows quite easily in this context: Corollary 2.5. Let M, N be two connected, oriented, boundaryless, smooth manifolds of dimension m, and let H : [0,1] x M ---t N be a proper smooth homotopy. Then deg(Ho) = deg(Hl)'

III. The Brouwer-Kronecker degree

108

Proof. Let w be an m-form with compact support on N, and consider H*w. By the properties of exterior diferentiation we have dH*w = H*dw. But dw = 0, because m = dim(N), so that

o= f

H* dw

f

=

J[O,l]XM

dH* w.

J[O,l]XM

Now we compute the latter integral using Stokes' Theorem. Since the boundary of [0,1] x M is a disjoint union of two copies Mo and Ml of M, with wrong and right orientations, respectively (11.7.8(3), p. 91), we have

o= f

dH*w

J[O,l]XM

Thus, deg(Ho) hence deg(Ho)

L

w

=

f

H;w -

JM

f

How.

JM

= 1M How = 1M H;w = deg(Hl)

L

W;

= deg(Hl).

D

Example 2.6. One of the deepest theorems in mathematics, the GaussBonnet Theorem, can be formulated as the computation of a degree. Let us describe this.

Fix a connected, closed, boundaryless smooth hypersurface M C Rm+1 of dimension m. Then M is orientable, disconnects R m +1, and has a global equation f = 0 (all of this is the Jordan Separation Theorem; see 111.6.4, p. 129). What is relevant here is that 11

= grad(f)/II grad (f) II : M

~

Sm C

R m +1

is a unitary global normal field (11.7.5(1), p.88). Replacing f by - f if needed, 11 is compatible with the orientation chosen on M (11.7.4, p. 87), and it describes the volume element of M: OM = det(lI, .). This 11 is the so-called Gauss mapping. Then, at each point x E M we have the derivative dx ll : TxM ~ Tv(x)sm

= hyperplane perpendicular to II(X) = TxM.

This is the Weingarten endomorphism, and it is self-adjoint. Indeed, a straightforward computation shows that for every u, v E R m +1 perpendicular to II(X), that is, for every u, v E TxM, 1 (v, dx ll(u)) = II grad(f)II (v, dx { grad(f)) (u))

2. The de Rham definition

109

and dx ( grad (f) ) is a self-adjoint endomorphism of Rm+1, its symmetric matrix being

(8!2Jx (x) ) . j

One can accept that this endomorphism dxv of the tangent space TxM measures the way M c R m twists at x, and its determinant K{x) = det{dxv) is the Gauss curvature of M at x. As dxv is self-adjoint, its eigenvalues are all real and the endomorphism diagonalizes. Those real eigenvalues are the principal curvatures of M at x, and their product is precisely K (x). Next, suppose M is compact. Then the number /'i,=

fMKnM

is well defined, and quite naturally, it is called the integral curvature of M. Thus we find the integral considered by Dyck in 1888 (I.3, p. 31). To explore this integral curvature further, consider the volume element nsm of the sphere. A straightforward computation shows that KnM v* nsm, and we have

Now the Gauss-Bonnet Theorem says that if m is even,

where X{M) is the Euler characteristic of M. We conclude that the GaussBonnet Theorem just says: The degree of the Gauss map of an even-dimensional compact hypersurface is one half of its Euler characteristic.

This statement can be proved by means of the Poincare-Hopf Index 0 Theorem. We will discuss this in V.7.

Exercises and problems Number 1. Compact supports are essential. Show that any smooth form Q = fdxl ... 1\ dXm in IR m has some primitive, but this may well not have compact support. Number 2. Let 'P : IR -t I-form on §l. Show that:

§l

be the local diffeomorphism t

t-t

(cost,sint). Let

Q

1\

be a

III. The Brouwer-Kronecker degree

110

I(t)

(1) There is a periodic smooth function 9 : ~ -+ ~ such that 'P*a = 9 is a smooth function such that I(t + 271") = I(t) + a. (2) If = 0, I induces a smooth function h: §I -+ ~ with = dh.

I;

Ist

Ist a

Deduce from the above that the linear map

Number 3. Consider the torus T m 0::; k::; m.

= §I

gdt, and

a

Ist : HI (§I ,~) -+ ~ is indeed injective.

x .. ·

X

§I. Prove that Hk(Tm,~)

=I 0

for

Number 4. Let T = §I X §I. (1) From each factor in T = §I X §I obtain a linear subspace of HI (T,~) isomorphic to HI(§I,~). (2) Use the usual periodic local diffeomorphism 'P : ~2 -+ T to represent the I-forms a on T by forms Idx + gdy on ~2, with I and 9 periodic, and prove that the linear mapping

is injective. Conclude that dimlR(HI(T,~))

= 2,

and so HI(§I x §I,~)

= ~2.

Number 5. Consider the smooth surface M C ~3 given by x 2 + y2 + Z4 = 1, and consider the smooth mapping I : M -+ §2 : (x, y, z) H (x, y, z2). Compute the integral /*w for the form w = x 2 dy 1\ dz - y 2 dx 1\ dz + z 2 dx 1\ dy on §2. Is the computation really necessary?

IM

Number 6. Consider in ~3 the two diffeomorphic surfaces x 2 +y2 Check that their Gauss curvatures are, respectively,

K=

(4X2

= z and X4 +y4 = z.

4 + 4y2 + 1)2

Study the extrema of both, as well as their lines of constant curvature.

Number 7. Let M C ~3 be the torus parametrized by x = (R + r cos u) cos v,

y = (R

+ r cos u) sin v,

z = r sin u.

Then: (1) Show that the Gauss mapping of M is

vex, y, z)

= (- cosucosv, -

cosusin v, - sin u).

(2) Obtain the matrix of the Weingarten endomorphism with respect to the basis

a/au, a/av of T(x,y,z)M. (3) Deduce that K(x, y, z)

= :\ (1 - v';t r .,2+ y 2).

(4) Compute the integral curvature of M.

3. The degree of a continuous mapping So far, we have defined the Brouwer-Kronecker degree for proper smooth mappings only. To have the notion for arbitrary proper mappings, it is

111

3. Tbe degree of a continuous mapping

enough to recall that every proper mapping is homotopic to a proper smooth mapping by a proper homotopy:

Proposition and Definition 3.1. Let f : M -+ N be a proper mapping of two oriented, boundaryless, smooth manifolds of dimension m; furthermore, N is connected. Then all proper smooth mappings 9 : M -+ N properly homotopic to f have the same degree, and we define the degree of f by deg(f) = deg(g). Proof. If g, g' : M -+ N are properly homotopic to f, then they are properly homotopic themselves, and by II.5.4, p. 79, there is a proper smooth homotopy H t such that Ho = 9 and HI = g'. But then, deg(g) = deg(g') by III.1.9, p. 101. That indeed there are smooth mappings 9 homotopic to f follows from 11.5.2, p. 76, and II.5.3, p. 78. 0

As for smooth mappings, we have the product formula for composite continuous mappings:

Proposition 3.2. Let f : M -+ Nand 9 : N -+ P be two proper mappings of oriented, boundaryless, smooth manifolds, with Nand P connected. Then deg(g 0 f) = deg(g) . deg(f). Proof. Let f' and g' be proper smooth mappings properly homotopic to f and g, respectively. Then g' 0 f' is a proper smooth mapping properly homotopic to 9 0 f. Thus:

deg(g 0 f)

= deg(g' 0 f') = deg(g') . deg(f') = deg(g) . deg(f).

D

We also have the following:

Proposition 3.3 (Boundary Theorem). Let X be an oriented, smooth manifold of dimension m + 1 with boundary ax = Y, and let N be a connected, oriented, boundaryless, smooth manifold of dimension m. Let H : X -+ N be a proper mapping. Then deg(Hly)

= o.

Proof. Let H' be a proper smooth mapping properly homotopic to H. Then H'ly is a proper smooth mapping properly homotopic to Hly, so that deg(Hly) = deg(H'ly), and the latter degree is zero by III.1.8, p. 101. 0

Similarly, we deduce invariance by homotopy:

III. The Brouwer-Kronecker degree

112

Proposition 3.4. Let H : [0, 1] x M -+ N be a proper homotopy of oriented, boundaryless, smooth manifolds, N connected. Then

deg(Ho) = deg(Hd. Proof. Let H' : [0, 1] x M -+ N be a proper smooth mapping properly homotopic to H, so that Hb and H~ are proper smooth mappings properly homotopic to Ho and HI, respectively. We have

deg(Ho) = deg(Hb)

and

deg(HI)

= deg(HD,

and by 111.1.9, p. 101, deg(Hb) = deg(Hi).

D

Thus we have an invariant attached to every proper mapping between two manifolds M and N, the Brouwer-Kronecker degree, which only depends on the homotopy class of the mapping. Example 3.5. Consider the degree -1 mapping f : §I -+ §I : z 1---7 liz (this is 111.1.6(3), p. 100, for d = -1). Then define on the torus T = §I x§I the mapping F = f x f : T -+ T. This map has degree 1, but it is not homotopic to the identity.

Indeed, to compute deg(F), look at the derivative of F at the fixed point a = (1,1):

daF(u, v) = (dIJ(u), dIJ(v)) = (-u, -v). This preserves orientation; hence deg(F) = 1. Next, suppose there is a homotopy Ht with Ho = F, HI = IdT. Denote by j : §I -+ T the injection z 1---7 (z, 1) and by p: T -+ §I the projection (z, z') 1---7 z. Then ht = poHtoj is a homotopy with ho = f and hI = Id~p. But such a homotopy cannot exist, because deg(f) = -1 and deg(Id§l) = 1. Thus we see that, in general, degree does not fully classify homotopy classes. We will see that for spheres it does classify: this is the content of the famous Hopf Theorems (V.2.1, p. 191, and V.3.1, p. 196). D

Exercises and problems Number 1. Prove that a proper mapping f : 1R ~ 1R has degree 0, +1, or -1. Show that the three cases occur. Compute the degree of a real polynomial mapping f(x) = x d +alxd- 1 + ... + Cd. What does this say about the solutions of the equation f(x) = O?

3. The degree of a continuous mapping

113

Number 2. Construct two proper mappings f,g : IR2 -+ IR2 which are homotopic but have different degree (of course, they will not be properly homotopic). Number 3. Prove a product formula for the degree of continuous mappings, like that of Problem Number 1 of 111.1. Number 4. Construct a proper mapping f : IR2 -+ IR2 of degree 1 such that f-1({y = O,x =I O}) is the graph of the topological sinus y = sin(l/x), x =I O. Extend fto §2 via stereographic projection from the north pole, and describe the inverse image by that extension of the meridians corresponding to the coordinate axes. Number 5. Fix integers m, n ::::: O. Every point z in §m+n+1 C IR m+ n+2 = IRm+1 x IR n+ 1 can be written uniquely as

z = cos(t)x + sin(t)y,

where x E §m, y E §n, 0

~

t

~

7r/2,

except that x (resp., y) is not determined for t = 7r/2 (resp., t = 0). Given two continuous mappings f : §m -+ §m and f : §n -+ §n, their join is defined by

f Prove that

* 9 : §m+n+1 -+ §ffi+n+1

: z>-+ cos(t)f(x)

+ sin(t)g(y).

f * 9 is continuous, and its degree is deg(f) deg(g).

Number 6. Consider the exponential mapping exp: IR -+ §1 : x >-+ exp(x) = (cos(27rx),sin(27rx)). Now, let f : §1 -+ §1 be a continuous mapping of degree d, and let h : [0,1] -+ IR be a continuous lifting of h = f oeXPilO,l] (that is, h = expoh). Show that d = h(l) - h(O).

Number 7. Let f, 9 : §m -+ §m be continuous mappings, m ::::: 2. Check that the mapping h = - f + 2(f, g}g : §m -+ §ffi is well defined and then compute its degree as follows: (1) Reduce to the case that f == (1,0, ... ,0) on the upper hemisphere and 9 == (1,0, ... ,0) on the lower one. (2) Note that h coincides on the upper hemisphere with (2g~ -1, 2g 1 g2, ... , 2g1gm +1) and compute the degree of the latter in terms of deg(g). (3) Note that h coincides on the lower hemisphere with (/1, - h, ... , - fm+I) , and compute the degree of the latter in terms of deg(f). (4) Conclude that deg(h) = (-l)mdeg(f) + (1- (-l)m)deg(g). Number 8. Let p,: §ffi X §m -+ §m be a continuous mapping. (1) Show that for every y E §m all mappings p,(., y) : §m -+ §m have the same degree, which we denote deg 1(p,). (2) Define a degree deg 2(p,) similarly. Now let g, f : §m -+ §m be two continuous mappings, and consider the mapping p,(f,g) : §m -+ sm. Show that

deg(p,(f,g» = deg(f) deg 1 (p,) How does this generalize the previous problem?

+ deg(g) deg 2 (p,)·

III. The Brouwer-Kronecker degree

114

4. The degree of a differentiable mapping So far, the degree of a mapping f is computed through some smooth approximation by the formula in IlL 1.4, p. 99. However, it is natural to expect that in case the mapping f is itself differentiable, the same formula works directly for f, without any interposed smooth approximation. This is indeed true but, somehow surprisingly, a quite delicate technical matter. Indeed, a first naive attempt would be to revise all constructions in I1Ll replacing smooth by differentiable cr mappings. Then, for homotopy invariance, we must apply the Sard-Brown Theorem to a Cr mapping [0, 1] x M -+ N: this requires the difficult version for finite class mappings, and even more, it needs class r > dim ([0, 1] x M) - dim (N) = 1. Thus we see the essential difficulty that differentiable Cl mappings are excluded by this approach. In fact, as we will see, this is the only obstruction to a Cl degree theory as neat as the smooth one we have previously presented. This is the part of the theory that relies on some results that we do not prove, and it is intentionally confined to this section. After this preamble, we get into the main result:

Theorem 4.1. Let f : M -+ N be a proper Cl mapping of two oriented, boundaryless, smooth manifolds of dimension m; furthermore, N is connected. Let a E N be a regular value of f. Then f-l(a) is a finite set and

deg(J)

L

=

signx(J)·

xEf-l(a)

Proof. We will use a close smooth approximation 9 : M -+ N as given by I1.5.2, p. 76, with some additional conditions. Since a is a regular value and f is proper, f-l(a) consists of finitely many points, say CI, ... , Cr. Then we pick r local coordinates that preserve orientations on disjoint domains Uk == jRm at each point Ck and another coordinate domain V == jRm at a such that Ck == 0, a == 0, and f(Uk) C V. Denote by fk : jRm -+ jRm the localizations of f at each Uk. We claim the following Let ( = fk' and let det(;;: (0))

{llxll

:S p} such that for

e:B -+

jRm

Ile(x) then:

E

>

°

=I- 0. There is a closed ball B =

small enough, if a smooth mapping

verifies

((x) II

< E,

1

8e · (x) _8(· _t (x) I <

_t

8xj

8xj

E,

for all i,j,

4. The degree of a differentiable mapping

115

(1) 0 = ~(b) for a unique b, and Ilbll (2) det (;;: (0)) . det

< p,

(;!: (b)) > 0 (that is, signo(() = signb(~)).

Assume this for the moment. Pick a smooth approximation 9 of f such that, for every x EM,

IIg(x) - f(x)11 < dist (J(M \ W), a), dist (J(T), N \ V), where W is the union of the sets Wk == {llxll < p} and T is that of the sets Tk == {llxll :s: p} (by construction these distances are not zero). These bounds give: (i) a

tf. g(M \ W). Indeed, if a = g(x) with x tf. W, we have dist (J(M \ W), a)

:s:

IIf(x) - g(x)ll,

which is impossible.

(ii) g(Tk)

C

V. For, if g(x)

tf.

V with x E Tk, we get

dist (J(T), N \ V)

:s:

Ilf(x) - g(x)ll,

a contradiction. Then, the approximation 9 has well-defined localizations gk : B -+ jRm, and the crucial point here is that 9 can be chosen so that each ~ = gk verifies the bounds on the derivatives required by the claim. This is the part we assume without proof. We also have that g-l(a) C W; hence

g-l(a)

== Ugk1(0), k

and by the conclusions of the claim, there are exactly r points bk E none is critical, and

We conclude that deg(f)

= deg(g) = Lsignbk(gk) = Lsignck(f). k

k

The proof is thus complete, except for the following:

gk 1 (0),

116

III. The Brouwer-Kronecker degree

Proof of the Claim. This is a review of the the proof of the Inverse Mapping Theorem. Consider the continuous mapping

Since ((0) = a ::f. 0, there is an 'T/ > 0 such that 1

a-det

(Z;i. (O)+tij)

1

<

J

~Ial

Then choose p > 0 such that a(i (Xi) 1-ax· J

for

1 -a(i (-)1 0 <-'T/

ax·J

2

(hence I((x) - al

~Ial),

<

II xIiI , ... , IIxmll < p.

Next, denote by 8 > 0 the minimum of 1I((x)1I on the sphere IIxll = p (this minimum is not zero, because by construction 0 is the unique zero of (). Then, choose E: > 0 smaller than ~1J and ~8. We will check that this E: does the job. Define [ mimicking (. Hence - _

~(x)

) a~i (_ ) a(i (-) = det (a(i ax. (0) + tij , where tij = -a Xi - -a 0, Xj Xj J

and on

IIXili < p we have

so that by (*), la - [(x)1

<

~Ial, which clearly implies a· a~·

~

a· det ( axt. (x))

[(x) > OJ hence

> 0 for IIxll < p.

J

This already guarantees (2) in the statement of the Claim, as soon as b is given. To find it, note that the b we sought is a minimum of the function IIxll = p we have

J..L(x) = 1I~(x)1I2 in the compact set B. But on its boundary 1I~(x)1I ~ 1I((x)II-II~(x)

-

((x) II

> 8-

E:

> E: >

II~(O)

-

((0)11 =

II~(O)II,

117

4. The degree of a differentiable mapping

which means that the minimum does not belong to the boundary. Hence, b is a critical point of the smooth mapping /-L on the open ball Ilxll < p, so that its gradient is zero at that point. But a simple computation gives gradb(/-L)

= 2~(b)

(:!i.

(b»),

J

and since the determinant of the latter matrix is not zero, we conclude = 0 as desired.

~(b)

Finally, we turn to the uniqueness of b. Suppose that ~(x) = 0 with IIxll < p. We apply the Mean Value Theorem to each component ~i on the segment [b, xl and find points Xi E [b, x], hence IIxili < p, such that

o = ~i(X) - ~i(b) = L. [)Xj [)~i (Xi)(Xj -

bj ).

J

This is a linear system in the unknowns Zj = Xj - bj, and its determinant is ~(x) -=1= 0 (see above). Hence the only solution is Zj = 0, that is, x = b. 0

(4.2) Degree theory for differentiable manifolds. Once we know that computations are the same for differentiable as for smooth mappings, the theory goes through for differentiable manifolds. As explained before (II.2.7, p.57), the key result needed is smoothness: every differentiable Cr manifold is Cr diffeomorphic to a smooth manifold. Such a smooth manifold is constructed by approximation, in a more delicate sense than the one developed in II.5.2, p. 76. Actually, it is the kind of approximation also used without proof for III.4.1, p. 114: approximation of derivatives. As mentioned already, here is where we put the limits on our presentation. Now, we sketch the construction of the Brouwer-Kronecker degree in the differentiable setting. Let f : M --+ N be a proper C1 mapping of oriented, boundaryless, C1 manifolds of dimension m, N connected. Then there are C1 diffeomorphisms r.p : M --+ M' and 'IjJ : N --+ N' onto smooth manifolds M' and N'; we choose in these smooth models the orientations compatible with r.p and 'IjJ and define deg(J)

= deg('IjJ 0 f

0

r.p-l).

Of course, this deg(J) does not depend on the diffeomorphisms r.p and 'IjJ. To prove it, pick any regular value a E N of f. Then, from the preceding theorem, III.4.1, p. 114, it easily follows that deg(J)

=

L xEf-l(a)

signx(J),

III. The Brouwer-Kronecker degree

118

which clearly does not depend on 'P or 'ljJ. After this, one extends the theory to C1 manifolds using smooth models: proper mappings are properly homotopic to proper C1 mappings whose degree we can compute by the sum of signs formula, the Boundary Theorem holds, and it follows that properly homotopic mappings have the same degree, and the degree of a composite mapping is the product of the degrees of the factors. All arguments follow without surprise.

Exercises and problems Number 1. Let ( : R m -+ R m be a differentiable mapping with ((0) is a linear isomorphism. Show that for p > 0 small enough

IIdoc

10

(d z (

-

do()

I <~

for

= 0 such that do(

Ilzll < p.

Pick a positive real number 8 < II doC 111- 1. Prove that for TJ smooth mapping ~ : R m -+ R m such that

>

0 small enough any

verifies the two conditions

Use these bounds and the Finite Increment Theorem to deduce that ~ has a unique zero c with Ilell < p and the signs at c of the Jacobian determinants of ~ and ( coincide.

Number 2. Show that the mapping

f : §1 -+ §1 defined by

f(x, y) = (cos(27rN), sin(27rN)) is C1 but not C2 , and compute its degree.

Number 3. Let f : M -+ N be a proper mapping of two oriented, boundaryless, smooth manifolds of dimension m, N connected. Suppose that there is an a E N such that f is smooth on some neighborhood U of f-1(a) and a is a regular value of flu. Prove that there is a close smooth approximation 9 : M -+ N of f that coincides with f on U (up to shrinking) and such that g-1(a) C U. Deduce that deg(f) =

L

signx(f)·

xEf-l(a)

What is the difference between this and III.4.1, p. 1147

Number 4. Compute the degrees of the following two mappings

(X2_y2,-2XY) () 1 9 ( x,y ) = { (X 2 _y2,2xy) ( 2) h(x

)_ ,y -

{(X' 2-y)

forx~O, for x:S; o.

((x _ 3y 2)X, (3x 2 - y2)y)

for x for x

~ 0,

:s; O.

§1

-+

§1:

5. The HopE invariant

119

Number 5. Complete the details of C1 mapping degree theory as suggested in 111.4.2, p.117. Number 6. Let M C 1R2 denote the graph of the function h: IR ~ IR given by

h(x)

=

{o

for x ::; 0, for x ~ 0.

X 3/ 2

Prove that AI is a C1 curve, but not C2 • Fix an orientation on !vI and find proper mappings f : IR ~ M of all possible degrees.

5. The Hopf invariant Although the notion of degree is mainly intended for discussing mappings of manifolds of the same dimension, it can be profitably used to treat some mappings between spheres of different dimensions. The pioneering ideas in this respect are due to Hopf (see 1.3, p. 30). We fix the following notation. Consider the north and south poles, p = (0, ... ,0,1) and -p = (0, ... ,0, -1), of the unit sphere §2m-l C 1R2m ,

m 2: 2, and let ¢ : §2m-l \ {-p} --+ Tp§2m-l == 1R2m - 1 be the stereographic projection from the south pole. Let n c §2m-l X §2m-l be the open set defined by the inequalities x =f. y, x =f. -p, y =f. -p, and consider the smooth mapping

iP:

n -+ §

2m-2

(

:

¢(y) - ¢(x)

)

x, y

t-t

1I¢(y) _ ¢(x)11 .

We have the following: Proposition and Definition 5.1. Let f mappzng.

§2m-l

--+

§m

be a smooth

(1) If f is not surjective, we let H(f) = 0.

(2) Iff is surjective, consider any two distinct regular values a, b =f. f( -p)

of f. Then f-l(a) x f-l(b) c n is a compact, oriented, boundaryless, smooth manifold of dimension 2m - 2, and iP restricts to a smooth mappzng ¢a,b : f-l(a) x f-l(b) -+ §2m-2, whose degree does not depend on the choice of a and b. We let H (f) = deg( ¢a,b)'

The integer H (f) is called the Hopf invariant of f.

III. The Brouwer-Kronecker degree

120

Proof. Suppose f surjective. By II.3.2, p. 62, and 11.7.6, p. 89, the inverse image of every regular value is an oriented manifold of dimension m - 1; hence the product f-1(a) x f-1(b) is an oriented manifold of dimension 2m - 2. On the other hand, since the compact sets f-1(a) and f-1(b) are disjoint and do not contain the south pole -p, their product is contained in D. All in all, the degree deg(4)a,b) is well defined.

Next, let us see why that degree does not depend on a (the proof for b would be the same). Let a' =1= f( -p) be another regular value of f, and let Ft be a diffeotopy of §m that fixes band f (-p) and is the identity off a neighborhood W of a, a' such that F1 (a') = a. We are to apply a boundary argument twice to conclude that deg(4)a,b) = deg(4)a l ,b). Suppose first that a' is close to a. Then we take We Rj, which implies that a is a regular value of the smooth mapping H(t, x) = Ft(f(x)). Indeed, if Ht(x) = a, we have Ft(f(x)) = a, and f(x) must be in W, so that x is a regular point of f. Hence dxf is surjective, and since Ft is a diffeomorphism, the derivative

dxHt = dj(x)Ft 0 dxf is surjective. With this settled, H-1(a) is a compact smooth manifold with -p (j.

Ht-1(a)

and

Ht-1(a)

n f-1(b) = 0

for all t.

For the first condition, notice that if a = Ht ( -p) = Ft(f( -p)), then f( -p) E W, a contradiction. For the second, if a = Ht(x) = Ft(f(x)), then f(x) E Wand b (j. W. Thus the smooth mapping Ai

~:

X

=

H- 1() f-1(b) a x

~2m-2

--t.:l)

:

(

t,x,y

)

I-t

4>(y) - 4>(x)

114>(y) _ 4>(x) II

is well defined, and by the Boundary Theorem (III.l.8, p. 101) 0= deg(4)lax),

where

ax = aH-1(a) x f-1(b).

Note that the boundary of H-1(a) consists of the two pieces {O} x f-1(a) and {I} X f- 1 (a'), one with the right and the other with the wrong orientation. Thus ax consists of

{O} x f-1(a) x f-1(b)

and

{I} x f-1(a') x f-1(b),

one with the right and the other with the wrong orientation. Clearly, 4> restricts to 4>a,b on the first component and to 4>a' ,b on the second, and by the remark concerning orientations we conclude 0= deg(4)lax) = deg(4)a,b) - deg(4)a l ,b).

121

5. The HopE invariant

Once this case is proved, assume a' is arbitrary. Then choose a diffeotopy Ft as above, but now a need not be a regular value of H t = Ft 0 f. Then, by the Sard-Brown Theorem, there is a regular value al of H close to a so that Fl1(al) = a~ is close to a'. By the case already proved, we have deg(cPa,b) = deg(cPal,b) and deg(cPa',b) = deg(cPa~,b).

On the other hand, the argument above with H-l(aI) instead of H-l(a) shows that

deg(cPal,b) = deg(cPa~,b)' and, as desired, we deduce

deg(cPa,b) = deg(cPa',b).

o

Remarks 5.2. (1) The degree of the mapping cPa,b is in fact the link coefficient of the two manifolds f-l(a) and f-l(b) as was introduced by Brouwer (see 1.2, p. 28). We use the stereographic projection to set the two manifolds in a Euclidean space. We will abuse notation and write

although this disregards the sign on the left-hand side; for our use here this is only a notation. (2) After the above definition of the Hopf invariant, it is only natural to ask whether for non-surjective mappings we can also compute it through link coefficients. The answer is that we can. Namely, if f : §2m-l -t §m is not surjective and a, b =1= f ( -p) are two regular values such that f- 1 (a) =1= 0 and f-l(b) =1= 0, then

To see this, just repeat the proof of II1.5.1, p. 119, by choosing a second regular value a' ~ f(§2m-l). Then f-l(a') = 0 and

so that ax

= aH-l(a)

x f-l(b) = {O} x f-l(a) x f-l(b).

Consequently, ~18X == cPa,b, and from deg(~18x)

= 0 we deduce

(3) We have fixed the center of projection at the south pole and need no more freedom for our purposes, but this choice is actually irrelevant. The

122

III. The Brouwer-Kronecker degree

argument to show this is quite similar to those above, using a rotation to move the center elsewhere. We will not give any details here. (4) In odd dimension, the Hopf invariant vanishes: if m is odd, H (J) = : §2m-l -+ §m.

o for every smooth mapping f

Indeed, we have the commutative diagram

f-l(a) x f-l(b) ~ T

f-l(b)

1~ X

§2m-2

~

f-l(a) ~

lu

§2m-2

T is the permutation (x, y) t---+ (y, x) and u is the antipodal mapping -z. But an easy computation shows that deg(T) = (_1)(m-l)2 = +1

where

z

t---+

(m is odd) and deg(u)

= -1 (111.1.6(1), p. 99), so that

H(J) = deg(<pa,b) = deg(u) deg(
o

= O.

Next we see that we have constructed a homotopy invariant: Proposition 5.3 (Homotopy invariance). Let f, g :

§2m-l

-+

§m

be two

homotopic smooth mappings. Then, H(J) = H(g). Proof. Since [0,1] X §2m-l is compact, the homotopy (which we can take to be smooth) is uniformly continuous; hence there is an c > 0 such that IIHt(x) - Hs(x) II


for all x E

§2m-l

and

It - 81


The result follows immediately if we see that for 0 < 8 - t < c, H t and Hs have the same Hopf invariant. In other words, we can merely assume

IIHt(x) - Hs(x) I


for all x, t,8.

Now, the result is trivial if H(J) = H(g) = 0; hence we suppose that one of the mappings is surjective, say f. We will use a boundary argument (as in the proof of 111.5.1, p. 119) to deduce first

and then

5. The Hopi invariant

123

Notice that what matters here is only that the smooth mappings ~.X .

= H-l(a) x

f-l(b) ---t §2m-2 . (t .

"

X

y)

H

¢(y)-¢(x) 1I¢(y)-¢(x) II

and

are well defined. But by the bound that we have assumed, this follows and are taken off immediately if the regular values verify Iia - bll ~ H ([0, 1] x {-p}) (and the latter is possible by the Easy Sard Theorem (11.3.4, p. 63), because m ~ 2).

!

Apart from this, we only recall what was stated in 111.5.2(1), p. 121, for non-surjective mappings. As mentioned there, if 9 is not surjective, we can take the regular value a off g(§2m-l), so that 8H-l(a) = {O} x f-l(a); hence in the first step we get

o

and this case is finished here.

As for degree, the preceding homotopy invariance result gives grounds for defining the Hopf invariant of arbitrary continuous mappings.

Exercises and problems Number 1. Let f : §2m-l --+ §m be a smooth mapping (m ;::: 2). We assume both spheres oriented. Let w be a form on §m of maximum degree m, such that w = l. Then j*w is a closed form in §2m-l, hence exact (because Hm(§2m-l, R) = 0), and there is a form a of degree m - 1 such that da = j*w. Consequently, we have on §2m-l the 2m - 1 form a 1\ da and can compute the integral

Ism

g(f)

=

1

a

1\

da.

s2m-l

Show that g(f) is the same for any choice of w and a under the conditions above. How does it depend on the orientations? This is in fact another definition of the Hopf invariant, but we cannot prove this now and consequently we use a different notation. Number 2. Show that the invariant g defined in the preceding problem is a homotopy invariant and thus it can be defined for arbitrary continuous mappings. Number 3. Compute the invariant g(f) of a continuous mapping f m is odd. Number 4. Compute H(f) and (!(f) for the mapping f(Xl,X2,X3,X4) = (2Xl\!1-

x~ - x~,

: §2m-l --+ §m when

f : §3 --+ §2 defined by

2x2V1- x~ -

x~,

1-

2x~ - 2x~).

124

III. The Brouwer-Kronecker degree

6. The Jordan Separation Theorem We devote this section to one very important application of degree theory: a proof of the Jordan Separation Theorem for closed differentiable hypersurfaces of the Euclidean space. In fact, this is the proof involving the Cauchy index that triggered Hadamard's, and then Brouwer's, research on the notion of degree (1.2, p. 14). In fact, this proof does not require the notion of degree in full, but only a simpler version which we quickly describe next. (6.1) Mod 2 degree theory. All constructions and definitions in the preceding sections can be carried over disregarding orientations, which means that manifolds need not be orientable. The resut is a mod 2 invariant defined as follows: Let f : M ---t N be a proper mapping of two boundaryless differentiable manifolds of dimension m; furthermore, N is connected. Let g : M ---t N be a proper differentiable mapping close to f and let a E N be a regular value of g. Then the parity

depends only on the homotopy class of f and is called the degree mod 2 of f.

We leave the reader the task of reviewing all proofs involved and only notice for the key result, 111.1.2, p. 96, that the curves r used there have either no boundary points or two boundary points; hence their union C has an even number of boundary points. Using this mod 2 degree, we can now prove the following: Theorem 6.2 (Jordan Separation Theorem). Let M c jRm+l be a compact, boundaryless, differentiable hypersurface. Then M disconnects jRm+l . Proof. Pick any point p

¢. M and define the differentiable mapping x-p

fp : M

---t §m :

x

t---+

Ilx _ PI! '

which is proper because M is compact. The mod 2 degree of this mapping is called the mod 2 winding number of M around p and is denoted by w2(M,p) (compare IV.4.1, p.156). It is easy to see that the winding number is constant on each connected component of jRm+ 1 \ M.

125

6. The Jordan Separation Theorem

Indeed, if p and q are in the same component, that component, which is an open set in Rm+1, contains a continuous (even piecewise linear) path , = ,(t) with ,(0) = p and ,(1) = q. Thus we can define a homotopy by H: [0,1]

xM

x-,(t)

m

-t §

:

x t-+

Ilx _ ,(t)ll·

Hence, by homotopy invariance for mod 2 degree,

Thus, to show that M disconnects, it is enough to find two points p and q with different mod 2 winding numbers. To that end, pick any point a EM and line Rthrough a whose direction is perpendicular to TaM. Then pick pER \ M. We see that a is a regular point of JP by direct computation. For v E TpM we have daf (v) p

because (a - p, v) phism, as desired.

= Iia - pl12v - (a - p,v)(a - p) =

lIa - pll3

v

Iia - pll '

= 0 by the choice of R. Thus, daJp is a linear isomor-

Consequently, by the Inverse Mapping Theorem, JP is a local diffeomorphism at a. This implies that Jp(M) has non-empty interior in §m, and by the Sard-Brown Theorem, JP has some regular value U E Jp(M). We denote by L the half-line x = p + AU, A > o. Then, if p is far enough from our compact hypersurface M, we have J;l(u) = M n L, and

126

III. The Brouwer-Kronecker degree

Now let x be the first point we find in M n L when starting from p, and pick a second point q E L \ M before the next (we can do this, because M n L = f;l(u) i= 0). In this situation, f;;l(u)

= M n L \ {x},

and u is a regular value of fq. Indeed, the first assertion is clear, and for the second note that

y - q = /-l(Y - p) with /-l > 0 for every y

E

f;;l(u).

Then, for v E TyM we have

d.f, (v) = Ily-qI12v- (y-q,v)(y-q)

Ily -

y q

qll3

= Ily - pl12v - (y - p, v)(y - p) = Id f: (v) /-lIly which shows that dyfq

pl13

tL Y P

,

= tdyfp is an isomorphism.

Thus,

Obviously this implies w2(M,p)

i= w2(M, q),

and the proof is finished.

D

As it stands, the above proof cannot be used for non-compact hypersurfaces: for them, fp is not proper any more. However, the idea behind it can be put to work in that case too. In fact, we have the following: Theorem 6.3 (Jordan Separation Theorem). Let M c jRm+1 be a closed, boundaryless, differentiable hypersurface. Then M disconnects jRm+l. Proof. We will use the following notation. Let D c jRm+1 be an open ball with boundary a sphere S = D \ D, and let p E jRm+1 \ D. We denote by 7rp the conic projection with center p into the sphere S. This mapping is defined on a compact cone Kp with vertex p and its image is the spherical region Kp n S. The boundary of Kp is the tangent cone to the sphere with vertex p, and the interior Tp of Kp in jRm+1 is the open cone generated by p and the interior Np of Kp n Sin S; note that Np = 7rp(Tp). We need the following simple property of 7rp: dx 7rp(u)

= 0 if and

only if u is parallel to x - p.

127

6. The Jordan Separation Theorem

Indeed, consider the curve germ ,(t)

= x + tu.

Then

for some smooth >.(t), and

dx 7rp (u) = (7rp 0,)'(0) = >.'(O)(x - p)

+ >'(O)u = v.

Since 7rp (x) = p + >'(O)(x - p) E Sand p ~ S, we have >'(0) -:j= 0, and v is vanishes if and only if u is parallel to x - p. We can reformulate the above remark by saying that d x 7rp is injective on any linear hyperplane L transversal to x - p. Now, pick a point e E M and a line e through e not tangent to M (that is, e ¢. e + TeM). This choice guarantees that e is isolated in en M (II.2.12, p. 60): there is a segment [a,e] C econtaining e(-:j= a,e), such that [a, e] n M = {e}. In particular, c is not in the closed set M, and there is an open ball D centered at e with M n D = 0.

p

L~~q [a,q]

Note that for D small enough, 7ro: restricts to a diffeomorphism ho: : Mo: = M n To: -+ No:. Indeed, for a first choice of D, the preceding remarks on the derivatives of conic projections tell that deho: is a linear isomorphism. Consequently, ho: is a local diffeomorphism at the point e, and the claim follows by shrinking D. Once this setting is ready, consider an arbitrary point p ~ M u D and the corresponding 7rp , K p, Tp, Np. If there is some point a E Np such that [a,p] n M = 0, we write

128

III. The Brouwer-Kronecker degree

Otherwise, Mp = Tp n M =1= 0 is an m-manifold open in M and 7fp induces a differentiable mapping hp = 7fp IMp : Mp = Tp n M -+ N p. This mapping is proper (the restriction of 7fp to the compact set Kp n M is proper), and we write W2(p) = deg2(hp). To compute this W2(p), we use regular values. Now, a E Np is a regular value of hp if and only if h;l(a) is finite and for each x in that inverse image, the vector a - p is not tangent to M at x. Again this follows from what was earlier remarked concerning the derivatives of conic projections. In short we say that [a,p] meets M transversally, and then

W2(p) = #([a,p]

n M)

mod 2.

Note that there is some overlapping here. Indeed, Mp may be non-empty and hp will be defined, even if [a, p] n M = 0 for some a E N p. But then a is a regular value of hp, so that deg2(hp) = o. Of course, the computation through any other regular value will give the same degree o. Next, note that our careful choice of data guarantees w2(a) = 1. On the other hand, take any !3 in the segment (e, 7fa(e)). Clearly, M{3 = 0, and W2(!3) = o. Thus, the invariant W2 is not constant. We claim that it is locally constant. To prove this, fix p. If [a, p] n M = 0 for some a E Np, as [a, p] is compact and M is closed, we see immediately that [a, q] n M = 0 for q very close to p. Hence suppose we have the mapping hp well defined, with a regular value a E Np so that [a,p] meets M transversally. We want the same for q close enough to p and also that

#([a,q] n M) = #([a,p] n M), to conclude W2(p) = W2(q). For this we argue as follows. For every x E [a,p] n M, let "fa,x be the conic projection with center a onto x + TxM: the same remarks on derivatives of conic projections apply to "fa to deduce that its restriction to a neighborhood V of x in M is a diffeomorphism from V onto an open set of x +TxM. This implies that for q close to p, the segment [a, q] meets M exactly as [a, p] does, and we are done. Summing up, W2 is locally constant, but not constant, on the open set jRm+l \ M U D: thus this set cannot be connected. Finally, if the open set jRm+l \ M were connected, no closed ball would disconnect it, and we conclude as desired that the complement of the hypersurface M is not connected. 0

6. The Jordan Separation Theorem

129

From the fact that hypersurfaces disconnect, much more information can be deduced. We gather all of it into a single statement: Proposition 6.4. Let M c Rm+! be a connected, closed, boundaryless, differentiable hypersurface. Then M disconnects Rm+! into two connected components D and E, whose closures D and E are closed manifolds with boundary M. Furthermore, M has a global equation f such that

M = {J = O},

D

= {J > O},

and E

= {J < O}.

In particular, M is orientable and has a unitary global normal vector field

= grad(J) I II grad(J)II·

1/

If M is compact, one of the two connected components is bounded (the interior of M) and the other is not (the exterior of M). Proof. As we know (II.2.3, p. 55), M has local equationsj hence every point in M has an open neighborhood U in R m +! such that Un M = {J = O} for some differentiable function f : U --+ R with 0 a regular value. In fact, by II.2.3, p. 55, we can assume that there is a diffeomorphism cp : V --+ U from an open ball V centered at 0 E Rm +!, such that cp-l(U

n M) = {x E V

: Xm+!

= O}.

In this situation:

(a) If a differentiable function h : U --+ R vanishes on is a well-defined differentiable function on U.

un M,

then hi f

(b) If h : U --+ R is another equation of M in U, then a = hi f has no zeros in U.

(c) The set U \ M has two connected components, U+ = {J > O} and U- = {J < O}, so that M n U c U+ n U-. For (a), we note that h h

0

cp(x}, ... , Xm , 0) = OJ hence

0

CP(Xl, ... , Xm, tXm+l)) dt = h*(x)xm+!,

(I a

0

cp(X) = Jo at (h

with h*(x)

=

1 1

o

a

a Xm+l

(h

0

cp(Xl, ... , Xm , txm+d) dt.

III. The Brouwer-Kronecker degree

130

Similarly, f 0 cp( x) = J* (x )xm+1' but in this case we can say more: J* never vanishes. Indeed, we have

where

7r :

IRm+1 -+ IR is the last coordinate projection. If J*(x) = 0, then

f(cp(x)) = 0; hence cp(x) EM and Xm+1 = O. Thus

and cp(x) E Un M would be a critical point of f, a contradiction. Consequently * h*(x) h 0 cp(x) = h (x)xm+1 = f*(x) (J 0 cp)(x), which shows that (hi J)

0

cp

= h* I J* is differentiable on V.

For (b), note that if h is an equation itself, the same argument above shows that h* has no zeros, and neither does h* I J*. Finally, assertion (c) is evident for the equation h = 7r 0 cp -1 and follows for f = ah, because a never vanishes, hence has constant sign on the connected set U. After these three local properties, we obtain a covering of M by open sets Ui C IRm+1 with equations Ii as above, to which we add a covering of IRm +1 \ M by open connected sets with trivial equations = 1. Recall that by the Jordan Separation Theorem, 111.6.3, p. 126, we know that M disconnects IRm+1. In fact, (d) IRm + 1 \ M has exactly two connected components D and E, and

Indeed, we claim first that W \ W = M for every connected component W of IRm+1 \ M. Note that any other component W' of IRm+1 \ M is open in IRm +1 and does not meet W; hence it does not meet W either. Thus, W \ W eM. On the other hand, the open set W cannot be closed in IRm+1; hence there is some point x E W \ W eM. Let y E M be arbitrary. As M is connected, there is a chain Uio' ... ,Uir with x E Uio' Y E Ui r , and Uik n Uik+l n M 1= 0, Uik \ M = Ui~ U Ui~' for every k:

6. The Jordan Separation Theorem

131

M

!

!

Ui U Ui~ and x E W, we deduce Ui C W or Ui~ C W. It follows that M n Uio C Ui n Ui~ C W, and looking at a point Xl E M n Uio n Uill we see Xl E Uh n W; hence Ui~ C W or Ui~ C W. Thus MnUil C W.

From Uio \ M

=

!

Repeating the argument, we conclude that y completes the proof of our claim.

E

M

n Uir

C

W. This

Now, let D = W be a component as in the argument above, and let E be a second component (recall that M does disconnect Rm+1). Arguing as above, we find Ui C E or Ui~ C E. As D and E are disjoint, there are two possibilities:

!

which gives no extra room for anymore components! Once we have (d), we choose signs Case Ui n M Case U·

~

nM

{+-11

Ci

as follows:

= 0:

Ci

=

0·

Co

= { +1 if Ui+ cD, -1 if Ui+ ¢. D (that is, Ui- CD).

-I..

r·.

With these signs fixed, set 9i

if Ui cD, if Ui ¢. D (that is, Ui

n D = 0).

= cili, so that

Consequently, the function 9k/9i, which is defined on Uk zeros by (a) and (b), is > O.

n Ui

and has no

Finally, let {(}i} be a differentiable partition of unity for the Ui's. The formula

132

III. The Brouwer-Kronecker degree

gives the equation

f : ~m+1

-t ~ that we sought.

First note that the sum is locally finite, and for x E Ui and Ok(X) =f. 0, it is x E Uk n Ui , and in that intersection 9k/9i > 0; hence the logarithm exists. Moreover, in every non-empty intersection Ui n Uj we have

By all of this, (*) is a consistent definition. Now, on Ui we have

f = edi' where ei = Ei exp[· .. J has no zero. Thus,

{f = O} n Ui = {Ii = O} n Ui = Ui

n M,

and for every x E Ui n M, derivation gives

Summing up, f is a global equation for M. Furthermore, as exp > 0, we have D = {f > O} and E = {f < O}, and by (d),

{f > O} \ {f > O} = M = {f = O}, that is, {f > O} = {f ~ O}. This is a manifold with boundary {f (II.3.2, p. 62). The same works for {f < O}.

= O} = M

The other assertions in the statement come from II.7.5(1), p. 88.

0

Exercises and problems Number 1. Complete the details for a mod 2 degree theory as suggested in 111.6.1. Note that for oriented manifolds deg 2 = deg mod 2. Number 2. Let. denote a (bilinear) multiplication in ]Rm+l such that x. y = 0 only if x = 0 or y = O. Fix any nonzero u E ]Rm+l and define the mapping

f : §m --+

§m :

x

t-+

f(x)

=

1\:: :1\.

Show that f is a homeomorphism whose homotopy class does not depend on u. Deduce that f is homotopic to - f and that m is odd.

Number 3. Let. denote a multiplication in ]Rm+l as above that furthermore is commutative. Consider f : lH'm --+ §= : x t-+ f(x) =

1\:: :1\.

7. The Brouwer Theorems

133

Show that: (1) f is a well-defined injective continuous mapping. (2) deg 2 (f) = 1; hence f is surjective. Deduce that U m is homeomorphic to §m. As is well known, this is the case only for m = 1; hence for m ~ 2 such a multiplication does not exist. Number 4. Let M c IR m +1 be a connected, compact, boundaryless, differentiable hypersurface. Denote by D (resp., E) the bounded (resp., unbounded) connected component of IR m +1 \ M. For every point p ¢ M set fp : M -t §

m

:

x >-t

x-P

IIx _ pil .

Prove that deg(fp) is 0 if pEE and ±1 for p E D, the sign depending solely on the orientation of M. Number 5. Let N C IRm +1 be a closed smooth manifold of dimension n, N =I IRm+1. Show that: (1) If n < m, the complement IRm +1 \ N is connected. (2) If n = m + 1, N has a boundary M = aN =I 0, and if M is connected, so is IR m +1 \ N. What if M is not connected? Number 6. Let M c IR m + 1 be a closed, boundaryless, differentiable hypersurface. Suppose that M has r connected components. How many does its complement IR m +1 \ M have? Number 7. Let M c IR m +1 be a connected, boundaryless, smooth hypersurface. Let U c IR m +1 be a tubular neighborhood associated to vM, and let p : U -t M be the corresponding retraction. Let v : M -t IR m +1 be a unitary global normal vector field. Prove that there is a smooth function 8 : U -t IR such that

8(x)v(p(x))

=x-

p(x).

Check that 8 is an equation of Min U and that grad(8) = v.

7. The Brouwer Theorems We will prove here two famous theorems due to Brouwer. We start with the following immediate consequence of the Boundary Theorem: Proposition 7.1. Let M be an oriented differentiable manifold with boundary aM = N. Then N is not a proper retraction of M; that is, there is no proper mapping p : M -+ N whose restriction to N is the identity. Proof. Suppose there is such a retraction p : M -+ N. Let N' be a connected component of N, and let M' be the connected component of M that contains N'. Since N' = p(N') c p(M') c Nand p(M') is connected, we conclude N' = p(M') = M' nN = aM',

134

III. The Brouwer-Kronecker degree

because M' is open in M. Furthermore, plM' is proper, because M' is closed in M. Now apply the Boundary Theorem, III. 1.8, p. 101, with X = M', Y = 8M', and H = pIM', We get 0 = deg(Hly) = deg(pIN') = deg(Id N,), which is impossible. 0 The condition that the retraction is proper is essential: for instance, look at the (non-proper) continuous retraction §l x [0, +00) -+ §l : (x, t) f----t x. From the preceding result we deduce (see 1.2, p. 20 and p. 28): Proposition 7.2 (Brouwer Fixed Point Theorem). Let M c homeomorphic to a closed ball. Then every continuous mapping M has some fixed point.

jRm+l

be

f : M -+

Proof. Clearly, it is enough to prove the statement for M = {x E jRm+l : Ilxll ::::; I}, which is a compact manifold with boundary 8M = §m. Suppose there is a continuous mapping f : M -+ M without fixed points. Then define a mapping p : M -+ §m as shown in the picture p(x) = f(x)

+ A(X)(X -

f(x))

with '\(x) > 0 such that IIp(x)11 = 1. This mapping is proper and fixes every point in §m. Thus §m is a proper retraction of M, contrary to the preceding proposition. This contradiction means that f must have some 0 fixed point. This theorem will be refined in the next chapter (IV.2.9, p. 152) using the Euclidean degree. For the moment, here there is a nice application of it: Proposition 7.3 (Perron-Fr6benius Theorem). If a regular real matrix has non-negative entries, then it has some positive eigenvalue, corresponding to an eigenvector whose coordinates are all ~ O.

7. The Brouwer Theorems

135

Proof. Suppose A is an (m + 1) x (m + 1) matrix. Since A is regular, we can define the following continuous mapping:

h§m :

§m ---t

:

x

A·x

HilA. xII .

Now the hypothesis on the entries of A guarantees that h keeps invariant the quadrant Q = {x E §m : Xl ~ 0, ... , Xm+l ~ O}. But this quadrant is homeomorphic to a closed ball, and by the Brouwer Fixed Point Theorem, it must contain some fixed point x of h. Consequently A . x = AX, where A = IIA . xii> O. Thus, A is the eigenvalue and x is the eigenvector we sought.

o

We finish the section with the important fact, also proved by Brouwer, that only on spheres of odd dimension can we find tangent vector fields without zeros (1.2, p. 20 and p. 28): Proposition 7.4 (Hedge-hog Theorem). A sphere §m C ]Rm+1 has a continuous tangent vector field without zeros if and only if m is odd.

Tangent vector field on

§2

vanishing at both poles

~=(-y,x,O).

Proof. If m is odd, then m vector field ex

=

+1 =

2k, and we have the smooth tangent

(-X2' Xl,··" -X2k, X2k-l) E Tx§m.

Next, suppose that x H ex E Tx§m is a continuous tangent vector field on §m without zeros. After dividing by its norm, we have Ilexll = 1 for all x E §m; in other words, we have a continuous mapping §m ---t §m such that (x, ex) = 0 for all x E §m. By this condition,

136

III. The Brouwer-Kronecker degree

is a well-defined homotopy, and deg(Ho) = deg(Hl)' Clearly, Ho is the identity and HI the antipodal diffeomorphism; hence we conclude that the latter has the same degree as the former, that is, +1. But in III.1.6(1), p. 99, we saw this is possible only for m odd. This finishes the proof. 0

Exercises and problems Number 1. Let M be an oriented differentiable manifold with boundary 8M = N. Prove that there is no proper mapping p : M -+ N whose restriction to N is homotopic to the identity. Number 2. Let D denote an open ball in IRP, D its closure, and S its boundary. Produce the following examples: (1) a continuous mapping D -+ D with no fixed point, (2) a continuous mapping D -+ D with no fixed point in D, (3) a continuous mapping D x S -+ D x S with no fixed point. Here, where does the proof of III. 7.2, p. 134, fail? Number 3. Let K C IR m +1 be a convex compact set. Prove that every continuous mapping f : K -+ K has some fixed point. Can convex be replaced by connectecl! Number 4. Let D C IRm +1 be the unit open ball, and let f : D -+ IR m +1 be a continuous mapping such that (x, f(x)) < 1 for all x E §m. Show that f has some fixed point a E D. Number 5. Let D c IRm +1 be the unit open ball, and let f : D -+ IRm +1 be a continuous mapping such that (x, f (x)) < 0 for all x E §m. Show that f has some zero a ED.

Chapter IV

Degree theory in Euclidean spaces In this chapter we develop Euclidean degree theory. This could be done in full generality for proper mappings on closures of arbitrary open sets. However, here we have chosen the simplest way, enough for the applications, and restricted ourselves to bounded open sets. The reader will find in §§1-3 that everything can be done with a little calculus (and some very ingenious arguments); on the other hand, it is quite straightforward to extend the construction to the general unbounded case, a profitable exercise we recommend to the reader. Along the way we include Gauss's proof of the Fundamental Theorem of Algebm and a refinement of the Brouwer Fixed Point Theorem. Next, in §4, we come back to the origins and define the winding number, which is the agent behind the scenes for the equivalence of the Brouwer-Kronecker and Euclidean degrees. In §5, we deduce the famous Borsuk- Ulam Theorem and an important consequence: the Invariance of Domain Theorem. After this, §6 contains one fundamental computation: the Multiplication Formula for the Euclidean degree of a composite mapping. Finally, that formula is used in §7 to deduce a most general purely topological version of the Jordan Sepamtion Theorem and from it the Invariance of Domain Theorem once again.

1. The degree of a smooth mapping From now on, D stands for a bounded open set in lRm+1 , and X = D \ D stands for its (topological) boundary. Our purpose here is to define the degree of a continuous mapping D --+ lRm+1. This we will do by the usual method: first we consider smooth mappings, and then we approximate by them any given continuous mapping. Here, the reader will recognize some familiar arguments, which are repeated to make the construction fully selfcontained. Note that, D being compact, a continuous mapping f : D --+ lRm +1 is proper (hence closed), and in particular f(D) and f(X) are closed in lRm+1. Furthermore, the norm

Ilfll

=

max{IIf(x) II : x E D}

-

137

IV. Degree theory in Euclidean spaces

138

is well defined. As usual, this norm measures the approximation to functions on D. As mentioned above, we will consider the case when f is smooth. In particular we will be dealing with the set R flD C ]Rm+l of regular values of fiD. Recall that a E RflD when the derivative dxf is bijective at every point xED with f(x) = a (trivially true if a tJ. f(D)). The Sard-Brown Theorem, 11.3.4, p. 63, says that RflD is a residual set, which alows us to find regular values everywhere. On the other hand, although the set C flD of all critical points of flD (the points at which the Jacobian determinant vanishes) is closed in D, the set RflD = ]Rm+l \ f(CfI D) need not be open in ]Rm+l. However we can say something useful: C flD U X is closed in D; hence f(Cfl D U X) is closed in ]Rm+l (f is proper), and

RflD \ f(X)

= ]Rm+l \ f(Cfl D U X)

These remarks will often be implicit in our arguments. Now, down to the matter of discussion, we introduce the degree of a smooth mapping:

Proposition and Definition 1.1. Let f : D -+ ]Rm+l be a smooth mapping, and let a E ]Rm+l \ f(X) be a regular value of fiD. Then f-l(a) is a finite set (possibly empty), and we define the degree of f by

d(f, D, a) =

L

signx(f)

xEf-l(a)

(of course, the sum is 0 in case f-l(a) signx(f) = signdet

= 0), where

(;:i,

(x))

= ±1

J

is the sign of f at

X.

Proof. Note that f-l(a) C D, and the derivative dxf is a linear isomorphism at every xED with f(x) = a. Thus, by the Inverse Mapping Theorem, all points of f-l(a) are isolated, and f-l(a) is discrete. But D is compact, and f-l(a) does not meet X = D \ D; hence f-l(a) is finite. 0 We stress that signA!) tells whether the derivative dxf : ]Rm+l -+ ]Rm+l preserves orientation (sign = +1) or reverses it (sign = -1) as seen when we discussed orientations in the general setting of manifolds (11.7.2). Note also that from this definition it is clear that:

1. The degree of a smooth mapping

139

(1) If flD is a diffeomorphism, then d(f, D, a) (2) If r : jRm+1.-t

jRm+1

= ±1.

is a translation, then d(f, D, a) = d(rof, D, r(a)).

Now, we prove a first property of this degree.

Proposition 1.2 (Degree is locally constant). There exists an open neighborhood of a, W C RflD \ f(X), such that d(f, D, a) = d(f, D, b) for all bEW.

Proof. As remarked above, D is closed in jRm+1, and jRm+1 \ f(D) is open in jRm+1 and is contained in RflD \ f(X). Now, if f-l(a) = 0, then a E jRm+1 \ f(D), and there is an open neighborhood W C RflD \ f(X) of a that does not meet f(D). Consequently, d(f, D, b) = 0 for all b E W.

After this, we can suppose f- l (a) = {Xl, ... , Xr }. Then, by the Inverse Mapping Theorem, we find a connected open neigborhood V C jRm+1 \f(X) of a and r disjoint open neighborhoods UI, .. . ,Ur of Xl .... , Xr, respectively, such that by restriction we have r diffeomorphisms

whose sign is constant (because Uk is connected) and is denoted by Uk. We claim that W = V\f(D\UUk) k

is the open neighborhood we seek. Indeed, we have: 1. W is open, because D \ closed.

Uk Uk

is a compact set; hence its image is

2. a E W, because all its preimages Xk are in

Uk Uk'

3. Every b E W has exactly r preimages Yk E Uk. Indeed, b has at least these r preimages, and if it had others, say Y ¢ Uk Uk, then Y E D \ Uk Uk, so that f(y) E f(D \ Uk Uk) and b = f(y) ¢ W. Thus, for every b E W we deduce as claimed

d(f,D,b) = Lsignyk(f) = LUk = L signxk (f) = d(f,D,a). k k k

0

Next, we turn to homotopy. We start with a somewhat technical statement:

IV. Degree theory in Euclidean spaces

140

Proposition 1.3 (Homotopy invariance, 1st). Let H : [0,1] x D ---+ IRm +1 be a smooth mapping, and consider a point a E IRm +1 \ H([O, 1] x X) that is a regular value of the three restrictions HolD, HIID, and HI[o,ljxD' Then

d(Ho, D, a) = d(HI, D, a). Proof. Denote as usual Ht (x) = H (t, x) and let H(t, x) = (CPl(t, x), ... , CPm+l(t, x)). Since a is a regular value of H!rO,ljxD and C this inverse image

= H-l(a)

does not meet X,

is, by 11.3.2, p. 62, a smooth compact curve possibly with boundary

oC = (C n ({O} x D)) U (C n ({1} x D)) == Hi) 1 (a) U HII (a).

rr

We denote by H, ... , the connected components of C. By the classification theorem for smooth curves, each = is either a compact interval or a circle. In any case, 11.3.3, p. 63, gives a smooth parametrization of 'Y: s

r-t

r rk

r:

(t(S),Xl(S), ... ,Xm+1(s)) = (t(s),x(s)),

0 ~ s ~ 1.

In particular, :: never vanishes. Moreover, from the identity CPi(r(S)) it follows that

(1)

== ai

dt ~ 0CPi dx j O -= -d (CPi°'Y ) (s ) -_ -0CPi -+ L..J . ds

at ds

. OXj ds

J

Now consider the determinant

.1(s) =

dt ds

~ ds

CIS

0e.!.

0e.!.

~ Xm+l

{J
----ax;-

{J
{J
at

{JXl

dXm+l

We claim that .1( s) never vanishes. Indeed, the last m + 1 rows are independent because a is a regular value of H, and by (1) the first row is

1. The degree of a smooth mapping

141

perpendicular to the others, hence independent. Thus we conclude that .1(s) has no zero; hence it has constant sign: (2)

.1(s) has the same sign at s = 0 and s = 1.

Next, we look again at the equations in (1), which tell how to combine the columns in our determinant to get

(3)

dt ..1 = ds

112112

~

dxm±l

ds

ds

0

0fl

~

0

8Xl

8
8 Xm±1

=

11~;112 det (:~;b(S))).

8
After this preparation, we compute degrees:

(4)

{

d(HO, D, a) = ~k ~(O,Y)Erk signy(Ho), d(Hl' D, a) = ~k ~(1,z)Erk signAHl).

For that computation, we are only interested in the components r = rk that meet the levels t = 0, 1. Those components are the bordered components among the rk'S, that is, the components diffeomorphic to a compact interval; hence they have two distinct boundary points, ,,(0) = p and ,,(I) = q, both at those t-levels. The following picture shows the various possibilities:

Clearly, the required equality of degrees d(Ho, D, a) come from the following facts:

= d(Hl, D, a)

will

IV. Degree theory in Euclidean spaces

142

1. If p and q are at the same t-Ievel, they contribute to the same degree in (4) with opposite signs; hence they cancel each other. 2. If p and q are at different t-Ievels, they contribute to different degrees in (4) with the same signs in both. This is easily proved. Consider the first component t : [0,1] -+ [0,1] of the parametrization 'Y = (t,x). Since (~("((s))) is the Jacobian matrix of J Ht(s) and a is a regular value of Ho and HI, the right-hand side in (3) is not zero for t(s) = 0 and 1; hence ~!(O) i= 0 and ~!(1) i= o. We have: l(i) If p and q are at the same level t = 0, that is, t(O) = t(l) = 0, then t is increasing at 0 and decreasing at 1, so ~!(O) > 0 and ~!(1) < o. From (2) and (3) we deduce that the signs of Ho at x(O) and x(l) are opposite. l(ii) If p and q are at the same level t = 1, that is, t(O) = t(l) = 1, then t is decreasing at 0 and increasing at 1, so ~! (0) < 0 and ~! (1) > O. By (2) and (3) again, the signs of HI at x(O) and x(l) are opposite. 2(i) If p and q are at levels t = 0 and t = 1, that is, t(O) = 0 and t(l) = 1, then t is increasing at 0 and 1, so ~! (0) > 0 and ~! (1) > O. By (2) and (3) as before, we see that the signs of Ho at x(O) and HI at x(l) are the same. 2(ii) If p and q are at levels t = 1 and t = 0, that is, t(O) = 1 and t(l) = 0, then t is decreasing at 0 and 1, so ~! (0) < 0 and ~! (1) < o. Thus, from (2) and (3) we deduce that the signs of Ho at x(l) and HI at x(O) are the same. This completes the proof.

D

The previous statement can be simplified a bit: Proposition 1.4 (Homotopy invariance, 2nd). Let H : [0,1] x D -+ ~m+1 be a smooth mapping, and consider a point a E ~m+1 \ H([O, 1] x X) that is a regular value of the two restrictions HolD and HIID. Then

d(Ho, D, a) = d(H I , D, a). Proof. This follows from the preceding results. By IV.1.2, p. 139, there is an open neighborhood W of a that does not meet the compact set H([O, 1] xX) and consists solely of regular values of HolD and HIID and is such that

d(Ho, D, a) = d(Ho, D, b) and d(H I , D, a) = d(H I , D, b)

1. The degree of a smooth mapping

143

for all b E W. Now, by the Sard-Brown Theorem, W contains some regular value b of the mapping HI[o,ljXD, and applying IV.1.3, p. 140, we get d(Ho, D, b) = d(Hl' D, b), which jointly with the preceding equalities proves the statement. 0 Once this is established, we can improve on IV.1.2, p. 139, and define degree for non-regular values: Proposition and Definition 1.5. Let f : D -+ jRm+1 be a smooth mapping, and let a E jRm+l \ f(X). Let fl be the connected component of jRm+1 \ f(X) that contains a. Then d(J, D, b) is constant for all b E fl n R(JID), and we define the degree of f by

d(J, D, a) = d(J, D, b) for any such b {which exists by the Sard-Brown Theorem}. Proof. Let b, b' E fl n R(JID)' Since fl is an open connected set in a Euclidean space, there is a smooth arc "( : [0,1] -+ fl with "((0) = b, "((1) = b', and we define H: [0,1] x D -+

jRm+I :

(t, x)

M

f(x) - "((t).

We have: (1) 0 rt. H([O, l] x X): if f(x) = "((t), for some x E X, then f(x) E f(X) n fl, which is a contradiction. (2) 0 is a regular value of HolD: since Ho = f - b, the derivatives dxHo and dxf coincide, and we know that b is a regular value of f. (3) 0 is a regular value of HIID: same argument for HI = f - b'. Thus, by IV.1.4, p. 142,

d(Ho, D, 0)

= d(Hl' D, 0).

Finally, since both Ho and HI are translations of f, we obtain

d(Ho, D, 0)

= d(J, D, b) and d(HI' D, 0) = d(J, D, b').

All together, we get what we want. For this extended definition, IV.1.4, p. 142, remains true:

o

IV. Degree theory in Euclidean spaces

144

Proposition 1.6 (Homotopy invariance, 3rd). Let H : [0,1] x D -+ jRm+l be a smooth mapping, and consider a point a E jRm+ 1 \ H ([0, 1] x X). Then

d(Ho, D, a) = d(HI, D, a). Proof. Let Q be the connected component of jRm+l \ H([O, 1] x X) that contains a. By the Sard-Brown Theorem, Q contains some regular value b of both HolD and HIID, and then d(Ho, D, a) = d(Ho, D, b) and d(Hl' D, a) = d(Hl' D, b) (IV.1.5, p. 143). But by IV.1.4, p. 142, d(Ho, D, b) are done.

= d(HI, D, b), and we D

Exercises and problems Number 1. Consider the quadratic transform

and the open set D a r-+ d(f, D, a).

c

IR n defined by 01 < Xl

< (31, ... ,On < Xn < (3n.

Describe the function

Number 2. ([Picard 1891]; 1.1, p. 12) Let D c IR m +1 be a bounded open set, let X = D \ D, and let/1, ... ,fm+! : D --+ IR be smooth functions. Suppose that 0 ¢ f(X) is a regular value of flD, where f = (/1, ... ,!m+1) : I5 --+ IRm+l. Consider the bounded open set E = D x (-1,1) C IR m+ 1 x IR = IR m+2 and the smooth function

-

fm+2 : E --+ 1R: (x, t) r-+ t . det

(Bh Bx (x) ). J

Finally, consider

Prove that d(g, E, 0) is well defined and that it coincides with the number of solutions in D of the system /1 (x) = ... = fm+l(X) = o.

Number 3. ([Heinz 1959]; lA, p. 38) Let D C IR m +1 be a bounded open set, let X = D \ D, and let f : D --+ IR m+! be a smooth mapping. Let a E IR m+ 1 \ f(X) be a regular value of f. Let 'Pe : IR m+ 1 --+ IR be a smooth mapping whose closed support Ke is contained in the open ball of radius c > 0 centered at the origin and such that IRm+l 'Pe = 1 (called a convolution kernen. Prove the Heinz Integral Formula:

d(f, D, a) for c small enough.

=

JDr 'Pe(f(X) -

a) det

(~(x»dx 3

145

2. The degree of a continuous mapping

Number 4. Explain and prove the following product formula for smooth mappings: d(f x g, D x E, (a, b))

= d(f, D, a) . d(g, E, b).

Number 5. Let D c ll~? denote the open disc of radius 2, and consider the smooth mapping j = (II, h) : D -+]R2 given by 4 + y)e XY + x 2y { II (x, y) = (_x 3 12 (x, y) = x y - 4xy.

Study the variation of d(f, D, (a, 0)) for a E

]R,

x 2 + y + 1,

when defined.

Number 6. Let X C ]R2 be the curve union of the following pieces: (i) the graph of the topological sinus y = sin(l/x), 0 < x :::; y'3, (ii) the segment x = 0, -1 :::; Y :::; 1, (iii) the arc of the circle x 2 + (y + 1)2 = 4 off the first quadrant, and (iv) the segment x = y'3, 0:::; y :::; sin(l/y'3). Show that X is not homeomorphic to §1, but nonetheless it bounds an open set. Then prove that the mappings j, 9 : X -+ ]R2 \ {(O, O)} given by

j(x, y) = (x 3 + y3 + 1, x - y)

and

g(x, y) = (x 3 + y3 + 1, x + y)

are not homotopic. Number 7. Let D C IR m +1 be a bounded open set and let X be its boundary. Suppose we are given two smooth functions j, 9 : D -+ IR m +1, a homotopy H : [0,1] x D -+ IRm +1 with Ho = j, HI = g, and a point a E ]Rm+1 \ H([O, 1] x X) that is a regular value of j and g. Prove that the fibers j-I(a) and g-l(a) have the same parity. Is it always true that they have the same number of points?

2. The degree of a continuous mapping After the discussion of smooth mappings in the preceding section, we are ready to define degree for arbitrary continuous mappings: Proposition and Definition 2.1. Let f : D --+ jRm+1 be a continuous mapping and let a E jRm+l \ f(X). Then there exists a smooth mapping g: D --+ jRm+l such that IIf -gil < dist(a, f(X)). For all such 9 's the degree d(g, D, a) is defined (a E jRm+1 \ g(X)) and is the same, and we define the degree of f by d(f, D, a) = d(g, D, a). Furthermore, 9 can be chosen such that a is a regular value of giD' and then d(f, D, a) = signx(g)·

L

xEg-I(a)

IV. Degree theory in Euclidean spaces

146

Proof. First of all, by the Weierstrass Approximation Theorem, we find a polynomial (hence smooth) mapping P such that

liP -

fll = max {IIP(x) - f(x)11 : x E D} <

~ dist(a, f(X))

(recall that a ¢. f(X) and f(X) is closed). Next, by the Sard-Brown Theorem, there is some regular value b of PID such that Iia - bll < One easily checks that 9 = P

+a -

! dist(a, f(X)). b is the function we sought. To see that

d(g,D,a) = d(g',D,a) for two mappings g,g' that are close to f, consider the smooth homotopy

H: [a, 1] x D --+

]Rm+1 :

(x, t)

H

(1 - t)g(x)

+ tg'(x).

Clearly, Ho = 9 and HI = g'; hence to apply IV.1.6, p. 144, we only must check that a ¢. H([a, l] x X). But suppose there is an x E X such that a = Ht(x). Then Iia - f(x)11

= IIHt(x) - ((1- t)f(x) + tf(x)) II :S (1- t)llg(x) - f(x)11 + tllg'(x) - f(x)11 < (1 - t) dist(a, f(X)) + tdist(a, f(X)) = dist(a, f(X)), D

a contradiction.

Remark 2.2. As is to be expected, the approximations 9 used above to define the degree of f are homotopic to f off a on X. Indeed, the homotopy

Ht(x) = tf(x)

+ (1 - t)g(x)

does not touch a. For, if there is an x E X with Ht(x) the segment between f(x) and g(x); hence

= a, then a lies in

Ilf(x) - g(x)11 2 Ilf(x) - all 2 dist(a, f(X)), contrary to the choice of g.

D

From the definition, we get the following: Proposition 2.3. Let f : D --+ ]Rm+1 be a continuous mapping. Then the degree d(f, D, .) is constant on every connected component of ]Rm+1 \ f (X).

2. The degree of a continuous mapping

147

Proof. Let a E lR,m+l \f(X), and, according to IV.2.1, p. 145, pick a smooth mapping 9 : D -+ lR,m+1 such that Ilf -gil < dist(a, f(X)). We can compute the degree d(f, D, a) through g, but in fact, we can also compute d(f, D, b) for any other bE lR,m+1 \ f(X) close enough to a: namely whenever Iia - bll < TJ = dist(a, f(X)) - Ilf - gil· Indeed, this implies dist(b, f(X)) ~ dist(a, f(X)) - Iia - bll > dist(a, f(X)) - TJ = Ilf - gil, and since TJ:::; dist(a,g(X)) (check it!), the two points a,b are in the same connected component of lR,m+l \ g(X). Then, by IV.1.5, p. 143, we see that

d(f, D, a) = d(g, D, a) = d(g, D, b) = d(f, D, b). This means that the function a r--+ d(f, D, a) is locally constant on lR,m+l \ f(X), and, since its values are integers, it must be constant on every con0 nected component of lR,m+1 \ f(X). Of course, we can improve on the homotopy invariance once the definition is most general: Proposition 2.4 (Homotopy invariance). Let H : [0,1] x D -+ lR,m+l be a

continuous mapping, and let 'Y : [0, 1] -+ lR,m+1 be a continuous path such that 'Y( t) ¢ Ht (X) for 0 :::; t :::; 1. Then d( Ht, D, 'Y( t)) does not depend on t. Proof. Fix 0:::; to :::; 1. The function (t,x) r--+ Ib(t) - Ht(x)11 is continuous and> 0 on the compact set [0, 1] x X and hence has a minimum c > O. On the other hand, H is uniformly continuous on the compact set [0,1] x D; hence there is an TJ > 0 such that IIHt(x) - Ht'(X') II <

!c

if

It - til,

IIx - xiII < TJ.

Then, by the Weierstrass Approximation Theorem, there is a polynomial mapping 9 such that Ilg(x) - Hto(x)11 < We deduce that

!c

for

x E D.

148

IV. Degree theory in Euclidean spaces

for x E D, It - tol

< TJ. Thus,

Ilg(x) - Ht(x) II < dist(r(t) , Ht(X))

for

x

E

D, It - tol < TJ.

In particular, for x EX, t = to, we get

hence g(x)

=1=

l'(to). As g(X) is compact, we deduce that dist(r(to), g(X))

> o.

Thus, for a possibly smaller TJ, we can assume that It - tol < TJ implies

Ib(t) -1'(to)11 < dist(r(to), g(X)). Now we are ready to compute the degree. Suppose It - tol < TJ. First, in view of the construction, we can apply IV.2.1, p. 145, to get

d(Ht,D,I'(t)) = d(g,D,I'(t)). Then, by translation ... = d(g -1'(t), D, 0).

Next, ... = d(g - l'(to), D, 0),

because II(g -1'(t)) - (g -1'(to))11 = Ib(t) -1'(to)11 < dist(r(to),g(X)) = dist (0, (g -1'(to))(X)) and IV.2.1, p. 145, applies again. We continue the computation, and by translation

... = d(g, D, l'(to)). Finally, by a third use of IV.2.1, p. 145, we obtain

Hence, d(Ht, D, l'(t)) is locally constant and consequently constant in the connected interval [0,1]. 0 We have thus defined the degree and proved its homotopy invariance in full generality. As an easy consequence we deduce the other properties that characterize degree axiomatically:

149

2. The degree of a continuous mapping

Corollary 2.5 (Nagumo, Fiihrer, Deimling Axiomatizations). (1) Normality: If f is the inclusion IdD : D -+ D c IRm +1, then

d(f D a) = "

{I

if a E D, if a fJ. D.

0

(2) Existence of solutions: Given a continuous mapping f : D -+ IRm +1 and a point a fJ. f(X), if d(f, D, a) =1= 0, the equation f(x) = a has some

solution in D. (3) Additivity: Given two disjoint open sets Dl, D2 CD, a continuous mapping f: D -+ IRm +1 , and a point a fJ. f(D \ Dl U D2), then

d(f, D, a) = d(f, D 1 , a)

+ d(f, D2, a).

Proof. For (1) use IV.I.1, p. 138.

For (2) we argue by way of contradiction. Suppose f-l(a) dist(a, f(X))

~

dist(a, f(D))

= 0. Then

= E > 0,

and by the Weierstrass Approximation Theorem there is a polynomial mapping 9 = g(x) such that Ilg - fll < in particular, a fJ. g(X). Then, d(f, D, a) = d(g, D, a). For x E D we have

!E;

Iia -

g(x)11 ~ lIa - f(x) II -lIg(x) - f(x)11 > E-

!E = !E.

!E.

This means that dist(a,g(D))> Now, we pick, in the connected comm ponent of IR +1 \ g(X) that contains a, a regular value b of glD such that lIa - bll < Then g-l(b) = 0 and d(g, D, a) = d(g, D, b) = 0, a contradiction.

!E.

For (3) notice first that D \ Dl U D2 contains the three boundaries

D\D,

Dl \Dl'

Now, dist(a, f(D \ Dl U D2)) = mapping 9 = g(x) with Ilg - fll follows that

d(f, D, a) = d(g, D, a)

E

<

and

D2 \D2.

> 0, and we can choose a polynomial Then a fJ. g(D \ Dl U D2), and it

E.

{

d(f, Dl, a) : d(g, D 1 , a), d(f, D2, a) - d(g, D2, a).

Finally, let [} be the connected component of Jlrl+1 \ g(D \ Dl U D2) that contains a, and pick in [} a regular value b of giD. Then

d(g, D, a) = d(g, D, b)

and

{

d(9' Dl, a) : d(g, D 1, b), d(g, D2, a) - d(g, D2, b).

IV. Degree theory in Euclidean spaces

150

All in all, assertion (3) follows from IV.2.1, p. 145, because

o We complete the construction of the Euclidean degree with another very important fact: Proposition 2.6 (Boundary Theorem). Given two continuous mappings f,g : D -+ ~m+1 such that fix = glx and given a point art. f(X) = g(X),

then d(f,D,a) = d(g,D,a). Proof. Apply IV.2.4, p. 147, to the homotopy Ht(x) and the path 'Y == a. Note that H t = f = 9 on X.

= tf(x) + (1 - t)g(x) 0

Example 2.7 (Gauss). We come back to the solution of complex algebraic equations, already discussed earlier (III. 1. 10, p. 102). There we used the Brouwer-Kronecker degree for proper mappings, but we can instead apply the more elementary Euclidean degree, along Gauss's lines (see 1.1, p. 2). Let P(z) = zP + CIZp-1 + ... + cp be a polynomial with complex coefficients. If ( E C is a root of P with 1(1 ~ 1, then

hence

+ ... + Icpl < Ic 1+··· + Ic I. 1.,1"1 -< ICIII(IP-l1(lp-1 - I p Thus we set p = 1 + ICII + ... + ICpI, and consider in ~2 == C the open disc D = {Izl < p}, with boundary X = {Izl = pl· Then f = PI D is a continuous mapping with 0 rt. f(X), and we are to compute the degree d(P, D, 0). To that end consider the homotopy Pt(z) = zP + tCIZp-1

+ ... + tcp, 0::; t ::; 1.

Using the same bound on the coefficients, we see that Pt has no root in X; hence (IV.2.4, p. 147)

d(f, D, 0) = d(PI , D, 0) = d(Po, D, 0). Now the mapping Po : z f---+ zP has degree p, for, any small enough a i= 0 is a regular value of Po with p different roots c in D. Moreover, at each c, Po preserves orientation: dcPo is multiplication times the complex number pcp-I.

2. The degree of a continuous mapping

151

We have thus seen that the degree of a complex polynomial is exactly its ordinary degree as a polynomial. Quite predictable, but very important, is the fact that degree is not zero; hence f = PID is surjective onto the connected component [l of C \ f(X) that contains 0 (IV.2.5(2), p. 149). In particular, P has some root in D: this is again the Fundamental Theorem of Algebra. Furthermore, we can even see that degree counts the number of roots of P. Indeed, the derivative of f = PID at any given point z is the linear mapping defined by complex multiplication by P'(z); such a multiplication preserves orientations. Thus, at every regular value a E [l of f we get p

= d(j, D, a) =

L

signA!)

= #f-1(a);

zEf-1(a)

that is, the equation P(z) = a has p roots. If a is not a regular value, the roots must be counted with multiplicities; details are left to the reader. 0

We close this section with a variation on the Brouwer Fixed Point Theorem as was stated in 111.7.2, p. 134. First we formulate a somewhat technical version: Proposition 2.8. Consider the unit open ball D m +1= {x E lRm +!: and the unit sphere §m c lRm + 1 .

Ilxll < I}

(1) Let f : yyn+l -+ lRm +! be a continuous mapping such that f(x) =F 0 for all x E yyn+ 1 . Then there is a z E

§m

such that f (z)

=

AZ for some

A> O. (2) Let g : yyn+! -+ lRm + 1 be a continuous mapping such that g(x) =F x for all x E DIn+!. Then there is a z E §m such that g( z) = AZ for some A> 1. Proof. (1) Suppose by way of contradiction that f(x) =FAX for all x E §n and A > O. Then we can apply IV.2.4, p. 147, to the homotopy Ht(x)

= (1 -

t)f(x) - tx,

so that d(j, D m +!, 0)

= d(Ho, D m +!, 0) = d(Hl' D m +1, 0) = d(-Id1r+l,D m +!, 0) = (_I)m+l =F O.

IV. Degree theory in Euclidean spaces

152

Hence, by IV.2.5(2), p. 149, there must be some x E Dm+l with f(x) a contradiction. (2) Apply (1) to the mapping f(x)

= g(x) -

= 0,

o

x.

Now the mentioned variation on 111.7.2, p. 134, is immediate: Corollary 2.9. Let D c ffim+l be an open ball with boundary S. Then every continuous mapping f : D -+ ]Rm+l such that f(S) C D has some fixed point. Proof. Let Dm+l be the unit open ball, and consider the mapping

g:

=D+1

TlDm+l -+ll'Io.:

xt-+

f(rx

+ c) r

c

,

where c is the center of D and r is its radius. By the hypothesis on f, g(§m) C D, so that for all x E §m we have IIg(x)II ~ 1; hence g(x) =1= AX if A> 1. Consequently, by IV.2.8(2) above, 9 must have some fixed point x, and rx + c is a fixed point of f. 0

Remark 2.10. We have thus weakened the condition f(D) C D to f(S) D, and this is indeed weaker. For instance, look at g(x)

=

{3X (5 -

This mapping fixes

§m

411xll)x

for for

but sends all points in

Ilxll ~ Ilxll ~

c

!, !.

i < Ilxll < 1 off Ilxll ~ 1 .0

Exercises and problems Number 1. Extend Euclidean degree to unbounded domains starting with the following. Let a c Rm + l be open, possibly not bounded. Then consider continuous mappings f : ?i -+ Rm+l verifying sup IIx - f(x)11 < +00. ",En

Prove that for every a 1. f(8?i), f-l(a) is bounded and for all bounded open sets Dca containing f- l (a), the degrees d(J, D, a) are the same. This common value is the degree d(J, a,a). Number 2. ([Poincare 1883]; 1.1, p. 11) Let al, ... ,an be positive real numbers, and for each i = 1, ... , n, let Ii : [-al, all x ... x [-an, an] -+ R be a continuous function positive on Xi = ai and negative on Xi = -ai. Prove that the system

/l(X) = ... = fn(x) = 0

153

3. The degree of a differentiable mapping

has some solution in (-al,al) x ... x (-an, an). Number 3. Let DC ]Rm+1 be a bounded open set, let X = D\D, and let J : D --t ]Rm+1 be a continuous function such that J(D) is contained in a hyperplane. Then d(f, D, a) = for all a E ]Rm+l - J(X).

°

Number 4. Let a < b be two real numbers and let D be a bounded open set of [a, b] x ]Rm+1 that meets all sets {t} x ]Rm+1. Let H : D --t ]Rm+1 be a continuous function and let c E ]Rm+1 \ H(8D). For a ~ t ~ b and X C [a, b] x ]Rm+1 we denote X t = {x E ]Rm+l: (t,x) EX}.

Show that D t C ]Rm+l is a bounded open set, that 8D t C (8Dh (equality not true in general), and that the degree d(Ht, D t , c) is well defined and does not depend on t. Number 5. Let D be a bounded open subset of [0, 1] x ]Rm+l meeting all sets {t} x ]Rm+1 , and let H : D --t ]Rm+1 be a continuous function. Let a E ]Rm+1 off H t (8D t ) for all t E [0,1]. Is it true that d(Ht, D t , a) depends on t? Number 6. Write]Rm =]RP x ]Rq and consider the closed balls D: = {x E ]RP :

Jlxll

~ c}

and

D~ = {y E ]Rq :

IlyJl

~ a}.

Let there be a continuous mapping g: D: x D~ --t]Rm: (x,y)

f-t

(x,gy(x))

with g(O, y) = y. Show that Ugraph(gy) = {(x,gy(x)) : x E D:,y E Dn y

is a neighborhood of (0,0) in ]Rm. Number 7. Let D denote the unit open ball in ]Rm+1, and let J : D --t ]Rm+l be a continuous function without fixed points. Prove that the angle a(z) determined by the two vectors z, z - J(z) reaches all values from to 11" for z E §m.

°

3. The degree of a differentiable mapping Only after all properties of the Euclidean degree are available, can we deduce that for an arbitrary differentiable mapping degree can be computed by the same regular value formula that gives the degree of a smooth mapping. In any case, it is interesting to remark how simple this is if we compare it to the Brouwer-Kronecker theory (III.4.1, p. 114). We need the following lemma, interesting in its own right: Lemma 3.1. Let D c ]Rm+1 be the open ball Ilx - ell < c, and let f : {l --t ]Rm+1 be a C1 mapping defined on a convex open neighborhood {l of D. Then the mapping f(tx

+ (1 -

Ht(x) = {

t)c) - f(c)

t dcf(x - c)

for t

=1=

for t

=0

0,

IV. Degree theory in Euclidean spaces

154

n

is a continuous homotopy on

Proof. Since f(x) - f(c)

of f - f(c) and de! - de!(c).

n is convex, we can write =

11 1

!f(tx + (1- t)c)dt

1 m+1

=

of

m+l

+ (1 -

L - . (tx o i=1 dx~

t)C)(Xi - ci)dt

=

L (Xi - Ci)!i(X), i=1

where the !i's are the continuous functions fi(X)

We see that fi(C)

=

1 1

o

of

-d' (tx x~

+ (1 -

t)c)dt.

= ~(c), and after a little computation we get m+l

Ht(x)

=

L (Xi - Ci)fi(tx + (1 - t)c), i=1 D

which shows that H is continuous. Now we can prove the following:

Proposition 3.2. Let f : D -+ IRm +1 be a continuous mapping whose restriction flD is C1, and let a E IRm +1 \ f(X) be a regular value of fiD. Then f-l(a) is finite, and d(J,D,a)

=

signdet (;:i. (x)).

L XE!-l (a)

J

Proof. That f-l(a) is finite follows as usual from the Inverse Function Theorem, say f- 1 (a) = {Cl, ... , Cr }. We pick disjoint open balls Di centered at the points Ci with Di c D, such that each restriction flDi is bijective. Since a tj. f(D \ Ui Di), from additivity (IV.2.5(3), p. 149), we deduce that d(J,D,a)

= Ld(J,Di,a). i

Then, by the previous lemma and the invariance of homotopy, we have d(J, D i , a)

= d(J - J(Ci), Di, 0) = d(dcJ - dcJ(Ci), Di, 0) = d(deJ, Di, dcJ(Ci)).

As (~( cd) is the matrix of the linear isomorphism deJ, we get what we J wanted. D

3. The degree of a differentiable mapping

155

Exercises and problems Number 1. Let hI, ... ,hr : JR. -+ JR. be differentiable functions, and define

f : JR.2 -+

JR.2 : (x, Y) H (xy, xy2

+ x 2 I>i(xy)2).

Compute all posible values of d(f, D, a). Number 2. Let f : D -+ JR. m+1 be a continuous mapping and consider a E JR. m+ l \f(X). Suppose that f is differentiable on an open set containing f- l (a). Prove that d(f,D,a)=

Signdet(;~i(x))

L xEJ-1(b)

J

for all regular values b close enough to a. Number 3. Let f : D -+ JR. m+ l be a continuous mapping whose restriction flD is differentiable, with Jacobian determinant 2 0 all through D. Let a E JR. m+ l have finite preimage f-l(a) c D. Is it true that d(f,D,a)

=

L

signdet

(;~i(X))

xEJ-1(a)

(where sign 0

J

= O)?

Number 4. Let f : D -+ JR. m+ l be a continuous mapping whose restriction flA to an open set A c D is differentiable, and let a E JR. m+1 \f(X) be a regular value of fiA. Find conditions on the fiber f- l (a) for the following formula to hold true: d(f,D,a)

=

L xEJ-1(a)nA

signdet

(;~i (x)). J

Number 5. Let f : D -+ JR.m+l be a continuous mapping whose restriction flD is differentiable. Suppose that the open set A consisting of all regular values of f in JR. m+1 \ f(X) is connected. Prove that

for any two values a, b E A. Number 6. Let Dk C JR.k denote the unit open ball as usual. Let f : I5 P -+ JR.P be a continuous mapping whose restriction flDp is differentiable with 0 ~ f(f5P\DP) a regular value. Consider g : D P + q -+ JR.p+q defined by

Prove that d(g, Dp+q, 0) is well defined and that it coincides with d(f, DP, 0). Number 7. ([Fiihrer 1971]; 1.5, p. 39) We already know that Euclidean degree theory verifies all properties of Fiihrer's characterization. Prove now that those properties give uniqueness of the theory, by the following steps: (1) By approximation and homotopy invariance, reduce uniqueness to the case of differentiable functions.

IV. Degree theory in Euclidean spaces

156

(2) By additivity, reduce to mappings defined on open balls and regular values with a single preimage. (3) By IV.3.1, p. 153, reduce to linear forms, and then use paths in the space of regular matrices.

4. Winding number Here we look at the relationship between the Brouwer-Kronecker degree and the Euclidean degree, where our X in this chapter is a hypersurface. The basic notion on which the whole comparison stands is that of winding number, and thus we come back to the origins of the theory (1.1, p. 5, and 1.2, p. 19). First of all, for D C ~m+1 bounded open and X have the following:

=D

\ D as usual, we

Proposition and Definition 4.1. Let f : X -+ ~m+l be a continuous mapping and consider a point a E ~m+1 \ f(X). Then, for all continuous extensions J: D -+ ~m+1 the degree d(J, D, a) is the same, and we define the winding number of f around a by w(j, a) = d(J, D, a) for any such

J (which exists by the

Tietze Extension Theorem).

This follows immediately from the Boundary Theorem, IV.2.6, p. 150. As is the degree (IV.2.3, p. 146), the winding number is also locally constant on the target:

Proposition 4.2. Let f : X -+ ~m+l be a continuous mapping and consider two points a, b in the same connected component of ~m+1 \ f(X). Then w(j, a) = w(j, b). Of course, the winding number is invariant by homotopy:

Proposition 4.3. Let H : [0,1] x X -+ ~m+1 be a continuous mapping, and let "( : [0,1] -+ ~m+l be a continuous path such that "((t) Ht(X) for 0:::; t:::; 1. Then w(Ht, "((t)) does not depend on t.

tt

Proof. By the Tietze Extension Theorem, H extends to a homotopy H : [0,1] x D -+ ~m+l, and IV.1.3, p. 140, applies. D

Another easy fact is the following consequence of IV.2.5(2), p. 149:

157

4. Winding number

Proposition 4.4 (Boundary Theorem). Let f : X -t ]Rm+1 be a continuous mapping and consider a point a E ]Rm+l \ f(X). If f has a continuous extension f: D -t ]Rm+1 such that a ¢:. f(D), then w(j,a) = O.

Next we want to represent the Brouwer-Kronecker degree as a winding number in this Euclidean setting. The key result needed to do this follows. Proposition 4.5. Let X be the boundary of a compact, oriented, differentiable manifold W of dimension m + 1. Let f : W -t ]Rm+1 be a differentiable mapping and let a E ]Rm+1 \ f(X) be a regular value of f, and consider the differentiable mapping g : X -t

§m :

x

f(x) - a

H

.

Ilf(x) - all Then f-l(a) is finite, and the Brouwer-Kronecker degree of g is

L

deg(g) =

sign x (/)'

xEf-l(a)

Proof. That f-l(a) is finite follows as usual, say f-l(a) = {Xl, ... .x r }. Then there are disjoint neighborhoods Uk of the points Xk such that the restrictions fluk : Uk -t B are diffeomorphisms onto an open ball B centered at a. Then, we choose a smaller closed ball DeB with boundary a sphere S and again restrict f to have diffeomorphisms flvk : Vk

= f-I(D) n Uk

-t D,

with

f(aVk)

= S.

Let V = W \ Uk Vk. This is a compact manifold with boundary Y = X U Uk aVk; here X and the aVk'S carry the orientation as boundaries of V. This means that each aVk carries the wrong orientation as boundary of Vk·

158

IV. Degree theory in Euclidean spaces

Since a 1. j(V), 9 is in fact defined on V, and by III. 1.8, p. 101, deg(gI8v) = 0; hence deg(glx) - Ldeg(gI8Vk )

= 0,

k

where the negative sign corresponds to the above remark on the orientations of the 8Vk'S. Moreover, if c is the radius of D, we have 1

-

gl8Vk = e(j1 8Vk - a), and it follows readily that deg(gI 8vk)

= deg(J18vk). Now:

(1) If signxk(J) = +1, the diffeomorphism JIVk : Vk -+ D preserves orientation. Hence the restriction JI 8Vk : 8Vk -+ S preserves orientation too and has Brouwer-Kronecker degree +1. (2) If signxk(J) = -1, the diffeomorphism JIVk : Vk -+ D reverses orientation. Hence the restriction JI 8Vk : 8Vk -+ S reverses orientation too and has Brouwer-Kronecker degree -1. Consequently, deg(glx)

= Ldeg(gI8vk) = Ldeg(j18vk) = Lsignxk(J), k

k

k

o

as desired.

Now we conclude the section with the announced representation of the Brouwer-Kronecker degree: Proposition 4.6. Let X C ]Rm+l be a connected, compact, boundaryless, differentiable manifold of dimension m, and let f : X -+ §m be a continuous mapping. Then X is the boundary of a bounded open set D c ]Rm+l, and

deg(j) = w(j, 0). Proof. The Jordan Separation Theorem for compact hypersurfaces (III.6.2, p. 124, and III.6.4, p. 129) says that X bounds a bounded open set D as claimed. On the other hand, since both deg and ware invariant by homotopy, we can suppose f is smooth, and then, by the smooth Tietze Extension Theorem, f has a smooth extension J : D -+ ]Rm+l. Finally, we pick a regular point a of J such that Iiall < 1. By IV.4.2, p.156, w(j,O) = w(j, a); hence we are reduced to showing that

deg(j) = w(j, a).

159

4. Winding number

But in this situation w(f, a) = L:xE/-1(a) signxU), and from IVA.5, p. 157, we get w(f, a) = deg(g), where 9 : X --+ §m : x

H

f(x) - a Ilf(x) - all·

~.:.......:------::-

Finally, the homotopy

f(x) - ta Ht(x) = Ilf(x) - tall is well defined, and by 111.304, p. 112, we conclude that deg(f)

= deg(Ho) = deg(H1 ) = deg(g) = w(f, a).

o

Exercises and problems Number 1. ([Poincare 1886] and [Bohl 1904]; 1.2, p. 19) Let X be a compact smooth hypersurface of jRm+1 and let

f

= (h, ... ,jm+1), 9 = (gl, ... ,gm+1) : X

-+ jRm+1

be two continuous mappings with 0 tJ. f(X),g(X). Prove: (1) If w(j, 0) '" w(g, 0), then there is at least one point x E X such that

hex) _ ... _ fm+1(x) < 0 gl(X) - gm+1(X) . (2) If w(j, 0) '" (_1)m+1w(g, 0), then there is at least one point Y E X such that

hey) = ... = fm+1(Y) > o. gl(Y) gm+1(Y) (3) If w(j,O) '" ±w(g,O), then the function hg1 constant sign on X.

+ ... + fm+1gm+1

cannot have

Number 2. ([Kronecker 1869a] and [Hadamard 1910]; I.1, p. 10, and 1.2, p. 19) Let f : X -+ jRm+1 be a smooth function on a compact smooth hypersurface X of jRm+1 , such that 0 tJ. f(X). Prove the following assertions: (1) w(j,O) = deg (m). (2) Denote by 0 the volume element of §m, and set rex) = x/llxll. Then

w(j,O) = (3) Let x

= (Xl, ... ,xm ) 1

vOl(~m)

Ix

(r

0

frO.

be local coordinates on X. Then {

w(j,O) = vol(§m) Jx IIf(x)1I

-m-1

(

8f(x)

8 f (x))

det f(x), 8X1 , ... , 8X m

dx.

160

IV. Degree theory in Euclidean spaces

Number 3. ([Cauchy 1855]; 1.1, p.6) Fix a E C == ]R2. Let f : §1 -t C \ {a} be a smooth mapping that has a holomorphic extension to some open neighborhood of §1 in C. Prove the Argument Principle: w(f, a)

1 = 27ry-1 r-1

1 §1

f'(z)

f() dz. z - a

Check in this case the geometrical idea that the winding number tells how many times f wraps §1 around the point a (1.1, p. 5): Given a half-line from the origin that meets f(§l) transversally at finitely many points, w(f,O) is the difference between the number of them at which the argument of f increases and the number of them at which the argument of f decreases.

Number 4. ([Gauss 1813]; 1.1, p. 11) Let S C ]R3 be a compact surface. Compute the winding number at a point a If. S of the inclusion S C ]R3, and deduce the Gauss Theorem: Let v(x) = 11:~~13 be the electric field generated by a unit charge placed at a point a. Then the flow of v through S is

1( ).-. s

V,Vs "s-

{47r 0

if a is inside S, if a is outside S

(vs and Os stand for a normal vector field and the corresponding volume element of S).

Number 5. [Gauss 1833] Let f,g : [0,1] -t ]R3 be two smooth disjoint knots (f(0) = f(l), g(O) = g(l), f(s) i- g(t) for all s, t). Consider the torus T C ]R3 parametrized by T: x = (2

+ cos(27rt)) cos(27rs),

and define h : T -t w(h,O)

]R3 :

(x, y, z)

f-t

Y = (2

+ cos(27rt)) sin(27rs),

z = sin(27rt),

f(s) - g(t). Prove that

= - 4~ 10 1 10 1 IIf(s) -

g(t) 11- 3 det(f(s) - g(t), J' (s), g' (t))dsdt,

which is the link number of the two knots (1.2, p. 28). Number 6. Let A be an orthogonal matrix of order m + 1. Use a winding number for the computation of the degree of the smooth mapping f : §m -t §m : x f-t Ax. Also do the direct computation in the Brouwer-Kronecker definition. Which is quicker?

5. The Borsuk-Ulam Theorem Our primary goal here is to obtain the important Borsuk-Ulam Theorem. Recall that a mapping f defined in some subset of a Euclidean space is called even if f(x) = f( -x) and odd if f(x) = - f( -x). The famous Borsuk-Ulam Theorem concerns the degree of such mappings. Afterwards, we will deduce several purely topological results, including the Invariance of Domain Theorem. We start with a technical statement:

5. The Borsuk- Ulam Theorem

161

Lemma 5.1. (1) Let K c M be compact subsets of~n and let f : K ~ ~m, m > n, be a continuous function such that 0 ct. f(K). Then f has a continuous extension J: M ~ ~m such that 0 ct. J(M). (2) Let D c ~n be a bounded open set, and let X = D \ D. We suppose that 0 ct. D and D is symmetric with respect to 0 (that is, x f---t -x induces a homeomorphism on D). Now let f : X ~ ~m, m > n, be an even (resp., odd) continuous mapping such that 0 ct. f(X). Then f has an even (resp., odd) continuous extension J: D ~ ~m such that 0 ct. J(D). (3) Assume the same hypotheses as in (2), except that here m = n. Then f has an even (resp., odd) continuous extension J : D ~ ~n such that 0 ct. J(D n {x n = O}). Proof. (1) Henceforth, norms of mappings are always intended on the compact set M. Let e = dist(O, f(K)). First, by the Tietze Extension Theorem, f extends to a continuous mapping 9 : ~n ~ ~m, and by the Weierstrass Approximation Theorem there is a polynomial mapping h = h(x) such that IIg - hll < Since m > n, the Easy Sard Theorem (II.3.4, p. 63) says that the image h(~n) has empty interior in ~m, and we can find a point a ct. h(~n) such that Iiall < Now consider h' = h - a : ~n ~ ~m. Clearly 0 ct. h' (~n), and furthermore

le.

le.

II h' -

gil::; II h' - h II

+ II h - gil < ! e.

Next consider the continuous mapping h"(x)

We have

IIh"lI

~

=

Eh'(x) { 2I1h'(x) II h'(x)

for IIh'(x)1I ::; otherwise.

!E. On the other hand, f(x) = g(x) for x E K, and

IIh'(x)1I ~ IIg(x)II-lIg(x) - h'(x) II >

so that h"(x)

!E,

= h'(x),

E- !E = !E,

and we conclude

IIh"(x) - f(x) II


for x E K.

Now, we can apply the Tietze Extension Theorem to the function

h" - f: K ~ B,

!E

where B is the open ball of radius (homeomorphic to ~m), and h" - f extends to a continuous mapping h'" : M ~ B. We claim that J = h" - h'" is the mapping we seek. Indeed, for x E K we have J(x)

= h"(x) -

h"'(x)

= h"(x) -

(h"(x) - f(x))

=

f(x),

162

IV. Degree theory in Euclidean spaces

and taking norms on M,

Ilfll

~

IIh"II - IIh"'II

~

!E - IIh"'II > !E - !E = a.

This completes the proof of (1). The proof of (2) goes by induction on n. For n = 1 the set D is a union of disjoint open intervals 1 = (a, b) with a > a and their symmetric copies -1 = (-b, -a), and X consists of the end points of those intervals. To extend f to D, we map each 1 onto a polygonal path in jRm \ {a} joining f(a) to f(b) (here we use m > 1), and for x E -1 define f(x) = f( -x) in the even case and f(x) = -f(-x) in the odd case. jRn

Now suppose the result holds for n-l. We identify jRn-1 and consider the set D1 = D n jRn-1,

== {x n = a} c

which is symmetric. Clearly, the set Xl = D1 \ D1 C jRn-1 is contained in X, and by induction we find a continuous mapping f1 : D1 -+ jRm that extends h = fix! and is never a. Now, since D1 n X = 0, we can define a nowhere zero continuous mapping 12 : D1 U X -+ jRm by

{ 12 ~ f1 12 = f

on D1, on X.

After this preparation we can apply (1). We already have one compact set, namely K=D1 UX, and to define M :J K, set

D+ = D n {x n > a},

D_ = D n {x n < a},

and take M

= D1 U X

U D+

= D1 U xu D+.

Now (1) gives a continuous mapping f2 : M -+ jRm that extends 12IK and is never a. To conclude the proof of (2), one extends f2 to D_ by symmetry. Finally, the proof of (3) is analogous, but one uses (2) instead of induction. The argument ends with an application of the Tietze Extension Theorem instead of (1), which gives the condition a rJ. f(D n jRn-1) instead 0 of a rJ. f(D). After these extension lemmas, we are ready to prove the famous BorsukUlam Theorem:

5. The Borsuk-Ulam Theorem

163

Theorem 5.2 (Borsuk-Ulam Theorem). Let D c IRn be a symmetric bounded open set, with 0 E D. Set X = D \ D, and let f : X ---+ IRn be a continuous mapping with 0 ¢ f(X). (1) Assume f(x)

f(-x)

Ilf(x)11

=1=

Ilf( -x)11

for all x E X.

Then the winding number w(f, 0) is odd, hence not zero. (2) Assume f(x)

f( -x)

Ilf(x)11

=1=

-llf( -x)11

for all x E X.

Then the winding number w(f, 0) is even, and zero for n odd. In particular, (1) applies if f is an odd function, and (2) applies if f is an even function Proof. We start with case (1). By the Tietze Extension Theorem, f extends to a continuous function on D, again denoted f : D ---+ IRn. But we need that extension to be odd. To amend that, consider the function 9 : D ---+ IRn

: X f---t

f(x) - f( -x),

which is clearly an odd function. Then we define the homotopy Ht(x)

= f(x)

- tf( -x),

Ho

= f,

HI

= g.

We claim that 0 ¢ H([O, 1] x X). Indeed, suppose 0 = f(x) - tf( -x) for some x E X, 0 ~ t ~ 1. Since 0 ¢ f(X), it must be true that t > 0, and from f(x) hypothesis.

=

tf( -x) it follows that

II~~:~II = II~~=:~II' contrary to the

We conclude that f and 9 have the same degree, and consequently we can substitute 9 for f, or merely suppose that f : D ---+ IRn is odd, which we do henceforth. Pick c > 0 small enough so that U mapping

f

1:

UUX

TT'Iln ---+~ :Xf---t

= {llxll

{f x

(x)

~

c} c

D and define the

if x EX, ifxEU.

Set DI =D\U, Xl =DI \D}, and /2=hlxl; note that Xl =xu{llxll =c}. The following figure depicts the setting:

IV. Degree theory in Euclidean spaces

164

~n-l

We can apply IV.5.1, p. 161, and obtain an odd continuous extension 12 : Dl --+ ~n of 12, such that 0 ~ 12(D 1 n ~n-l). Next, we set

f 3: D --+IN..

llJln

It is clear that

hlx = fix;

:xt---+

{12(X) x

if x E D 1 , if x E U.

hence by IV.2.6, p. 150,

d(f, D, 0) = d(h, D, 0). Now we will use the additive property of the degree, IV.2.5(3), p. 149. We write (see the figure above)

D1+

=

Dl n {x n > O},

D 1-

=

Dl n {xn < O}.

As 0 ~ 12(D 1 \ D1+ U D 1 -), from additivity it follows that

We make the change of variables (a linear isometry in fact)

Then, by direct computation,

and consequently,

165

5. The Borsuk-Ularn Theorem

To conclude, we look at the interior Uo = {llxll < c-} of U. We have D => Uo U (D \ U), and 0 ff- h(D \ Uo U (D \ U)), which again by additivity, gives

d(f, D, 0) = d(h, D, 0) = d(hl uo ' Uo, 0) + d(f3, D \ U,O)

= d(f3, Uo, 0) + d(f3, D I , 0) = d(Id, Uo, 0) + d(J2, D I , 0) = 1 + 2N, which is an odd number, as desired. Thus we have proved part (1) of the theorem. For (2), the same argument works, with the modifications we describe next. To reduce to the case where f is defined on the whole D, the function + f(-x). The function II must be defined as

9 needed is g(x) = f(x)

1[])n f 1: U U X -tJN..

:XH

{f(X) Ixl

= (IXII, .. ·, Ixnl)

if x E X, if X E U.

Accordingly,

Then, Thus,

and consequently,

Finally

d(f, D, 0)

d(f3, D, 0) = d(h, Uo, 0) + d(f3, D \ U,O) = d(f3, Uo, 0) + d(f3, D I , 0) = d(1 Id I, UO, 0) + d(J2, DI, 0) = 0 + (1 + (-1)n)N, =

which is the required number in part (2). Note here that d(1 Id I, UO, 0) = 0 by IV.2.5(2), p. 149. Indeed, d(1 Id I, UO, 0) = d(1 Id I, UO, a) = 0 for a = (-c-,O, ... ,O) ff-IIdl(X). 0 The above theorem has a very nice converse:

166

IV. Degree theory in Euclidean spaces

Theorem 5.3 (Hirsch Theorem). Let D c ~n be a symmetric bounded open set, with 0 ED. Set X = D \ D, and let f : X ~ ~n be a continuous mapping with 0 1. f (X) .

(1) If the winding number w(j, 0) is odd, then there is some x E X such that - f( -x) f(x) IIf( -x)II' IIf(x)11 (2) If the winding number w(j, 0) is even, then there is some x E X such that f(-x) f(x) Ilf( -x)II' Ilf(x)1I Proof. (1) Assume by way of contradiction that

f(x) Ilf(x)1I for all x EX. We extend define a homotopy by

-f(-x)

of: Ilf(-x)11

f to a continuous mapping 1 : D

Ht(x) = (1 - t)J( -x) By assumption, Ht(x)

of: 0 for x E X,

~ ~n and

+ tJ(x).

and by IV.2.4, p. 147,

w(j,O) = d(/,D,O) = d(Hl,D,O). 2

Now note that

Hdx) = !J(-x) 2

+ !J(x)

is an even mapping, and by the Borsuk-Ulam Theorem, IV.5.2(2), p. 163, the degree d(H 1, D, 0) must be even. This is a contradiction, because 2 w(j,O) is odd. Part (2) is proven similarly: one assumes

f(x) Ilf(x)1I for all x EX, extends

of:

f( -x) IIf(-x)11

f to 1, and defines Ht(x) = (1 - t)J( -x) - tJ(x).

Then the odd mapping H 1 has even degree, contrary to the Borsuk-Ulam 2 Theorem, IV.5.2(1), p. 163. 0

5. The Borsuk- Ulam Theorem

167

Note that from the Borsuk-Ulam Theorem we see readily that a function like f in case (1) on p. 163 (for instance an odd J) must have some zero in D after any extension. It is also easy to obtain a fixed point result:

Corollary 5.4. Let Dc ]Rn be a symmetric bounded open set, with 0 E D. Set X = D \ D, and let f : D -+ ]Rn be a continuous mapping whose restriction fix is odd. Then there is some x E D such that f(x) = x. Proof. The function 9 : D -+ ]Rn : x

glx: g( -x)

=

-x - f( -x)

f---+

x - f(x) also has an odd restriction

= -x + f(x) =

-g(x)

for x E X.

If 0 E g(X), we are done, so we assume 0 tJ. g(X). Then, by the BorsukUlam Theorem, IV.5.2(1), p. 163, d(g, D, 0) =I 0; hence there is some xED such that g(x) = 0 and f(x) = x. 0

Here is a second interesting consequence:

Corollary 5.5. Let Dc ]Rn be a symmetric bounded open set, with 0 E D. Set X = D \ D, and let f : X -+]Rm be a continuous mapping with m < n. Then there is an x E X such that f(x) = f( -x). Proof. Set ]Rm == {Xm+l = ... = Xn = O} C ]Rn, and consider the odd mapping 9 : X -+ ]Rm : x f---+ f(x) - f( -x). We assume 0 tJ. g(X) (otherwise there would be nothing to prove). By the Tietze Extension Theorem 9 has a continuous extension 9 : D -+ ]Rm, and we then define 9 : D -+ ]Rn : x f---+ (g(x),O). Like g, the preceding function is odd on X, and by assumption 0 tJ. g(X). Thus, by the Borsuk-Ulam Theorem, IV.5.2(1), p. 163, the degree of 9 is not zero: d(g, D, 0) =I O. However, pick c > 0 small enough so that af; = (0, ... ,0, c) E ]Rn \ g(X). Then, by IV.2.3, p. 146,

d(g, D, 0) = d(g, D, af;), and the degree in the right-hand side is zero, because af; p. 149).

tJ. g(D) (IV.2.5(2),

This contradiction comes from our assumption that 0 tJ. g(X); thus there must be some x E X with f(x) = f( -x). 0 Now we give a very important theorem (which for dimension 1 is an immediate consequence of the Bolzano Theorem):

Theorem 5.6 (Invariance of Domain Theorem). Let W c ]Rn be an open set, and let 9 : W -+]Rn be a locally injective continuous mapping. Then 9 is an open mapping.

IV. Degree theory in Euclidean spaces

168

Proof. Let a

E

Wand consider the translated open set D

=W

- a

= {x - a: x

E W}

(which contains 0) and the continuous mapping

f :D

~ ~n :

x

~

g(a + x) - g(a).

By hypothesis, for c > 0 small enough, f is injective on the closure B of the open ball B = {llxll < c} c Dj we set X = B \ B. Then we define the following homotopy:

Ht(x) =

fe: t) - f(;~Xt)'

x

E

B.

We see that 0 rt H([O,lJ x X): if H(t, x) = 0, since fiB is injective, l~t = ~!~ and x = 0 rt X. Thus, by homotopy invariance, IV.2.4, p. 147,

d(f, B, 0)

= d(Ho, B, 0) = d(Hl' B, 0).

But the mapping

Hl(X)

= f(!x) - f(-!x)

is odd, and by the Borsuk-Ulam Theorem, IV.5.2(1), p. 163, its degree is not zero. Let V denote the connected component of~n\f(X) that contains O. We deduce (IV.2.3, p. 146) that

d(f, B, y)

=1=

0 for every y E V,

which by IV.2.5(2), p. 149, means y E f(B). Consequently,

o EVe f(B). Thus, there is an "1 > 0 such that

if Ily'll < "1, then y' = f(x') for some x' E ~n with Setting x'

IIx'll < c.

= x - a and y' = y - g(a), we deduce that

if Ily - g(a) II < "1, then y = g(x) for some x

Ilx - all < c. is a neighborhood of g(a), and since the a + B's E ~n

with

This means that g(a + B) for c small enough form a neighborhood basis of a, we conclude that 9 maps every neighborhood of a onto a neighborhood of g(a). Since a E W is arbitrary, 9 is open. D

169

5. The Borsuk-Ulam Theorem

Remark 5.7. Suppose W =

~n

in the above theorem. Then, if

lim IIf(x)11 = +00, lixll-Hoc the mapping f is surjective.

Indeed, that condition implies that f is proper, and then only open as the theorem says, but also closed in ~n.

f(~n)

is not D

Variations of the previous theorem follow: Corollary 5.8. (1) (Invariance of Domain Theorem) Let W c ~n be an open set, and let g : W --+ ~n be an injective continuous mapping. Then V = g(W) C ~n is open, and g is a homeomorphism onto V. (2) (Invariance of interiors) Let S, T c ~n be arbitrary sets, and let g : S --+ T be a homeomorphim. Then

(of course, Intn means interior in ~n). (3) (Invariance of dimension) Let W c ~n and V c open sets. If Wand V are homeomorphic, then n = m.

~m

be non-empty

Proof. Assertion (1) is an immediate consequence of the previous theorem.

For (2), note that since f is injective, f(Intn(S)) is an open subset of~n; hence f(Intn(S)) C Intn(T). Arguing for f- 1 , we get the other inclusion. Equality concerning closures follows by complementation, because

A n ~n \ A = A \ Intn(A)

for any A.

For (3) suppose n < m, and put W' = W x {O} C ~n X ~m-n = ~m. Of course, W' is homeomorphic to W, hence to V by hypothesis. Then, by (2), Intm(W') is homeomorphic to Intm(V) = V. But this is impossible, because W' C {xm = O} has empty interior in ~m. D We conclude the section with a common improvement to the different fixed point theorems seen so far (III.7.2, p. 134, and IV.2.9, p. 152): Proposition 5.9. Let D C ~n be a bounded open set whose closure D is homeomorphic to a closed ball, and denote X = D \ D as usual. Then every continuous mapping f : D --+ ~n such that f(X) C D has some fixed point.

IV. Degree theory in Euclidean spaces

170

Proof. Let h : D -+ yr be a homeomorphism onto the unit closed ball yr c ]Rn; by IV.5.8(2), p. 169, h(X) is the unit sphere §n-l. Since h(f(X)) c h(D) = yr, we can define a homotopy Ht : X -+]Rn by

0 ~ t ~ 1.

Ht(x) = x - h-1(th(f(x))), Now suppose Ht(x) that

= 0 for some x

E X. Then

th(f(x)) = h(x) and as h(f (x)) E

E

yr, it must be true that t

x = h-l(th(f(x)))

E X, so

§n-l = 1. Consequently,

x = h-l(th(f(x))) = h-l(h(f(x))) = f(x), and we have a fixed point in X. Otherwise, we in fact have H t : X -+ ]Rn \ {O} and by the definition and homotopy invariance of winding numbers, d(Id - f, D, 0) = w(Id - fix, 0) =

= W(Hl' 0) = w(Ho, 0)

w(Id-h-1(0)lx,0) = d(Id-h-1(0),D,0) = 1,

because h-1(0) E D. Thus, d(Id - f, D, 0) i= 0, and by IV.2.5(2), p. 149, there is an xED such that x - f(x) = 0; that is, we have a fixed point in D. D

Exercises and problems Number 1. Suppose we have a decomposition

into non-empty closed subsets with Ai n -Ai = 0. (1) Show that for every i = 1, ... , m + 1 there is an odd continuous function Ii : §"' --+ ]R such that J;(x) = 2 if and only if x E Ai and J;(x) = -2 if and only if x E -Ai. 1 Ai t= 0. (2) Deduce that

n::i

Number 2. Let AI, ... , A m+ 1 be non-empty closed subsets ofthe sphere §m. Prove the following: (1) If §m = Al u··· U A m+ l , there is a point x E §"' such that x E Ai and -x E Ai for some i. (2) If §m = (AI U ... U Am) U (-AI U··· U -Am), there is a point x E §"' such that x E Ai and -x E Ai for some i. Number 3. Let /1, ... , 1m : ]Rm+l --+ ]R be homogeneous continuous mappings of odd degree (that is, li()..X) = )..Pi Ii (X) with Pi odd). Prove that the system /1 = 0, ... ,1m = 0 has some non-trivial solution. Number 4. Let I : ]Rn --+ ]RP be a homogeneous continuous mapping of odd degree, without zeros. Prove that every linear subspace L c ]RP of dimension n - 1 is perpendicular to I(x) for some x E ]Rn.

171

6. The Multiplication Formula

Number 5. An odd continuous mapping f : sm -+ sn is also called equivariant, expressing that it commutes with the antipodal involution. Prove the following: (1) There are equivariant mappings f : sm -+ sn if and only if m S n. (2) Every equivariant mapping f : sm -+ sm has odd degree; hence it is not nullhomotopic. (3) Every equivariant mapping f : sm -+ sn, m < n, is not surjective, hence nullhomotopic. Number 6. Here we consider topological spaces X equipped with a continuous involution ax (that is, ax 0 ax = Idx); in particular, asm is the antipodal involution. A continuous mapping f : X -+ Y of spaces with involution is called equivariant when ay 0 f = f 0 a x. For a space with involution X, define the following: (i) the level s(X), as the infimum (possibly 00) of all integers m ~ 1 such that there is an equivariant continuous mapping X -+ sm, (ii) the colevel s'(X), as the supremum (possibly 00) of all integers m ~ 1 such that there is an equivariant continuous mapping sm -+ X. Prove that s'(X) S s(X). Compute both invariants when X is the boundary of a symmetric bounded open neighborhood D of the origin in Rn, equipped with the antipodal involution.

Number 7. Consider the polynomial algebra A = R[Xl, ... ,xn ]/(l + x~ + ... + x~). Complete the details of the following proof that -1 is not a sum of less than n squares in A. (1) Otherwise, let -1 = h (X)2 + ... + f~-l (x) + fo(x)(l + x~ + ... + x~) for some polynomials A(x) E R[x], and write

fk( yCI x) = Pk(X)

+ yCIqk(X),

(2) Replacing x with obtain

A

Pk(X), qk(X) E R[x], Pk(X) even and qk(X) odd.

x in the equation for -1 above and comparing real parts,

n-l

-1

= I)Pk(x)2 -

qk(X)2)

+ po(x)(l - xi - ... -

x~).

k=l

(3) Show that the mapping q = (ql, ... , qn-d : D n -+ R n - 1 collapses two antipodal points in aD n = sn-l, and being odd, it has a zero a E sn-l. (4) Substituting x = a in the equation for -1 in (2), get a contradiction.

Number 8. Invariance of Domain holds for manifolds, but no further: (1) Prove that a locally injective continuous mapping f : M -+ N of boundaryless manifolds of the same dimension is open. (2) Consider the lemniscate X C R2 with polynomial equation (X2 + y2)2 = xy. Define a continuous bijection f : R -+ X which is not a homeomorphism.

6. The Multiplication Formula In this section we prove the general formula for the computation of the degree of a composite mapping. This is much more delicate than the corresponding resuit for the Brouwer-Kronecker degree, but always elementary. Here is the statement of the formula:

172

IV. Degree theory in Euclidean spaces

Proposition 6.1 (Multiplication Formula). Let D, E c jRn be bounded open sets with boundaries X = D \ D and Y = E \ E. Let f : D -+ jRn and 9 : E -+ jRn be continuous mappings such that f(D) C E. The open set E\f(X) decomposes into countably many connected components El, and we arbitrarily choose one point Cl E El in each. Now, let a E jRn\g(YUf(X)). Then d(g 0 f, D, a) = d(g, El, a)· d(f, D, Cl).

L l

Proof. We split the argument into several steps. Step I: The formula is well defined. First of all, the degree d(g 0 f, D, a) exists, because jRn \ g(f(X)) ~ jRn \ g(Y U f(X)). Next, since El is a connected subset of E\f(X) C jRn\f(X), the degree d(f,D,·) is constant on it (IV.2.3, p. 146). For the other factor d(g, El, a) we must see that a rJ- g(El \ El), but we have El \ El c (E \ E) U f(X) = Y U f(X) and a rJ- g(YUf(X)). Finally, the sum Ll is finite. Indeed, since g-l(a) C E \ f(X) is a compact set, it is contained in the union of finitely many Ee's, which are disjoint; hence g-l(a) n El = 0 for, say, f ~ fo. Thus, by IV.2.5(2), p. 149, d(g, Ee, a) = 0 for f ~ fo.

Step II: Approximation data. By the Weierstrass Approximation Theorem, there is a polynomial mapping j such that on D Ilf _

jll < {dist(f(D),jRn \

E) dist(g-l(a), f(X)).

and

From the first bound we deduce that j(D) C E, and the composite mapping go j does exist. From the second bound, we see that a rJ- g(j(X)), hence a rJ-jRn \ g(Y U j(X)), and c = dist(a,g(Y U j(X))) > O. Again by the Weierstrass Approximation Theorem, there is a polynomial mapping gl such that IIg - gIll < !c on E, and by the Sard-Brown Theorem, there is a regular value b of the mapping gl 0 jlD with lIa - bll < !c. We consider now the polynomial mapping g = gl + a-b. We have:

(i) On E, IIg hence a E

jRn \

gil

~ IIg - gIll

+ IIgl - gil < !c + !c = c;

g(Y U j(X)).

(ii) By translation, a is a regular value of g 0 jlD.

6. The Multiplication Formula

173

Step III: Reduction to the approximation data. We claim that d(g 0 f, D, a) = d(g 0 j, D, a) = d(9 0 j, D, a). For the first equality, we consider the homotopy

4>t(x) = (1- t)f(x) + tj(x),

x

E

D.

We have

Ilf(x) - 4>t(x) I = tllf(x) - j(x) II < dist(f(D),lRn

\

E);

hence 4>t(x) E E, and the homotopy go 4>t is well defined. Furthermore, art g(4)([O, 1] x X)). Indeed, suppose there is an x E X such that 4>t(x) E g-l(a). Then

l14>t(x) - f(x) II = tllj(x) - f(x) I < dist(g-l(a), f(X)), which is impossible. Thus, we can use homotopy invariance (IV.2.4, p. 147) and

d(g 0 f, D, a)

= d(g 0

4>0, D, a)

= d(g 0

= d(g 0

4>1, D, a)

j, D, a).

For the second equality of our claim we use the homotopy

llit(x) = (1 - t)g(j(x)) Again, the key fact is that a

lIa -

+ tg(j(x)),

rt lli([O, 1] x X).

x E D.

But if x E X, we have

llit(x) I = I (a - gj(x))

+ t(gj(x) - gj(x)) II 2: lIa - gj(x)lI- tllgj(x) - gj(x) I 2: dist(a, g(j(X))) -lig - gil > dist(a, g(j(X))) - c 2: 0,

by (i) in the preceding step. Thus, we can indeed apply homotopy invariance.

Step IV: Multiplication. By IV.2.1, p. 145, we know that

L signx(g j) = L signj(x) (g) signx(j) = L L signy(g)· signx(j)

d(g 0 f, D, a) =

0

xE(fjo/)-l(a)

YEfj-l(a)

=

L

xE(fjoj)-l(a)

xEj-l(y)

L

signy(g) ( Signx(j)) xEj-l(y)

yEfj-l(a)

=

L

signy(9). d(j, D, y).

yEfj-l(a)

IV. Degree theory in Euclidean spaces

174

Now, for every k the set Wk~ = {y E E \ j(X): d(j,D,y)

= k}

is a union of connected components of E \ j(X), whose boundaries are contained in Y U j(X) (as for the Ep's). Hence, by (ii) in Step II, a rt 9(Wk~\ Wk), and we can continue the above sequence of equalities as follows: d(goj,D,a)

= ... = L = Lk( k

signy(§)' d(j, D, y) L

signy(g))

= Lk. d(g, Wk~a).

yEg-l(a) nWk-

k

Step V: Second multiplication. We repeat the same computation for j. We set Wk = {y E E\j(X): d(j,D,y) = k} = UEf;, i

where the Efi's are the connected components of E\j (X) on which d(j, D, .) == k. Since W k \ Wk c YUj(X), we have art g(W k \ Wk), and by additivity (IV.2.5(3), p. 149) we get d(g, Wk,a) = Ld(g,Efi,a). i

Indeed, as remarked in Step I, the sum is in fact finite: L

d(g, Efi' a)

= L

because g-l(a) n Ef =

d(g, Efi' a)

i: fi<eO

i

0 for f 2: fa; moreover, in this situation

art 9(Wk \ Wk) = 9(Wk \

U Eei)

i: ei
and the additivity property does apply. Notice now that the Eei's in this sum are the Ee's contained in Wk. We conclude that L f

d(g, Ee, a) . d(j, D, ce)

= L k· L k

d(g, Ee, a)

f:EecWk

= Lk. d(g, Wk,a). k

6. The Multiplication Formula

175

Step VI: Simplification of mappings. Comparing the formulas in the preceding two steps, we are reduced to showing that

for every k. But by (i) in Step II, Ilg - gil <

E

= dist (a,g(YU j(X)))

:S dist (a,g(W;\ W k-)).

Hence and we must show the equality of the following two degrees: d(g, Wk, a)

= d(g, Wk~ a).

Step VII: Computation of the left-hand side degree. We claim that

1- g(Wk \ W;). Suppose a = g(z) with z E Wk \ Wk~ Since a 1- g(YUf(X)) and Wk \ Wk a

C

Y U f(X), it must be true that z E Wk, so that dU, D, z) = k. Moreover, II f - jll

< dist(g -1 (a), f(X)) :S dist(z, f(X))

and we get d(j, D, z) = dU, D, z) = k, that is, z E W;, a contradiction. Thus the claim is proved, and, by additivity (IV.2.5(3), p. 149), d(g, Wk, a)

= d(g, W; n Wk, a).

Step VIII: Computation of the right-hand side degree and conclusion. The argument is analogous. First, a 1- 9(Wk-\ W k ), because a 1- g(Y U j(X))

and Wk-\ Wk- C Y U j(X). Then, additivity gives d(g, Wk-,a) = d(g, Wk-n Wk,a).

Thus, we have the equality of degrees stated at the end of Step VI, and the proof of the Multiplication Formula is finished. 0

Exercises and problems Number 1. Find two continuous functions f : D --+ ]Rn and 9 : E --+ ]Rn verifying the hypothesis of the Multiplication Formula (IV.6.1, p. 172), with D and E connected, such that the sum in that formula has at least two different summands.

176

IV. Degree theory in Euclidean spaces

Number 2. Let D (resp., E) be a bounded open subset ofli", and let I: I5 -+ lin (resp., 9 : E -+ lin) be a continuous function. Suppose E is connected and that I(D) c E, I(8D) C 8E. Prove that

d(g 0 I, D, a)

= d(g, E, a) . d(j, D, b)

for a E lin \ g(8E) and bE I(D).

Number 3. Let D (resp., E) be a bounded open subset of lin , and let I: D -+ lin (resp., 9 : E -+ lin) be a continuous function. Suppose D is connected and that I(8D)nI(D) = 0 and I(D) C E. Prove that

d(g 0 I, D, a)

= d(g, E, a) . d(j, D, b)

for a E lin \g(8EU I(8D)) such that g-l (a) C I(D) and bEE. Show that the condition g-l(a) C I(D) is necessary for this conclusion.

°

Number 4. For p> we denote by D~ C lik the open ball of radius p centered at the origin and by and S;-l the corresponding closed ball and sphere; if p = 1, we just get Dk, Dk, and §k-l. Fix n 2: 1, and set Ep = (-p,p) x D;. (1) Compute the degree of the mapping

D:

!pp: Ep

-+ n;+l : (t,x) t--+ (t, ~Jp2

(2) Show that for every continuous mapping and (Id[-l,l) Xf)(El) C Er with r > 1,

d(j,D n , 0)

= d(!Pr 0

I: D n

- t 2 x).

-+ lin such that J(§n-l) C §n-l

(Id[-l,l) xf),El,O).

(3) Prove that for every continuous mapping 9 : §n-l -+ S',-l there is a unique continuous mapping E(g) : §n -+ §n (the suspension of g) such that !PI 0

and prove that deg(E(g))

(Id[-l,l) xg)

= E(g) 0 !PI,

= deg(g).

Use the above construction to produce by induction on k a mapping §k -+ §k of any given degree d.

7. The Jordan Separation Theorem The core of this section is to prove the following purely topological statement: Theorem 7.1 (Jordan Separation Theorem). Let K, L c ]Rn, n ~ 2, be two compact sets. If K and L are homeomorphic, then their complements ]Rn \ K and]Rn \ L have the same number N ::; 00 of connected components. Proof. Note first that both complements ]Rn \ K and ]Rn \ L are open sets with exactly one unbounded connected component each, which we denote

177

7. The Jordan Separation Theorem

£1 0 and To, respectively. In fact, we consider the decompositions into connected components

{

IRn \ K = Ui2:0 £1i' a = number of bounded £1i's :::; 00, lRn \ L = U j 2:0 Tj , (3 = number of bounded Tj's :::; 00.

Pick points ai E £1 i (i the boundaries

~

1) and bj E Tj (j

~

1) . Also, we consider as usual

By hypothesis, we have a homeomorphism h : K ---+ L, and by the Tietze Extension Theorem we find continuous mappings cp, 'l/J : lRn ---+ lRn such that cplK = hand 'l/JIL = h- 1 and both cp and 'l/J are the identity off an open ball containing K and L. In this situation, we easily compute the degrees

Indeed, by construction and Finally, since cp 0 'l/JIYi = Id h'i and 'l/J Theorem, IV.2.6, p. 150, d( cP 0

.1.

1"'

'f" 1. i,

b) j

= d(Id ,r,i , bj ) =

0

cplxi = Id lXi' by the Boundary

{I if bj E ri, that is, if i 0 otherwise

=j

}

= 8.oJ'

and

Aaj) = d(Id , ..:...li, A aj ) = {I if aj E Lli' that is, if i = d( 'l/J 0 cp, ..:...li, o otherwise

j} =

>:. '.

uoJ

Next, we compute d(cp 0 'l/J, Ti, bj) by using the Multiplication Formula.

Step I: Multiplication setting. We decompose the complement of the compact set 'l/J(Yi) into connected components, the first one unbounded:

Now, this complement contains

178

IV. Degree theory in Euclidean spaces

hence each connected component of the last union is contained in a unique De, and we can write and Do \ Uo C K, and Dl \ Ul C K, C ~n

Next we choose a big enough open ball E conditions: (i) bj

tt. 'P(E \ E)

(recall that 'P is the identity off an open ball).

(ii) E:J 'l/J(ri) U (~n \ Do); hence E (iii) Eo

= En Do

to have the following

:J Ue~l

De .

is connected, and Eo ct'l/J(ri).

(iv) E \ 'l/J(Yi) = Eo U El U E2 ... , where Ee = De for P 2: 1.

In this situation we can apply the Multiplication Formula, IV.B.1, p. 172, to the mappings 'l/Jlri : r i -+]Rn and 'PIE: E -+]Rn to compute d('P °'l/J, ri, bj). Step II: Computation by multiplication. Using the Multiplication Formula, we obtain

d('P0'l/J,ri,bj ) = "Ld('P,Ee,bj)d('l/J,ri,Cf), e~o

for any chosen Ce E Ee. Here we can immediately discard the P = 0 term by picking CO E Eo \ 'l/J(ri) (condition (iii) above). Thus,

d('P ° 'l/J,ri,bj ) = "Ld('P,Ee,bj)d('l/J,ri,ce), e~l

and Ce E De = Ee.

Step III: Additional expansion. Here we further decompose some terms of the last sum. In fact, we are to show that r

First, this is a well-defined finite sum, for, since

bj

E

rj

C ~n \

L=

]Rn \

'P(K)

C ]Rn \

'P(Xkr(f») ,

every degree d( 'P, Llkr(l) , bj ) exists. Furthermore it is zero for r big enough. Indeed, since bj tt. L = 'P(K), 'P-1(bj ) n K = 0; hence

'P-l(bj) \ .10 c

ULlk · k~l

179

7. The Jordan Separation Theorem

But 'P-l(bj) \ ..10 is compact (closed and bounded), so that for r large, say r > ro, 'P- 1 (bj) n Llkr(f) = 0; hence d( 'P, Llkr(f) , bj) = 0, and ro

Ld('P, Llkr(f) , bj) = Ld('P, Llkr(f),bj). r

r=l

Now, by construction we know that

¢ 'P(Df \

bj

ULlkr(e)), r

and since 'P-l(bj ) does not meet Llkr(f) for any r > ro, we get ro

bj

¢ 'P( Df \

ULlkr(e)). r=l

Thus we can apply additivity (IV.2.5(3), p. 149), to see that the degree d( 'P, Ee, bj ) is the finite sum above. Step IV: End of the computation. Putting Steps II and III together, we have d('P 0 'ljJ, ri, bj )

=L

( L d('P, Llkr(f) , bj ) )d('ljJ, r i , Cf)

e~l

r

=L

d('P,Llkr(f),bj). d('ljJ,ri,akr(f))·

f,r~l

The last equality follows by rearranging the sums and taking Ce = akr(e) E Llkr(f) C De for every e, i. But the last sum can also be rewritten as

... =

L

d('P,Llk,bj)·d('ljJ,ri,ak)

LlkrtDo

= L

d('P, Llk' bj) . d('ljJ, r i , ak).

k~l

This is because if Llk

c

Do, then

for any ale E Do (IV.2.3, p. 146). But we can find ale far from 'ljJ(ri ) so that the equation 'ljJ(x) = ale has no solution in r i and the last degree is zero (IV.2.5(2), p. 149). With this, we have finished the computation of d( 'P 0 'ljJ, ri, bj ).

180

IV. Degree theory in Euclidean spaces

All that has been done so far for r.po7jJ can be done analogously for 7jJor.p, and one ends up with the following formulas:

(*)

L d( r.p, Llk' bj) . d( 7jJ, n, ak) = d( r.p L d( 7jJ, re, aj) . d( r.p, Lli' be) = d( 7jJ

0

7jJ, r i, bj )

= 6ij ,

0

r.p, Lli' aj)

= 6ij .

k~l

(**)

e~l

With these formulas, we can finally prove

O!

= (3.

To that end, we define two linear mappings. Let R(-r) stand for the direct sum of 1 :S 00 copies of R, and let {ei} stand for its canonical basis. Then

{

A: R(a) -+ RCB) : ek B : RCB) -+ R(a) : ee

t---+ t---+

(d(7jJ, r i , ak) : i ~ 1) = (Aik : i ~ 1), (d(r.p, Lli' be) : i ~ 1) = (Bie : i ~ 1).

Now, we can rewrite the formulas above as

""' -- 6·· t tJ' L.J Bk·J . A-k k~l

(**)

L Aej . Bu = 6ij e~l

and we see that A and B are inverse to each other; hence both linear spaces 0 have the same dimension, and O! = (3, as desired.

Examples 7.2. (1) A compact set KeRn that is homeomorphic to a differentiable hypersurface disconnects Rn. (2) A compact set KeRn that is homeomorphic to a connected differentiable hypersurface disconnects R n into two open connected components, one bounded, whose boundaries are both K. No proper subset T c K disconnects Rn. (3) An arc (that is, a compact set homeomorphic to an interval) never disconnects Rn for n ~ 2. (4) A topological closed ball never disconnects R n for n

~

2.

(5) A compact set KeRn homeomorphic to a proper subset of §n-l does not disconnect Rn. (6) A sphere is not homeomorphic to any of its proper subsets (otherwise we would have two homeomorphic compact subsets of a Euclidean 0 space, one disconnecting it and the other not).

7. The Jordan Separation Theorem

181

The Jordan Separation Theorem, IV.7.1, p. 176, also holds when we replace Euclidean spaces by spheres: Proposition 7.3. Let K, L c §n, n ~ 2, be two compact sets. If K and L are homeomorphic, then their complements §n \ K and §n \ L have the same number N ~ 00 of connected components. Proof. If one of the two compact sets is the sphere, then the other is the sphere also, by IV.7.2(6) above. Hence we can pick two points a E §n \ K and b E §n \ L and consider the stereographic projections from them, denoted by x : §n \ {a} -+ ~n and y : §n \ {b} -+ ~n, which are diffeomorphisms. Then x{K) and y{L) are compact homeomorphic subsets of ~n; hence they disconnect it in the same number of connected components. Pulling them back to §n via x and y, we see that §n \ K and §n \ L have the same number of connected components too. Indeed, it suffices to remark that the unbounded component U of ~n \ x{K) (resp., V of ~n \ y{L)) corresponds to the connected component U' = {a} U x-I (U) of §n \ K that contains a (resp., V' = {b} U y-I{V) of §n \ L that contains b). 0

We can finally generalize IV.7.1, p. 176, to closed sets: Proposition 7.4. Let G, G' c ~n, n ~ 2, be two closed sets. If G and G' are homeomorphic, then their complements ~n \ G and ~n \ G' have the same number N ~ 00 of connected components. Proof. Let j : G c ~n be the canonical inclusion. Since j is a proper mapping, it extends to the one-point compactifications G* of G and §n of ~n, by mapping 00 f---t 00. Similarly, j' : G' c ~n extends to G'* -+ §n. Of course, G* and G'* are homeomorphic as G and G' are. Thus, we have two homeomorphic compact subsets G* and G'* of the sphere §n, and by the preceding proposition, they disconnect §n in the same number of connected components. Note here that none of those components contains 00; hence they pull back to ~n without surprise, and the result is proved. 0

For instance, we can improve the statement of IV.7.2(1), p. 180, to the following: a closed set G c ~n that is homeomorphic to a closed differentiable hypersurface disconnects ~n. We close this section with a new proof of the Invariance of Domain Theorem: Proposition 7.5. Let W c continuous mapping f : W -+

~n ~n

be an open set. Every locally injective is open.

182

IV. Degree theory in Euclidean spaces

Proof. Pick any a E W. Consider an open ball B centered at a of radius > 0 and the compact sets

E

K =

{llx - all

~

c}

and

L=

{llx - all

=

c};

by hypothesis, for E small enough, f is defined and injective on K. Then, the restrictions flK and flL are injective closed mappings (by compactness of their domains of definition), hence homeomorphisms onto their images f(K) and f(L). Consequently, jRn\f(K) is connected (and unbounded) and jRn \ f(L) splits into two connected components Do, Dl, with Dl bounded. It follows that jRn \ f (K) c Do. Hence Dl

c f(K) = f(B) U f(L),

and we conclude that Dl C f(B). But f(B) is a connected subset of jRn \ f(L), so it cannot be bigger than D 1 . Thus f(B) = Dl is an open set. This shows that f maps all small enough open balls centered at a onto 0 open sets. Since a is arbitrary, f is open.

Exercises and problems Number 1. Prove the following general facts for a compact set K in Rn. (1) The complement R n \ K has exactly one unbounded connected component. (2) The set K contains the boundaries of all connected components of R n \ K. (3) If R n \ K is not connected, but R n \ A is connected for every closed subset A s;: K, then K is the boundary of all connected components of R n \ K. Number 2. Exhibit two homeomorphic compact subsets of R3 whose complements are not homeomorphic. Number 3. Let C and C' be closed sets of RP and Rq, respectively. Show that if C and C' are homeomorphic, then RP x Rq \ ex {O} and RP x Rq \ {O} X C' are homeomorphic. Number 4. Let "(, "(' : [0, 1] ~ D2 be two injective continuous mappings with

"((0)

= (1,0),

"((1)

= (-1,0);

"('(D)

= (0,1),

"('(I)

= (0,-1).

Prove that they meet at some point. -2

-2

Number 5. Let C be a closed subset of the closed disk D . If C and D \ C are both -2 -2 connected, then en aD and aD \ C are both connected. Number 6. Let M be a boundaryless connected smooth manifold, dim(M) 2 2. (1) Find some smooth hypersurface HeM, diffeomorphic to a sphere, such that M \ H has two connected components. (2) Show that there may be two hypersurfaces Hl, H2 eM, both diffeomorphic to a sphere, such that M \ Hl has two connected components and M \ H2 is connected.

Chapter V

The Hopf Theorems In this chapter we formulate and prove the theorems that fully complete degree theory: the degree is the only homotopy invariant for spheres. This must be formulated for both the Brouwer-Kronecker and the Euclidean degrees, although in essence the two cases are the same. These are the Hopf Theorems. As these theorems refer to mappings into spheres, we start the chapter by constructing in §1 various basic examples of them. Then in §2 we prove the first Hopf Theorem, that two mappings into a sphere of the same (Brouwer-Kronecker) degree are homotopic; hence, in particular, the cohomotopy group 7rm(M) of any m-manifold and the homotopy group 7rm(sm) of the m-sphere are Z. In §3 we deduce the same result for the Euclidean degree, which has the virtue of determining the cohomotopy groups 7rm(x) for some more exotic spaces X of dimension m, like the topological circle. These Hopf Theorems require the use of diffeotopies. Next, in §4 we come back to the Hopf invariant and define the Hopf fibrations to study some homotopy groups 7rk (sm) for k > m. Then we turn to the theory of tangent vector fields, and in §5 we discuss the essential notion of index of a tangent vector field at an isolated zero, using both the Brouwer-Kronecker and the Euclidean degrees. This notion is best illustrated by the gradients of Morse functions, which we describe in §6. Finally, we conclude the chapter and the book by presenting in §7 another deep theorem also named after Hopf, the Poincare-Hopf Index Theorem, that computes the index of a tangent vector field. We also see how the Gauss-Bonnet Formula follows from this.

1. Mappings into spheres In this section we discuss several methods for constructing smooth mappings into spheres, or to modify one already given. To start with, we exhibit simple examples of mappings of arbitrary degree.

Example 1.1. Let §m C jRm+l be the standard unit sphere. Then, for every integer d there are mappings fd : §m -+ §m of degree d. Since all constant mappings have degree 0, we assume d =I O.

-

183

v.

184

The HopE Theorems

Case d ~ 1. We will generalize to dimension m the intuitive idea of winding a circle k times around another (III.1.6(3), p. 100). Any point x E §m can be parametrized as

x

= (Xl,X2,X') = (pcosO,psinO,x')

p = .j1-lix'11 2 .

where

Then, we define the winding by

9d(X) = (pcos(d·O), psin(d·O), x'). To eliminate the parameter 0, recall that

cos(d·O) = Pd(cosO,sinO),

sin(d·O) = Qd(cosO,sinO),

for suitable homogeneous polynomials Pd and Qd of degree d. We have

9d(X) = (PPd(~Xl,~X2),pQdGXl,~X2)'X')

= (pLl Pd(Xl, X2), i-I Qd(Xl, X2), x'). This shows that the map 9d is differentiable except when amend this, we use the deformation ji of p defined by

for a smooth function A(t) which is the figure below:

= t for t

~

lix'il =

1. To

! and < t for t > !, as in

Thus, we have the smooth mapping

Yd(X)

=

(jid~l Pd(Xl, X2), jiLl Qd(Xl, X2), x').

!

For Ilx'll ~ this coincides with 9d(X) E §m, but otherwise, we must normalize to get fd = Yd/IiYdll : §n -+ §n. We claim that deg(fd) = d. Indeed, pick a = (1,0, ... ,0) E §m. Suppose fd(X) = a. Then x' = 0 and p = 1, so that fd(X) = 9d(X) = (cos(d·O), sin(d·O), 0), and we obtain where Ok

k

= 27rd ,0

~

k < d.

1. Mappings into spheres

185

Near such a point we can use the parametrization x = (p cos 0, p sin 0, x'), which is compatible with the standard orientation of the sphere (straightforward computation). The local expression of our mapping is then fd

== 9d : (0, x')

t-+ (d·O, x').

This is clearly a diffeomorphism that preserves orientation, which shows that a is a regular value of fd, and d

deg(Jd)

= L

signx(Jd)

xEfil(a)

= L(+l) = d. k=1

One can use the same method for d negative, just noting that one winds around clockwise, instead of counterclockwise. However, here we have the following alternative argument. Case d

=

-1. As we saw in 111.1.6(2), p. 100, the symmetry

has degree deg(J-l) Case d

< 0. Define

= -1. fd

deg(Jd)

=

f-l

0

f-d, so that by 111.1.7, p. 100,

= deg(J-l) . deg(J_d) = (-1)( -d) = d.

o

We can extend the preceding result to arbitrary compact manifolds (compactness is required because the target §m is compact): Proposition 1.2. Let M be a compact, oriented, boundaryless, differentiable manifold of dimension m. Then for every integer d there is a differentiable mapping hd : M -t §m of degree d. Proof. First we remark that it is enough to find h : M -t §m of degree ±1. Indeed, given such an h, we can use the mappings f±d of the preceding example to define hd = f±dOh, which has degree d (111.1.7, p. 100). In fact, let us define a differentiable mapping h : M -t §m such that the north pole (0, ... ,0,1) E §m is a regular value with a single preimage.

Let U be an open subset of M diffeomorphic to smooth function for t ::; = for t ~ 1.

~(t)

G

!,

~m,

and consider a

v.

186

We identify U ==

JR.ffi

to define h : M -+

h(x)= (

by

the south pole, for x ct. U,

(O, ... ,0, -1) {

§ffi

The HopE Theorems

2T(X)X T(x)2 2 T(x)2 + Ilx11 ' T(X)2

-llxIl2 ) + IIxll 2

where

T(x)=p,(llxI1 2 ), for

x E U.

This mapping is well defined because for x E U with IIxll ~ 1 we have T(X) = and h(x) = (0, ... ,0, -1). On the other hand, using the equations of the stereographic projection 7r from the south pole (11.2.4, p. 55), for x E U with T(X) =f:: 0, we get

°

h(x) =

7r-I(T~X)).

!.

Thus, h is the diffeomorphism 7r- 1 for T(X) = 1, that is, for Ilxll ::; On the other hand, x = is the only preimage of (0, ... ,0,1). Indeed, suppose (0, ... ,0, 1) = h(x). Then T(X) =f:: 0, and

°

(0, ... ,0,1)=h(x)=7r

-l( T(X) x )

j

hence x = 0. We are done.

D

Remark 1.3. The existence of non-homotopic maps M -+ N between two manifolds depends on the nature of the given manifolds. In general we cannot expect the richness shown in V.1.2 above. To illustrate the opposite situation we recall that all continuous mappings §2 -+ §l X §l are null-homotopic, hence have degree zero.

Indeed, the reason is that any continuous mapping f : §2 -+ be lifted to §2 -+ JR. x JR., through the universal covering

1:

7r: JR.

x

JR.

-

-+ §l

X §l :

(a,{3)

H

§l X §l

can

(cosa,sinajcos{3,sin{3),

= (1- t)f(x)

so that f :::: 7r 0 f. Then the homotopy Ht(x) Ht = 7r 0 H t with Ho = f and HI constant.

goes down to D

Now we want to modify mappings already given. The modification must keep the homotopy type. For mappings into spheres this comes easily, as follows: Lemma 1.4. Let M be a possibly non-compact manifold. Let f, g : M -+ §ffi be two mappings that do not have antipodal images: f(x) =f:: -g(x) for all x E M. Then H x _ tg(x) + (1 - t)f(x) t( ) - Iltg(x) + (1 - t)f(x)11

1. Mappings into spheres

187

is a well-defined homotopy with Ho = f, HI = g. In particular, if f is not surjective, it is null-homotopic.

Proof. Let x E M. Then p = tg(x) + (1 - t)f(x) is a point in the segment joining f(x) and g(x). As both points belong to the sphere, p = 0 only if f(x) and g(x) are antipodal, which is excluded by hypothesis. Thus Ht is indeed well defined. Finally, if there is a p ¢. f(M), f is homotopic to the constant mapping 9 == -po D An immediate application of the lemma is that every differentiable mapping M -7 §m with dim(M) < m is null-homotopic. Indeed, with that hypothesis, no smooth mapping M -7 §m is surjective, by the Easy Sard Theorem. Now that we have this homotopy lemma, we will further explore the idea in the proof of V.1.2, p. 185:

(1.5) Antipodal extension, linearization, and normalization. Let M be an oriented, boundaryless, differentiable manifold of dimension m, possibly non-compact. (1) Suppose we fix some parametrization ~m == U c M and a smooth mapping h : ~m -7 ~m such that h-I(O) = {c}. Then we use the stereographic projection 7r : §m -7 ~m from the south pole as = (0, ... ,0, -1) E §m to identify ~m with the open set V = §m \ {as} C §m and E ~m I with the north pole aN = (0, ... ,0,1) E §m. Thus, h == 7r- 0 h is seen as a mapping into ~m == V C §m :

°

M

U U

§m C

~ O==aN

U

== IRm ~ IRm == V

What we want is to extend h : U -7 V from a neighborhood of c to Ii : M -7 §m by sending everything off another neighborhood of c to the south pole. For such an extension we modify h by a trick analogous to the one used in the proof mentioned above:

Ii(x) =

{

(O, ... ,0, -1)

for x ¢. U,

2T(X)h{x) T{x)2 - Ilh{x)1I2 (T(X)2 + IIh(x) 112 'T{x)2 + IIh{x) 112 )

for x E U,

188

V. The HopE Theorems

with a smooth bump function

T(X) =

J.L(llx - c1l 2) =

Ilx - cll :S~, Ilx - cll ~ 1.

I for {0 for

This defines well a smooth mapping, because Ilh(x)11 f:. 0 on a neighborhood of IIx - cll ~ 1, and Ii(x) = (0, ... ,0, -1) for IIx - cll ~ 1. After some straightforward computation, for x E U with T(X) f:. 0 we find

Ii(x) = 7r-1 (h(X)); T(X) hence h == 7r- 1 oh on the ball ofradius~. Note also that Ii(x) = (0, ... ,0,1) if and only if x E U and h(x) = 0, if and only if x = c. We will call this construction the antipodal extension. In view of the homotopy lemma, V.1.4, p. 186, we are also interested in possible antipodal images of Ii and h in §m. Suppose this is the case for some point x E U. Then

(

2T(X)h(x) T(x)2 T(x)2 + IIh(x)112 'T(x)2

-lIh(x)112 ) + Ilh(x)112

( = -

2h(x) 1 - Ilh(x)1I2 1 + Ilh(x)112 ' 1 + Ilh(x)112

)

Looking at the last component of these two points, we find that T(X) Ilh(x)112 and h(x) =1= O. Then, looking at the other components, we get

. =

1 + Ilh(x)112'

which is impossible. Hence we conclude that the mapping h and its extension Ii have no antipodal images.

(2) Next, suppose h is a diffeomorphism, and denote L = dch, which is a linear isomorphism. Consider the homotopy simplifies to h(tx + (1 - t)c) H t (X ) -_ { t L(x - c)

of IV.3.1, p. 153, which here

£ ...J. 0 or t r , for t

= O.

Clearly Ht(x) = 0 if and only if x = c, and we can apply the same antipodal extension to H t replacing h by H t all through the construction. We get a homotopy fIt such that fIl = Ii and 7r 0 fIo == L - L(c) near c. We have thus linearized the mapping Ii. (3) Now, the determinant of the linear isomorphism L can be either positive or negative. In the first case, let A denote the identity, and in the

189

1. Mappings into spheres

second, the symmetry with respect to the first variable in jRm. In any case, the determinants of L and A have the same sign. Now, m x m matrices are points in jRmxm, and the closed hypersurface det = 0 decomposes jRmxm into two connected components: det > 0 and det < O. Since L and A are in the same one of those two components, there is a continuous mapping tHAt with det(At) =I- 0 for all t, Ao = L and Al = A. Again we apply the antipodal extension to B t = At-At(c) and find a homotopy 13t . We remark here that the construction is possible because At is bijective (det(At) =I- 0): this guarantees that B t (x) = At (x - c) = 0 if and only if x = c. This normalization shows that the mapping h is homotopic to 9 = 131 , which is the antipodal extension of A - A(c). The construction we have just described is the method for classifying smooth mappings f : M -+ §m by homotopy. Indeed, suppose the north pole a E §m is a regular value of f with finitely many preimages Cl, ... , Cr. As usual, we find local coordinates around each Ci, on whose domains Ui the restrictions flui are diffeomorphisms. Choosing the Ui'S disjoint and small enough, we can glue the antipodal extensions of the restrictions flui into a smooth mapping J: M -+ sm. By (1) above, f and J have no antipodal images, and by V.1.4, p. 186, f and J are homotopic. This mapping J depends only on the preimages of the regular value and the signs of f at them: exactly what we use to compute the degree. Actually, this is almost a proof of the Hopf Theorem, and what remains is quite evident: (i) to move the preimages freely and (ii) to dispose of those pairs whose signs cancel each other. We devote the following section to settling these two matters, by means of diffeotopies (II.6, p. 80).

Exercises and problems Number 1. Points (x, y, t) in the sphere §2 C lR? == c x lR can be written as pairs (z, t), z = x +iy. For every integer k > 0 define a continuous mapping f : §2 -t §2 by a formula f(z, t) = (Zk, th(t)) involving a suitable function h > O. Compute the degree of f. Number 2. It is a fact that all smooth mappings Cpl x Cpl -t CP2 have even degree. Mappings of every possible degree can be obtained as follows. (1) Use the Segre embedding Cpl x Cpl -t Cp3: (xo: Xl;YO: Yl)

f-t (XOyo: XOYI : XIYO: XIYl)

and a conic projection in Cpa to produce a smooth mapping Cpl x Cpl -t CP2 of degree 2. (2) Compose the mapping in (1) with mappings Cpl x Cpl -t Cpl XCpl of arbitrary degree.

190

V. The HopE Theorems

Number 3. Let f : §m -+ §m be a continuous mapping. Show that: (1) If deg(f) =1= +1, there is an x E §m with f(x) = -x. (2) If deg(f) =1= (_1)m+1, there is an x E §m with f(x) = x. (3) If f is null-homotopic, then it fixes a point and sends another to its antipode. Number 4. Let f, 9 : §2n -+ §2n be two continuous mappings. Prove the following assertions: (1) At least one of the mappings f, g, or go f has a fixed point. (2) If f has no fixed point, there are x, y E §2n such that f(x) = y and f(y) = x. (3) If f has no fixed point, then it sends some point to its antipode. Number 5. Let r : §m -+ §m be a continuous involution (=1= Idsm). Show that r sends some point to its antipode by way of contradiction, as follows: (1) Prove that, otherwise,

f

:

§m

x+r(x)

§m

-+

:XH

IIx+r(x)11

is a well-defined continuous map of degree 1, hence surjective. (2) Let F C §m denote the set of fixed points of r, and write V = §m \ F, U = f-l(V) C §m. Show that flu: U -+ V is a well-defined proper map of the same degree as f. (3) Use the identification x == r(x) to define a surjective local difeomorphism 1T : U -+ M onto an abstract smooth manifold M (just as x == -x gives M = um). (4) Show that flu factorizes through M, and use that factorization to conclude that f has even degree.

Number 6. Let G be a non-trivial group of homeomorphisms acting on a sphere of even dimension. Suppose that no h E G distinct from the identity has fixed points. Then G==Z2.

Number 7. Let f, 9 : §m -+ §m be two continuous mappings which are never perpendicular (that is, (f(x),g(x») =1= for all x E §m). Then deg(f) = ±deg(g).

°

Number 8. Let P, Q denote two complex polynomials of degrees p and q, respectively, and consider the rational function F = P / Q : C == ]R2 -+ C == ]R2. This function is defined except at finitely many poles Cl, ... ,Cr : the zeros of Q which do not cancel with those of P. Show that via the sterographic projection from the north pole aN = (0,0,1), F extends to a smooth mapping f : §2 -+ §2 by setting

Compute the degree of f.

Number 9. Let M C ]RP be a connected, boundaryless, smooth manifold of codimension m. Show that there is a differentiable mapping f : ]RP -+ §m such that the north pole aN = (0, ... ,0, 1) E §m is a regular value of f and M = rl(aN).

2. The HopE Theorem: Brouwer-Kronecker degree

191

2. The Hopf Theorem: Brouwer-Kronecker degree Here we will show that an example like III.3.5, p. 112, cannot occur for mappings into spheres. Theorem 2.1 (Hopf Theorem). Let M be a connected, compact, oriented, boundaryless, differentiable manifold of dimension m. Two continuous mappings M -+ §m with the same Brouwer-Kronecker degree are homotopic.

Proof. We will prove the theorem by showing that all mappings f : M -+ of a given degree have a common normal form, in the sense described in the previous section. Suppose deg(f) = d ~ 0 (resp., d ::; 0) and fix a coordinate domain U == ~m, whose local coordinates preserve orientation, and any d points CI, ... , Cd E U (none if d = 0). We are to built up a mapping homotopic to f that solely depends on these points and d.

§m

We can assume f is differentiable and choose a regular value a E §m, which by a diffeotopy (11.6.5, p. 82) of the sphere we move to the north pole. Then a has 2r + d inverse images, so that (i) at d + r of the images the sign of

(ii) at

r of the images the sign of

f is positive (resp., negative),

f is negative (resp., positive).

Then, choose 2r + d points PI, qI, ... ,Pr. qr, CI, ... ,Cd E U, and use a diffeotopy to move the points in f- l (a) as follows (we do nothing if there are no points to move):

(1) r of them at which f is negative (resp., positive) to ql,· .. ,qr, (2) some r at which f is positive (resp., negative) to PI,··· ,Pr, (3) the remaining d of them at which f is positive (resp., negative) to cI,···, Cd·

Using the chosen coordinates in U, we identify U == ~m, and our points qk,Pk, Ci, are points in ~m. Then we use a diffeotopy of ~m which is the identity off a big enough ball (II.6.6, p. 83) to move those points so that

(1) qk = (-3k, 0, ... ,0), (2) Pk = (3k, 0, ... ,0), (3) Ci = (3(s + i), 0, ... ,0), for some fixed s

~

r.

192

V. The HopE Theorems

Now, using the antipodal extension, linearization, and normalization (V.1.5, p. 187), we find a mapping J homotopic to j, which is == -a off the open balls of radius 1 around the points qk,Pk, Ci, and inside those balls it is, respectively, the antipodal extension of (1) the symmetry: (Xl, y) f---t (-Xl, y) + qk, (2) the translation: (XI,Y) f---t (XI,Y) - Pk, and (3) the translation: (Xl. y) f---t (Xl, y) - Ci· What is important here is that J(-XI,Y) = J(XI,Y) for IXII ~ 3r+ 1. This implies that the following homotopy is well defined for ~ t ~ 3r + 1:

°

H t ( Xl, Y) -_ {J(XI,Y) j(t, y)

for IXII ~ t, for IXII ~ t.

Note that this homotopy on]Rm == U c M extends to M, because H t == -a off -3r - 1 ~ Xl ~ 3(8 + d) + 1, IIYII ~ 1. We have Ho = J, and for t = 3r + 1: H t (X )

_ {J( x) -a

-

on the balls centered at the points Ci, off those balls.

This 9 = H t is the mapping we seek. Of course, it is homotopic to J, hence to j, and, in addition, the north pole a is a regular value of g, g-l(a) = {CI, ... ,Cd}, and 9 is the antipodal extension of the translation (Xl, y) f---t (Xl, y) - Ci on the unit ball around each Ci. Note that if d = 0, this is the constant mapping == -a. Clearly, this construction can be carried along simultaneously for two mappings of degree d taking enough points Pk, qk, and a common 8, and the final form is the same for both. 0

2. The Hopf Theorem: Brouwer-Kronecker degree

193

Corollary 2.2. Let M be a connected, compact, oriented, boundary less, differentiable manifold of dimension m. Then

Proof. This follows immediately from V.2.I, p. 191, and V.1.2, p. 185, beD cause 7rm (M) = [M, §m].

Remarks 2.3. (1) The condition that M is boundaryless is essential to have different homotopy classes. In fact, if M has a boundary 8M f:. 0, then all mappings M -+ §m are null-homotopic.

We only present a quick sketch of the argument. First one glues along their boundaries M and a copy - M with the opposite orientation, to get a new oriented, compact, boundaryless manifold M* = M u - M of the same dimension (some smoothing is required around 8M == 8( -M) by means of a bump function). Then every continuous mapping f : M -+ §m extends in an obvious unique way to f* : M* -+ §m, and this f* has degree zero (pick a close smooth approximation 9 of f* and look at a regular value of 9 off g(8M)). By the Hopf Theorem, the mapping f* is null-homotopic, and so is its restriction f = f*IM. (2) After the above example, it is natural to ask for a degree for manifolds with boundary. This can indeed be done, using the ideas behind Euclidean degree. If f : M -+ N is a proper mapping from an (m + 1)manifold with boundary into a connected, boundaryless, (m + I)-manifold, there is a degree d(f, M, a) for a E N \ f(8M). In this setting homotopies do not touch the boundary and the situation in (1) does not occur. (3) In case M is non-compact, there is no proper mapping M -+ §m, so that there is no notion of degree for a mapping f : M -+ §m. But still, one can ask about the homotopy type of f. The dramatic answer to this question is that f is always null-homotopic. Actually, the argument is a kind of infinite degree computation. We only sketch the idea. First, replace f by some close smooth mapping, which will be homotopic. Then, pick a regular value a E §m, and move it to the north pole. Replace f with a homotopic mapping which is == -a off a compact neighborhood of f-l(a) (the antipodal extension is still posible because f-l(a) is discrete). Pick a countable discrete family of coordinate domains in M far from that neighborhood, and reversing the method used in the proof of the Hopf Theorem to cancel points with opposite sign, create a pair of

V. The HopE Theorems

194

them on each such domain. From this we are sure that f- 1 (a) consists of an infinite sequence (Pk) of points at which the sign of f is positive and another sequence (qk) at which the sign is negative. Then we can move each pair Pk, qk into a coordinate domain far from the other points, using a diffeotopy that does not move anything else. Thus they cancel each other separately as in the proof of the Hopf Theorem, and in the end, f-l(a) is empty. Thus, f is not surjective, hence null-homotopic. (4) For non-orientable manifolds, the mod 2 Brouwer-Kronecker degree (III.6.1, p. 124) provides the following parallel version of the above Hopf Theorem, V.2.1, p. 191: Let M be a connected, compact, non-orientable, boundaryless, differentiable manifold of dimension m. Two continuous mappings M --t §m with the same mod 2 Brouwer-Kronecker degree are homotopic. In particular, we see that for such an M, 7rm(M)

= Z2.

The proof is no surprise: one can cancel pairs of points p, q in the inverse image of any regular value. Indeed, after some diffeotopy, non-orient ability gives two parametrizations
From the Hopf Theorem we easily deduce for spheres the converse to the Boundary Theorem, III.3.3, p. 111: Proposition 2.4. Let X be an oriented differentiable Cr manifold of dimension m + 1 with connected boundary ax = M, and let f : M --t §m be a proper (resp., proper Cr ) mapping of degree O. Then f has a continuous (resp., Cr ) extension J : X --t §m. Proof. As f is proper, M is compact. By V.2.1, p. 191, there is a homotopy == a E §m and HI = f. Then, the set Z = ({O} x X) U ([0, 1] x M) is closed in [0,1] x X, and we have the continuous mapping ) for t = 0, §m ( F:Z--t : t,x H Ht(x) forti-O.

H t : M --t §m with Ho

{a

Now, by the Tietze Extension Theorem, this extends to a continuous mapping F : [0,1] x X --t ]Rm+1. Consider the set A = F- 1 (]Rm+l \ {O}), which

195

2. The HopE Theorem: Brouwer-Kronecker degree

is an open neighborhood of Z in [0,1] x X. In particular, there is an open set U C X such that U ~ M and [0, 1] x U c A, and we choose a bump function 0 which is == off U and == 1 on M. We define a continuous mapping J : X -+ §m by

°

F(O(x), x) f(x) = IIF(O(x), x)II' This is well defined, because F(O(x),x) x¢. U and O(x) = 0, so that 0= F(O(x), x)

a contradiction. Furthermore so that JIM = HI = f.

=

°

implies (O(x), x) ¢. A; hence

= F(O, x) = a E §m,

J is an extension of f,

since O(x)

= 1 on M,

We have thus proved the assertion for proper mappings. For proper Cr mappings the argument is the same using the Cr Tietze Extension Theorem, 11.1.5, p. 52. The only requirement is to change the first definition: F: Z -+

§m

(

)

: t, x t---+

{aHTf(t) (x)

°

for t for t

= 0, i= 0,

k

where'T/ is a smooth bump function == for t ~ and == t for t 2 ~. This guarantees F is Cr , and the proof follows readily. D In fact the above proof is an adapted version of the general argument used to extend homotopies from closed subsets of a metric space to the whole space (the Borsuk Theorem). This then implies that the property that a continuous mapping on such a closed subset extends to the whole space only depends on the homotopy type of the mapping.

Exercises and problems Number 1. Let f, 9 : §m --+ are homotopic.

§m

be two continuous mappings. Prove that fog and go f

Number 2. Let f, 9 : §l --+ §1 be smooth mappings. Then the (complex) multiplication f . 9 : §1 --+ §1 is well defined, and deg(f . g)

= deg(f) + deg(g).

Prove this directly from the Hopf Theorem and compare with Problem Number 8 of 111.3. Deduce that if a complex polynomial P(z) has m roots (counted with multiplicities) inside the circle §l C ]R2 == C and none on it, then the mapping f : §l --+ §l : z I-t l~~:ll has degree m.

196

V. The HopE Theorems

Number 3. Use the preceding formula deg(J . g) = deg(J) + deg(g) to review Problem Number 4 of IlIA, noting that for the 9 and h there, h = Id 'g. Number 4. Let X be a space with involution for which s(X) = s'(X) = m < (Problem Number 6 of IV.5). Then 7rm (X) contains Z.

00

Number 5. Let M and N be smooth manifolds, N boundaryless and connected, and fix a point c EN. For every proper smooth mapping f : M --+ N such that f(8M) = {c} define deg(J)

=

'E

signx(J),

xEf-l(a)

where a t- c is a regular value of f. Show that this degree is well defined and that it is invariant by homotopies H t such that H t (8M) = {c}.

Number 6. Two spaces X and Y have the same homotopy type if there are continuous mappings f : X --+ Y and 9 : Y --+ X such that go f and fog are homotopic to Idx and Idy. Prove the following: (1) Two spheres of different dimensions do not have the same homotopy type. (2) A sphere does not have the same homotopy type as the product of two others. Number 7. Let N C Me RP be differentiable manifolds, N closed in M. Let X be a topological space and let H : [0, 1] x N --+ X be a continuous homotopy. Suppose that f = H 0 : N --+ X extends to a continuous mapping ! : M --+ X. Then H extends to a continuous homotopy H: [0,1] x M --+ X such that! = Ho. Number 8. Let X be a metric space, C C X a closed subset, and H : [0,1] x C --+ M a continuous mapping into a differentiable manifold M. Suppose that f = Ho : C --+ M extends to a continuous mapping! : X --+ M. Then H extends to a continuous homotopy H: [0,1] x X --+ M such that! = Ho.

3. The Hopf Theorem: Euclidean degree In this section we prove the suitable version of the Hopf Theorem, V.2.1, p. 191, in the Euclidean degree setting: Theorem 3.1 (Hopf Theorem). Let X C ~m+1 be a compact set that disconnects ~m+1 into two connected components, DI bounded and Do unbounded, whose topological boundary is X. Let j, g : X --+ ~m+1 \ {O} be two continuous mappings with the same winding number: w(j,O) = w(g, 0). Then j and g are homotopic. Proof. The proof is a recollection of various results already at hand. After a translation, 0 E D 1 , and we choose an open ball B of radius p containing D 1 , whose boundary (the sphere of radius p) we denote by M = B \ B. Now recall that

w(j,O) = d(J, Db 0)

and

w(g,O) = d(g, D 1 , 0),

3. Tile HopE Tileorem: Euclidean degree

197

where J, 9 are arbitrary continuous extensions of f, 9 to Dl = D1 U X, which we can further extend to B, without zeros in M (Tietze Extension Theorem). By the same definition of degree (IV.2.1, p. 145, and IV.2.2, p. 146), we can assume that J and 9 are smooth on B and that 0 is a regular value for both on B. Actually, J and 9 are polynomials; hence they are defined on the whole Euclidean space ~m+1. Let us now concentrate on f. The discrete set J- 1 (0) n B is finite, and its points split into those in D1 and those in Do n B (note that 0 ~ f(X) U f(M)): say C1, ••. , Cr E DI, bI, ... , bs E Do n B. Now pick an open ball B' C D1 centered at 0 E D 1. Since D1 is connected, we can move the cj's into B' by a diffeotopy of ~m+1 that fixes X and all other zeros of J; then, we pick a ball B" :J B, so that J has no zeros in B" \ B, and since D1 n B" is connected, we move the bi'S into B" \ B by a diffeotopy of ~m+1 that fixes X and all other zeros of J (II.6.6, p. 83). o o o

-

1

(

.0

~J(X)

We have moved the points so that 0 has the same preimages in Band D1; hence d(J, D1 , 0) = d(J, B, 0). Furthermore the mapping

1: u = B \ B' -+ §m : x H ~(x)

IIf(x)11

is well defined. Now, by IV.4.5, p. 157, the Euclidean degree d(J, B, 0) is the Brouwer-Kronecker degree of the smooth mapping M; that is,

11

v.

198

The HopE Theorems

The same construction for 9 gives a smooth mapping

'9: u =

B \ B' -+

§m :

x

I-t

g(x) Ilg(x)11

with

d(g, Db 0) = deg('9IM).

Con~quently deg(fiM) = deg('9IM), and by the Hopf Theorem, V.2.1, p. 191, flM and '9IM are homotopic. It follows immediately that the two mappings

f* : X -+ §m

:

x

I-t

f(px/llxID

and

g*: X

are also homotopic or we can simply say that

-+

§m :

x

f* : X -+

I-t

'9(px/llxll)

jRm+l \

{O} and f* and

g* : X -+ jRm+l \ {O} are homotopic. Thus, it remains to show that g* are, respectively, homotopic to f and g.

We write the proof for f in two steps. First, since we can consider the homotopy

Ht(x)

=

f(tx

+ (1 -

t)px/llxll)

i= 0,

1is defined on B \ B', x

E X

(every segment joining x E X to px/llxll EM is fully contained in B \ B'). Clearly Ho = f* and HI = fix. Second, we show that mapping

fiX is homotopic to f. To that end, consider the

Ft(x) =

/(x) _ (1 - t) + tllf(x)11

If the denominator vanishes at some x E X, then t = 1 and /(x) = 0, which is impossible. Also, if Ft(x) = 0 for some x E X, then /(x) = 0, again impossible. Hence, Ft is a well-defined homotopy with Fo = fix = f and Fl = fix.

For 9 the argument is the same, and the proof of the theorem is thus finished. D As in the preceding section, this completely describes the cohomotopy group: Corollary 3.2. Let X C jRm+l be a compact set that disconnects jRm+l into two connected components whose topological boundary is X. Then

3. The HopE Theorem: Euclidean degree

199

Proof. The group 7l'm(x) classifies by homotopy the mappings X -+ §m. Using the inclusion §m C lR. m+1 and its canonical retraction lR.m+1 \ {O} -+ §m : x t-+ x/llxll, one sees immediately that 7l'm(x) = [X, lR. m+1 \ {O}l. Then, by the latter Hopf Theorem, V.3.1, p. 196, the winding number gives an inclusion [X, lR.m+1 \ {O}l c Z, and to conclude, it is enough to exhibit mappings f : X -+ lR. m+ 1 \ {O} of arbitrary degree. This follows easily by the technique used in the preceding proof. Let Dl be the bounded component of lR. m+1 \ X, and pick any big ball B ::J X U D 1 • Let M denote the sphere that bounds B and choose a smooth mapping h : M -+ §m C lR. m+1 of Brouwer-Kronecker degree deg(h) = d (V.1.1, p. 183); by IV.4.6, p. 158,

d

= deg(h) = w(h, 0).

Let h : B -+ lR. m+1 be a smooth extension of h. Then d = w(h,O) = d(h, B, 0) is computed through the preimages of any regular value a of h. Now, those preimages off Dl can be moved inside by a diffeotopy that leaves kf invariant. Consequently, we can assume all preimages are in D 1 , so that

d

= d(h, B, 0) = d(h, D 1 , 0)

= w(hlx, 0).

D

We are done.

Also, we can deduce the converse of the Boundary Theorem for winding numbers (IV.4.4, p. 157):

Proposition 3.3. Let f : X -+ lR.m+1 \ {O} be a continuous mapping with winding number w(j,O) = O. Then f has a continuous extension J: D -+ lR.m+1 \ {O}. The proof is a copy of that of V.2.4, p. 194, using V.3.1, p. 196, instead of V.2.1, p. 191. We leave it to the reader.

Exercises and problems Number 1. We have shown, by a combination of various results, that the Hopf Theorem for the Euclidean degree (V.3.1, p. 196) follows from the Hopf Theorem for the BrouwerKronecker degree (V.2.1, p. 191). Show that to the contrary, the converse implication is quite immediate for hypersurfaces. Number 2. Let h : IR -+ IR be a periodic continuous mapping of the form

hex + 2rr) = hex) + 2krr,

for a fixed integer k.

V. The HopE Theorems

200

Compute the winding number induced by h. Number 3. Let

Then identify

I : sm

sm-l

==

--t

w(j,O)

IRm+1

sm-l X

\

of the continuous mapping

I : Sl

--t

IR2 \ {O}

{O} be a continuous mapping such that

{O} and IR m == IRm x {O}, so that

9 = Ilsm-1 : sm-l --t IR m\ {O}. Prove the following: (1) w(g,O) = w(j, 0). (2) The suspension of glllgil (Problem Number 4 of IV.6)

I restricts to a mapping

is homotopic to

1/11111.

Number 4. Let X C R2 be the curve of Problem Number 50fIV.I. Show that although X is not homeomorphic to a circle, it verifies the hypotheses of V.3.1, p. 196, and find simple representatives for all homotopy classes in [X,IR 2 \ {O}l.

4. The Hopf fibration The Hopf Theorems in the preceding sections show how the degree fully determines the cohomotopy groups [M, §m] = Z when dim(M) = m. We also know that [M, §m] = {O} when dim(M) < m (V.1.4, p. 186). Thus the problem is the case dim(M) > m, in particular for spheres M = §k. However, the degree can still be used to obtain important information concerning mappings §2m-1 --+ sm. The tool is the Hopf invariant introduced in III.5.1, p. 119. The additional key property of this invariant that we give here follows. Proposition 4.1. Let 9 : §2m-1 --+ §2m-1 and f : §2m-1 --+ mappings. Then H(f 0 g) = H(f) deg(g).

§m

be smooth

Proof. We can assume m even, because otherwise H == O. Since the Hopf invariant and the degree depend only on the homotopy class, we can at any moment change every mapping by any other one homotopic to it. This said, we distinguish several cases. Case I: deg(g) = O. Then 9 is null-homotopic, and we can replace it with a constant function, so that fog is also constant, and the assertion is trivial. Case II: deg(g) = 1. Then 9 is homotopic to the identity, and for the identity the result is evident. Case III: deg(g) = -1. We can assume that 9 is the linear symmetry with respect to the hyperplane Xl = 0, which is compatible with the stereographic projection from the south pole, and then 9 = g-l. Pick regular

4. Tbe Hopi fibration

201

values a, b to compute H(f) and H(f gram

0

g). We have the commutative dia-

f-l(a) x f-l(b) gxg

1~

because 9 ¢(g(y)) - ¢(g(x)) ( ¢(y) - ¢(x) ) ¢a,b(g(x),g(y)) = 11¢(g(y)) - ¢(g(x))11 = 9 1I¢(y) - ¢(x) II .

Consequently,

H(f 0 g) deg(g x g) = deg(g)H(f). Here, we know that deg(g) = -1 (111.1.6(2), p. 100) and deg(g x g) Thus, we conclude deg(g)H(f) = H(f 0 g).

= +1.

Case IV: deg(g) = d > O. Replacing 9 with a homotopic mapping of the type obtained in the proof of V.2.1, p. 191, we can assume that: (1) The north pole p has a neighborhood V such that g-l(V) is a disjoint union of d open sets Ui and on each of them 9 is a diffeomorphism that preserves orientations.

(2) There are open sets Vi :J Ui , which do not meet the south pole, and after projection from the south pole, every two of them can be separated by a hyperplane. (3) Off the

Vi's everything goes to the south pole.

We denote by gi the mapping that coincides with 9 on Vi and maps everything else to the south pole. By construction, deg(gi) = +1. Now, let a, b i= f( -p) be regular values of f. By a diffeotopy of §2m-l that collapses everything to the north pole (11.6.3, p. 81), we can assume f-l(a) and f-l(b) are contained in V, and so a, b are also regular values of fog, both distinct from f(g( -p)) = f( -p). In this situation we have

(f 0 g)-l(a) = Ug;l(f-l(a)),

(f 0 g)-l(b) = Ug;l(f-l(b)),

i

i

so that

H(f 0 g) = £((f 0 g)-l(a), (f 0 g)-l(b))

= L£((fo9i)-1(a),(fogj)-1(b)). i,j

v.

202

The HopE Tlleorems

We look first at a pair i =I- j. Since the two inverse images involved are projected into Ui and Uj, respectively, there is a hyperplane L that separates those projections. Hence no line joining a point in one to a point in the other can be parallel to L, and the corresponding map into §2m-2 is not surjective and hence has degree o. Thus we are left with

H (f 0 g)

=

L e((f

0

gd -1 (a), (f 0 gi) -1 (b))

=

L H (f

0

gd.

i

i

As deg(gi) = +1, by Case II we know that H(f 0 gi) = H(f), and we conclude that H(f 0 g) = H(f) . d = H(f) deg(g). Case V: deg(g) = d < O. This runs like Case IV, using Case III instead of Case II, with the only modification that 9 reserves orientations on Ui and consequently deg(gi) = -1. 0

By the preceding proposition, given f : §2m-1 -+ §m with H(f) =I- 0, its compositions with mappings 9 : §2m-1 -+ §2m-1 of different degrees give non-homotopic mappings fog: §2m-1 -+ §m. Thus we get an injection Z = '7r2m-1 (§2m-1)

-+ '7r2m-1 (§m).

Clearly, the best 1's for this construction seem to be those with H(f) = ±1. The classical method to define them follows.

(4.2) Hopf fibrations. Let F : ]Rm x ]Rm -+ ]Rm be the multiplication of (i) complex numbers for m = 2, (ii) quaternions for m = 4, and (iii) oct onions for m = 8, and define the Hopf fibration associated to F by

f : §2m-l -+ §m : (x, y)

t--t (11x11 2 - IIYI12, 2F(x, y)).

Thus we obtain smooth mappings

whose Hopf invariant is 1. The fibers of these mappings are, respectively, §l, §3, and §7; that is, the three mappings are fibrations of spheres by spheres. We will content ourselves with the proof of this for m = 2, the first case settled by Hopf in 1931 (see 1.3, p. 30).

Proposition 4.3. The Hopf invariant of the Hopf fibration Proof. The explicit equations of this mapping are

§3

-+ §2 is 1.

203

4. The Hop£ fibration

An easy computation shows that the poles of §2, a = (0, 0, -1) and b = (0,0,1) (both =I- (-1,0,0) = j(O,O,O,-l)), are two regular values with inverse images:

{

Ca = j-1(a) = Cb

=

j-1(b)

§3

n {Y1 + X2 = Y2 + Xl = O},

= §3 n {Y1

- X2

= Y2

By projection from the south pole we get in

E . a'

{v(u-1)2+2v2=2, + w 0, =

Eb . .

]R3

- Xl

= O}.

two ellipses Ea and Eb:

{v(u+1)2+2v - w 0, 2 =2. =

They are depicted below, with the orientations (a and (b that they carry as inverse images. (3

Let us describe (a by means of II.7.6, p. 89. We localize j using (i) the projection from the south pole in §3 and (ii) the projection from the north pole in §2. Since both projections reverse orientation (II.7.5(3), p. 88), we get the right orientation in the ellipse Ea, which for this localization has the equations {

91 : t(4u 2 + 4v 2 - ~w2 - t 2) = 0, 92 - is (8uw - 4vt) - 0,

where {

t=1-U 2 - V 2 -W 2, 8 = (2 - t)2 - 8vw - 4ut.

The denominator 8 is important (at least its sign), because it affects the derivative of the localization 9 = (91,92) of f. Let VI and V2 be the gradients of 91 and 92, respectively. Then the inverse image is oriented by

204

V. The HopE Theorems

the vector product 111 x 112. Indeed, that vector product is perpendicular to the gradients, hence tangent to the inverse image, and of course det(lII, 112, 111 X 112) > 0. In addition, dpg maps 111,112 onto a positive basis in JR2 :

(Cauchy-Schwartz). With this settled, as orientation is determined by one single point, we compute 111 and 112 for an easy one, namely (1 + y'2, 0, 0). For this point, u > 0, v = W = 0, t = -2u < 0, 6> 0, and we have

{Ill: 112 -

t(8U(1- u)~,o) = (-,0,0), ;S(0,8u,8u) - (0,+,+),

so that the orientation (a is that of

as in the figure. For (b the argument and computations are most similar, but in this case we localize using in §2 the projection from the south pole, which preserves orientation. Thus 111 x 112 does not give the right orientation, and we must take the opposite -111 x 112. The details are left to the reader. Finally, we have to compute the degree of the mapping ~ : Ea x Eb

-7

lC'2 0)

( :

)

p, q

r-+

q- p

Ilq _ pil .

Now, the derivative of this mapping at (p, q) is quite simple: it is the linear mapping (a, {3) r-+ {3 - a, followed by the orthogonal projection onto the plane perpendicular to q - p, plus a scaling by the factor 1/ II q - pll. Then we pick p,q and a,{3 as in the figure. It is clear that ~-l(~(p,q)) = {(p,q)}; hence we only must check that the two vectors d(p,q)~(a, 0) = a',

d(p,q)~(O, {3) = {3'

form a positive basis of Tcp(p,q)§2. This last linear space is the plane perpendicular to q - p, and a', {3' form a positive basis if and only if det(q - p, a', {3') > 0. But this should be clear from the picture!

D

205

4. The Hopi fibration

As mentioned before, once we have this, composition with the Hopf fibration gives an inclusion Z C 7r3(§2), which shows that 7r3(§2) is infinite. As a matter of fact, it can be proved that the above inclusion is an equality; hence 7r3(§2) = Z. This is the first non-zero homotopy group of a sphere (excluding of course 7rm(§m) = Z), because 7rk(§I) = 0 for k > l. Concerning the other two Hopf fibrations, it is also true that their Hopf invariant is 1; hence we have injections 7r2m_I(§2m-l) --+ 7r2m_I(§m), and so Z C 7r2m_I(§m), but no more: it is known that

Furthermore, except for m = 2, the Hopf invariant does not determine the homotopy class: there are non-homotopic mappings §7 --+ §4 (and §15 --+ §8) with the same Hopf invariant. On the other hand, not every integer is a Hopf invariant for arbitrary even m (see 111.5.2(4)): even Hopf invariants always occur, but odd ones occur only for the crucial dimensions m = 2,4,8. All of this enters into the wide open problem of computing homotopy groups of spheres, a beautiful topic which stands far beyond our reach here.

Exercises and problems Number 1. View the unit sphere §2m+1 in C m + 1 as defined by ZlZl + .. ·+Zm+lZm+l = 1. The mapping h: §2m+l -t ClP'm : Z = (Zl, ... ,Zl) -t (Zl : ... : Zm+l) is also called Hopi fibration. Explain this name for m = 1. Use the fact that 7l"2m+l (§l) = 0 to show that two continuous mappings I, g : §2m+1 -t §2m+ 1 such that hoi = hog must have the same degree. Compute the degree of all a : §2m+1 -t §2m+l such that h(a(z)) = h(z). Compare this with Problem Number 7 of 111.1. Number 2. Let that

I: §2m-l -t §m and g : §m -t §m be smooth mappings (m ~ 2). H(g 0 f)

Show

= deg(g/ H(f).

Number 3. Let I : §2m-l -t §m be a continuous mapping with non-zero Hopf invariant (not necessarily 1). Let 9 : §2m-l -t §2m-l be a continuous lifting of g : §m -t §m (that is, log = go f). Show that deg(g) = deg(g?, and deduce that few continuous mappings are liftings. Deduce also that liftings of homotopic mappings are homotopic. What about the other way around? Number 4. Let g : §2m-l -t §2m-l and I: §2m-l -t §m be smooth mappings (m Show that fl(f 0 g) = fl(f) deg(g), where fl is the invariant defined in Problem Number 1 of 111.5.

~

2).

v.

206

Number 5. Let f : §2m-l --+ the formula

§m

and 9 : §m --+

§m

The HopE Theorems

be smooth mappings (m ~ 2). Prove

Number 6. Compute the invariant (l(f) of the Hopf fibration f : §3 --+

§2.

5. Singularities of tangent vector fields In this section we describe the information borne out by tangent vector fields on a manifold. That information is disclosed by the computation of the so-called index of a tangent vector field. Let M sion m.

c

RP be an oriented, boundaryless, smooth manifold of dimen-

(5.1) Non-degenerate zeros of a tangent vector field. A (smooth) tangent vector field on M is a smooth mapping ~ : M -+ RP such that ~(z) E TzM for every z E M. We can represent ~ in local coordinates as follows. Let 'P : U -+ M be a parametrization of M. For x E U set z = 'P(x) E M. The tangent space TzM is generated by the basis of partial derivatives a a'P (x), 1 ~ i ~ m. -a = dx'P(ei) = -a Xi z Xi Hence we can write

I

where the functions ~i : U -+ R are uniquely determined and smooth. We will also consider the smooth mapping ~: U -+ R m

Now suppose z all i. Then

= 'P(x) is a

:

x

H

(6(x), ... ,~m(x)).

zero (or singularity) of ~, that is, ~i(X)

= ~ a~i (x) a'P (x) = ~ a~i (x)~1 ~ ax'J i=l

ax'~

~ ax'J i=l

ax'~ z

.

= 0 for

207

5. Singularities of tangent vector fields

Thus, we actually have a linear mapping dz~ : TzM --* TzM c '\R'P, whose matrix with respect to the basis of partial derivatives is the Jacobian matrix Jx1. of the localization f.. In particular, the determinant of dz~ is a welldefined invariant of ~:

-= (a~i -a (x) ).. '

Jx~

Xj

~,J

We will say that the zero z of ~ is non-degenerate when this determinant is not zero. This is equivalent to saying that the localization 1. : U -+ lR.m is a local diffeomorphism at x, hence locally injective, so that our zero z is isolated. If z is non-degenerate, the sign ±1 of det(dz~) i= 0 is called the index of ~ at z and is denoted by

Examples 5.2. Below there are some basic examples of vector fields on the plane, with a single non-degenerate zero, and the corresponding index. What is depicted is the flow generated by the field, which consists of the integral lines whose tangents are prescribed by the field. They come from an associated differential equation, and the investigation of their nature is a highly interesting topic beyond our purposes here. Sink (or attractor)

Source (or impulsor)

~ = -xtx - yty

'

- (-1 0)

d(o,o)~ =

Saddle

~ = x tx - y t y

1 - (1 d(o.o)~ =

indo ~

'

~ = (x, -y),

0)

0 -1 '

= signo ~ = -1

0 -1 '

~ = (-x,-y),

208

V. The HopE Theorems

Circulation (or center)

o But we want to deal with arbitrary isolated zeros, not only the non-degenerate. This is achieved as follows. Proposition and Definition 5.3 (Splitting of an isolated zero). Let ~ : M -+ IRP be a smooth tangent vector field. Let il be an open set where ~ has a unique zero z. Then there are an open neighborhood U of z, with U c il, and a smooth tangent vector field ( : M -+ IRP such that we have the following:

(1) U is a smooth compact manifold with boundary Z = U \ U. (2) (:= ~ off U. (3) ( has finitely many zeros in il, all of them in U, and all of them non-degenerate. We say that ( is a splitting of ~ at z. Proof. The question is clearly local; hence we can assume il = IRm , ~ = ~, and z = O. Then, pick a regular value a of ~ with Iiall < 11~(x)11 for 1 ::; Ilxll ::; 2. Also, let () we a smooth bump function with 0 ::; () ::; 1, () == 1 on Ilxll ::; 1, and () == 0 on Ilxll ~ 2. Then set ( = ~ - a(). We have:

(i)

(== ~ off U : Ilxll < 2 and hence has no zeros off U.

(ii) II~II >

lIall

~

Ila(}11 on 1 ::; Ilxll ::; 2; hence (

has no zeros off V :

IIxll <

1.

(iii) (= ~ -a on V; hence the zeros of (in V are the points in Vn~-l(a). As a is a regular value of ~, those zeros are finitely many and nondegenerate. Finally, U is a closed ball in IRm , hence a compact manifold with boundary the sphere Z : U \ U: Ilxll = 2. 0

5. Singularities of tangent vector fields

209

This splitting is the key step to defining the index at isolated degenerate zeros: Proposition and Definition 5.4. Let € : M -+ ffi.P be a smooth tangent vector field, and let z E M be an isolated zero of €. Let ( be a splitting of € at z in the domain n corresponding to a parametrization tp. Then

where the sum ranges over all (non-degenerate) zeros z* of ( and S is a small enough sphere centered at x = tp-l(z). Hence, this sum does not depend on either the splitting or the parametrization. This sum is the index of € at z. We write

z·

If € has finitely many zeros, then all of them are isolated, and the sum of the indices of € at the zeros is the total index of €. We write

z

(the sum is zero if the field has no zeros). Proof. Consider the open neighborhood U c n associated to the splitting, and its boundary Z = U \ U. Set X = tp-l(Z) and x* = tp-l(z*), so that

L ind z·

z• (()

=

L sign

x • (()

x·

= deg(

1I~lIlx)

= deg( II~II = deg(

Ix)

II~IIIJ

(by definition of index at a non-degenerate zero) (by IV.4.5, p. 157) (because € ==

( on

Z)

(by the Boundary Theorem, 111.3.3, p. 111)

Now, the right-hand side does not depend on (, and the left-hand side does not depend on tp; hence none depends on either ( or tp. 0

V. The HopE Theorems

210

Examples 5.5. Here we give two examples of degenerate isolated zeros with indices ±2: ~

-2xy JL ax

+ (x 2 -

f =

-L =

- 2xy, x

(;=

Di ole p

1

-

II~II

indo ~

(

y2)JL ay' 2

2

- y ) x 2 + y2 '

= deg(flllxll=/O) =

+2

o Exercises and problems Number 1. Show that the only possible indices at an isolated zero of a tangent vector field on a curve are 0, +1, and -l. Number 2. Produce smooth tangent vector fields on the plane of arbitrary indices.

jR2

with isolated zeros

Number 3. Let M be a compact, boundaryless, smooth manifold of dimension m.. Suppose M is parallelizable, that is, there are tangent vector fields 6, ... , €m such that {€I ,x, ... , €m,x} is a basis of Tx M at each point x E AI; this defines an orientation on PvI, Show that: (1) Each (smooth) tangent vector field € on M is defined by € = L::l J;€i, where every Ii : AI -+ jR is a smooth function. (2) A zero of € is a zero of the mapping 1= (/l, ... ,lm) : M -+ jRm, and it is non-degenerate if and only if it is a regular point of I. (3) If € has only non-degenerate zeros, then it has finitely many, and Ind(€) = deg(f). Conclude that on a parallelizable compact manifold, all tangent fields with isolated zeros have total index zero. Number 4. Let € be a smooth tangent vector field on a smooth manifold AI, and let I: M -+ jR be a smooth function. Consider the tangent vector field (. = I(z)€ •. Study the zeros of ( in terms of those of €. When are they isolated? When non-degenerate? Can something be said concerning indices in the latter case? Number 5. Use the dipole (V.5.5 above) and stereographic projection to obtain a tangent vector field on §2 with a unique zero of index +2. Number 6. Consider the vector field on 28

jR3

28

defined by 2

€ = xz 8x + yz 8y + z(z -

8 1) 8z'

6. Gradient vector fields

211

and let Melle be the surface x 2 +y2 = Z2 -1, Z > O. Show that erestricts to a tangent vector field on M with a unique zero, and compute its index.

e

Number 7. Let (resp., () be a tangent vector field on a manifold M (resp., N) with an isolated zero x E M (resp., YEN). Prove that ex ( is a tangent vector field on M x N, that z = (x, y) is an isolated zero of ex (, and that

6. Gradient vector fields We will discuss here a particular case of tangent vector fields and compute their indices. We start with the following:

Proposition and Definition 6.1. Let Me lRP be a boundary less smooth manifold of dimension m, and let f : M -+ lR be a smooth function. Then there is a unique smooth tangent vector field ~ such that dzf(u)

=

(u, ~z)

for all u E TzM,

Z

E M,

where (., .) stands for the Euclidean scalar product in lRP . This vector field

~

is called the gradient of f and is denoted grad(J).

Of course, if M is an open set of lRP , then grad(J) =

(It, ... ,/!; ).

Proof. First recall that for every Z E M the mapping v f-t (., v) is a linear isomorphism from TzM onto its dual space C(TzM, lR) (this is the Riesz Representation Theorem). Consequently, ~ is uniquely defined by the condition in the statement, and we must check that it is indeed a smooth tangent vector field, which is a local matter. Thus, consider any parametrization 'P of an open neighborhood U of z E M and the corresponding localization

and let us show that the

~k 's

are smooth functions. We have

V. The HopE Theorems

212

Thus we obtain a linear system with unknowns the ~k 's, and the matrix Gxtp of this system is the Euclidean Gramm matrix of the basis of partial derivatives, which is the matrix of the Euclidean scalar product with respect to that basis. Clearly such a Gramm matrix is positive definite and in particular has determinant > O. Consequently, we can solve the system and find smooth expresions for the ~k 'so We are done. 0 Note that the critical points of f : M -+ lR are exactly the zeros of its gradient (and this is why zeros of tangent vector fields are called singularities). We want to determine those which are non-degenerate zeros (V.5.1, p. 206). Such a non-degenerate zero is called a non-degenerate critical point of f. (6.2) The Jacobian of a gradient vector field at a zero. Let f : M -+ lR be a smooth function as above, and set ~ = grad(f). Let z be a critical point of f, that is, a zero of ~. We consider a parametrization tp of an open neighborhood U of z with, say, z = tp(x), and as in the preceding proof we have

We are interested in the following Jacobian determinant: Jx~-

= (8~i -8 (x) ) ... Xj ~,J

Let us derive the formulas above:

8 2 (f 0 tp) = '""' 8(B£, l!;) 8x·8x· ~ 8x'J J~ k Since z

~k + '""' / 8tp,

8tp )

8~k .

~ k \8x·~ 8Xk 8x'J

= tp(x) is a zero of ~, the first sum vanishes at x, and so 8 2 (f 0 tp) (x) = '""' / 8tp (x), 8tp (x)) 8~k (x). 8Xj 8Xi 8Xi 8Xk 8xj

7\

Again we find the Gramm matrix Gxtp, and introducing the symmetric matrix

Hx(f 0 tp) = (8 2 (f 0 tp) (x)) . . ' 8xj8xi ~,J we rewrite the sum in matricial form as follows:

213

6. Gradient vector fields

Since det( Gx
> 0, this completely determines the matrix dz~.

Jx~, hence the

Furthermore, we get

indz(~)

= signdet(dz~) = signdet (Hx(f 0
Here we include the case ind = sign = 0, meaning that z is a degenerate zero of ~, that is, z is a degenerate critical point of f.

In the previous paragraph we encountered the Hessian of the localization f 0 <po In fact, this is an intrinsic notion: Proposition and Definition 6.3. In the setting above and with all of the notation there, consider on TzM the quadratic form Qz(f) whose matrix z} is Hx (f 0
{Ixt I

= (f 0 ,),)"(0).

Qz(f)(u)

In particular, Qz(f) does not depend on the parametrization <po This quadratic form is the Hessian of f at Z. Proof. This is a straightforward computation. Let (U1,'" ,urn) be the coordinates of u with respect to the basis of the partial derivatives. Then 2

Qz(f)(u)

" 0 (f 0
J

t

Now, set

= b1,' .. ,')'m), so that the above coordinates are Ui = ')'~(O). We will compute (f 0,),)"(0). 1 =
0')'

First,

(f 0 ,),)'(t) = ((f

0


01)'

(t) =

L o(;;.
t

and then

(f 0 ,),)"(t) For t

= '" 02(f 0
= 0 we obtain (f

0 ')'

)"() 0

",02(fo
= L..J ax .ox. x ij

J

Uj Ui

t

(the second sum vanishes because x = 1(0) is a zero of all the partial °If! ) ). 0 derivatives 8U 8Xi

214

V. The Hopi Theorems

The Hessian is used to define the index of a critical point: Definition 6.4. Let f : M -+ lR be a smooth function and let z E M be a non-degenerate critical point of f. The index (= the number of negative eigenvalues) of the Hessian Q z (f) is called the index of f at z and is denoted by indz(f). With this we can compute indices of gradient vector fields: Proposition 6.5. Let f : M -+ lR be a smooth function and let grad (f) . Let z E M be a non-degenerate critical point of f. Then

~

Proof. Keeping all preceding notation, we already know that indz(~)

= signdet (Hx(f 0 cp)),

where Hx(f ocp) is the symmetric matrix of Qz(f) with respect to a suitable base. Recall now that the determinant of a matrix is the product of its eigenvalues. As indx (f) is precisely the number of those that are negative, the assertion follows. D

(6.6) Morse functions. A Morse function f : M -+ lR is a smooth function whose critical points are all non-degenerate, equivalently, such that all the zeros of its gradient vector field ~ = grad(f) are non-degenerate. Thus, they are isolated, and in case there are finitely many (for instance, if M is compact), the total index of ~ is

z

where the z's are the critical points of

f.

This can be rewritten as

Ind(~) = ~) -l)kak' k

where ak is the number of critical points with index k. Of course, the question is whether or not Morse functions exist. The answer is that they do. In fact, we will see that many a linear form is a Morse function. We formulate this as follows.

6. Gradient vector fields

215

(6.7) Height functions. Let M c lRP be a boundaryless smooth manifold of dimension m. For every a E lRP consider the smooth function fa: M --+ lR:

X t-+

(x,a).

This can be seen as a height function with respect to the linear hyperplane H perpendicular to a: the oriented distance to H is given by dist(x, H)

= faIT fa(x).

Also, note that after a linear change of coordinates in lRP , we have a = (0, ... ,0,1), and then fa(x) = xp. Finally, denote by M the set of all a E lRP such that fa is a Morse function. The set M is quite explicitly determined for hypersurfaces: Proposition 6.S. Let M c lRm+1 be a connected, closed, boundaryless, smooth hypersurface, and let 1/ : M --+ §m be a unitary global normal field. Let a E lRm+l. Then fa : M --+ lR : X t-+ (X, a)

is a Morse function if and only if and -1/.

faITa

E §m is a regular value of both

1/

In particular, the set M is a residual, hence dense, subset of lRm+l. Proof. Recall that 1/ exists by III. 6.4, p. 129. On the other hand, since f>..a = >'fa, we can assume II all = 1. Now, we look for the critical points z of f = fa. Clearly, dzf = (., a)ITzM vanishes if and only if a is perpendicular to TzM, that is, if and only if a = ±I/(z). Suppose this is true, and let us see when the critical point z is non-degenerate. We first claim that

Indeed, let, be a curve germ in M with ,(0) = z, ,'(0) = u. Then we derive (J 0 ,)(t) = (,(t), a) twice to get

Qz(J)(u)

=

(J 0 ,)"(0)

=

(,"(0), a).

But on the other hand, deriving (,'(t),I/(!(t»)

0= (,"(t) , I/(!(t»)

= 0, we see that

+ (,'(t) , dzl/(!'(t») ,

v. The HopE Theorems

216

which for t = 0 gives

b"(O), v(z)) = (u, -dzv(u)). But v(z) = ±a; hence b"(O), a) = (u, =r=dzv(u)) , and the claim follows. Now, the claim implies that the quadratic form Qz(J) has, with respect to any orthonormal basis of TzM, the same matrix as the self-adjoint endomorphism =r=dzv (III.2.6, p. 108), hence the same determinant. We have thus seen that z is non-degenerate if and only if it is not a critical point of ±v. From this, the result follows readily. The last assertion in the proposition is an easy consequence of the SardBrown Theorem. D For arbitrary manifolds one shows a slightly weaker result in a less explicit way: Proposition 6.9. Let M c 1l~.P be a boundaryless smooth manifold of dimension m. As above, let M be the set of all points a E ~p such that the mapping fa : M -t ~ : X H (X, a) is a Morse function. Then M is dense in

~p.

Proof. First we remark that every point in M has a neighborhood U on which some linear projection 1T : X H (Xii' ... ,Xim ) defines a local coordinate system. Indeed, dp1T = 1T must be injective on TpM for some choice of indices ik, and then the Inverse Function Theorem applies. Moreover, since M has a countable basis of open sets, it has a countable covering consisting of such open coordinate domains as U.

Now pick any U as above, and, to symplify notation, suppose the projection 1T corresponds to the indices ik = k; let X = (x', x") E ~p = ~m X ~p-m . We claim that the following statement holds true: Let a" E ~p-m, and denote by Mu(a") the set of all a' E ~m such that falu : U -t ~: x H (x', a') + (x", a") is a Morse function. Then the set ~m

\

Mu(a") has measure zero.

Assume this for the moment, and consider the set Mu of all a E ~p such that falu is a Morse function. What the claim says is that ~p \ Mu cuts every (p - m)-plane x" = a" in a measure zero set; hence, by the Fubini Theorem, ~p \ Mu itself has measure zero. Since there are

6. Gradient vector fields

217

countably many U's, the union Uu (~p \ Mu) also has measure zero, hence has empty interior, and its complement nu Mu is dense. As, obviously, M ::) nu Mu, we are done. To complete the argument, let us prove the claim. Since 7rlu is a diffeomorphism onto an open set W C ~m, its inverse has the form x' H (x', (x')), for a suitable smooth function : W -+ ~p-m. Then the localization of lalu is 9a: W -+ ~: x'

H

(x', a')

+ h(x'),

where h(x') = ((x'),a"),

and straight from the definition one sees that 9a is a Morse function if and only if -a' is a regular value of the gradient grad(h) : W -+ ~m. Consequently, the claim follows from the proof of the Sard-Brown Theorem 0 (see (*) on p. 64).

Exercises and problems Number 1. Let M c RP be a smooth manifold and let F : RP -t R be a smooth function; set f = FIM. Describe grad(f) in terms of grad(F) and orthogonal projections into the tangent spaces of M. Number 2. Is there any smooth function Number 3. Let that

f : §2

-t

R with gradient -y

f (resp., g) be a Morse function on a manifold h : M x N -t R: (x, y)

f-+

f(x)

:x + x :y ?

M (resp., N). Show

+ g(y)

is a Morse function on M x N, and describe its critical points, as well as their indices, in terms of those of f and g. Number 4. Let f and g be Morse functions on two compact manifolds M and N, respectively. Show that for any two positive real numbers a and b large enough, the function h : M x N -t R; (x, y) f-+ (a + f(x»(b + g(y» is a Morse function on M x N. Describe the critical points of h and their indices.

Number 5. Let M = 80(3) C R 3X3 be the rotation group in R 3 , consisting of all 3 x 3 matrices x = (Xij) whose columns form a positive orthonormal basis of R3. This M is a boundaryless, compact, smooth manifold of dimension 3. Show that

h(x) = 2xu

+ 3X22 + 4X33

is a Morse function on M and that it has four critical points with indices 0,1,2,3. Number 6. Study when the function f ; RlP'7n -t R : (xo : ... ; x m ) f-+ 11;11 2 a Morse function. Then, describe its critical points and their indices.

L:k

CkX~ is

V. The HopE Theorems

218

7. The Poincare-Hopf Index Theorem Now that the notion of total index has been described, we can turn in this last section to the main fact that the total index is an invariant of the manifold, independent of the tangent vector field used to compute it. To prove this, we must take a closer look at tubular neighborhoods.

(7.1) Normal vector fields on tubes. Let M c lRP be a compact boundaryless smooth manifold of dimension m. Then, M has a tubular retraction p : U ~ M (II.4.3, p. 71) such that x - p(x) is perpendicular to Tp(x)M. Consider the smooth function

r: U ~ lR:

X

t--+

IIx -

p(x)112.

As x - p(x) is perpendicular to dxp(lRP ) C Tp(x)M, it follows easily that gradx(r)

= 2(x - p(x)).

This gradient vanishes exactly for x = p(x), so that x E M and r(x) = O. Thus, any c > 0 is a regular value of r, so that N : r(x) ~ c is a smooth manifold with boundary aN : r(x) = c. We also know that gradx(r) is perpendicular to aN at x, and consequently gradx(r) 1 ) TJ(x = II gradx(r)II = yIc(x - p(x)) is a unitary normal vector field on

aN

(III.2.6, p. 108).

Now recall that (i) Ilx - p(x) II = dist(x, M) for every x E U and (ii) dist(M, lRP \ U) > O. Consequently, for c > 0 small enough N = {x E lRP

:

dist(x, M) ~

vE},

aN = {x E lRP

:

dist(x, M) =

vE}.

219

7. The Poincare-Hopi Index Theorem

Summing up, N is a compact manifold, and 'fJ is a normal vector field on its boundary aN, which we call the normal vector field on a tube around M. Using the above normal vector fields, we can state the main theorem: Theorem 7.2 (Poincare-Hopf Index Theorem). Let Me jRP be a compact oriented boundaryless smooth manifold, and let be a smooth tangent vector field with isolated zeros. Let'fJ: aN -+ §p- 1 be a normal vector field on a tube around M. Then Ind(e) = deg('fJ).

e

Proof. By V.5.3, p. 208, and V.5.4, p. 209, we can split all zeros of e to assume they are non-degenerate. Once this is the case, with the notation of the preceding paragraph, we define

f: N -+

jRP: X t-+ X -

p(x) +e(p(x)).

We claim that 0 is a regular value of f. Indeed, since x - p(x) and e(p(x)) are perpendicular, f(x) = 0 if and only if x = p(x) and e(p(x)) = O. This means that x E M is a zero a of e. Furthermore, given such a zero a, we have: (1) f

== eon

M; hence daflTa M

= dae.

(2) p == a on the orthogonal complement a + E of TaM; hence daflE IdE.

==

This means that daf = IdE El1dae, so that det(daf) = det(dae) =1= 0, since a is a non-degenerate zero of e. This shows that 0 is a regular value of f, its preimages are the zeros of e, and a

a

On the other hand, by IV.4.5, p. 157, deg

(1I~lIlaN)

= Lsigna(J); a

hence we only must show that deg('fJ) = deg (nfrrlaN)' But these two mappings from aN into §p- 1 are homotopic because they do not have antipodal images (V.1.4, p. 186). Indeed, if they had some, i.e., if

--X(x - p(x)) = x - p(x) then e(p(x))

+ e(p(x)),

with x E aN and -X

> 0,

= -(1 + -X)(x - p(x)). This is impossible for x ~ M.

0

v.

220

The HopE Theorems

Remark 7.3. If m is odd, this invariant has little significance. Indeed, first note that Ind(-e) = (-l)mInd(e) (this is immediate for the index at every non-degenerate zero and then follows for the total index). Thus, if m is odd, Ind(e) = o. 0 (7.4) Euler characteristic. The Poincare-Hopf Theorem also says that the total index is the Euler characteristic x(M) of M (hence = 0 if m is odd) and consequently depends only on the topological type of M: Ind(e)

= X(M).

However, to bring in X, we would need a rigorous definition, which is not our concern here. Instead we draw some pictures to see how X enters the scene. First notice that by the last theorem, it is enough to find a field e whose total index is indeed the Euler characteristic. We will describe some informal but very natural means to construct such a e. (1) Suppose we have a triangulation. Then we mark one point ak at each face of dimension k. These points will be the zeros of our field e, and all are non-degenerate. In fact, e can be defined on each face by a source at k a ak, that is, by the local form e = I:i=l Xi ax;. Of course, only on the faces of maximal dimension k = m do we get a true source. On the others the flow is incoming from higher-dimensional faces. Thus we start at the am '8 with sign +1, and the sign changes to -1 at the am-l's, and then again to +1 at the am -2 's, and so on. Thus we have Ind(e)

= Am - A m- 1 + Am-2 - ... + (_l)m Ao

where Ak stands for the number of faces of dimension k. If m is even, this sum is X, and if m is odd, it is -X, and we can change e to -e (although it does not matter much because in that case X = 0).

7. The Poincare-HopE Index Theorem

221

The preceding picture shows X in dimension 2: we get a source at every face, a saddle at every edge, and a sink at every vertex, adding up to the classical Euler formula X = F - E + V. (2) We can imagine the construction above in a physical way, as the flow of a liquid poured over the manifold from some vintage points. We can use this image to guess the Euler characteristic of a torus with 9 holes. The figure below shows this: sources have index +1, saddles -1, and sinks +1.

t

t

source: +1

sink: +1

E=O

E=-2

E=O

E=-2

When we add up all indices, the four zeros around each hole cancel each other, and each pair of zeros in between two consecutive holes gives -2. Since there are 9 holes, there are 9 - 1 of those pairs, and the final result is what it must be: X = Ind = Lind = - 2(g - 1) = 2 - 2g.

This 9 is the genus of the surface. The reader can try different forms of pouring the liquid and check that, of course, the result is always the same. (3) For the sphere §2 C IR3 (g = 0), we have just a source and a sink, which gives index 2 and X(§2) = 2 = 2 - 2g. Or we can simply use the field e(x, y, z) = (-y, x, 0) depicted after the Hedge-hog Theorem, III.7.4, p. 135: its two zeros are circulations and hence have index +1. Also, we understand now what that theorem says for arbitrary compact, boundaryless manifolds: if there is a tangent vector field e without zeros, then X = Ind(e) = o. (7.5) The Morse inequalities. This is a description of the Euler characteristic by means of gradient vector fields. Again let M be an oriented, com-

222

V. The HopE Theorems

pact, boundaryless, smooth manifold of dimension m, and let f : M --+ lR be a Morse function (which always exists; V.6.S, p. 215, and V.6.9, p. 216). We have already computed the total index of the gradient ~ = grad(f) of f: Ind(~) = -llok,

2) k

where Ok is the number of critical points of index k of Thus we get X(M) = ~)-l)kok'

f (V.6.6, p. 214).

k

This formula for the Euler characteristic in terms of the critical points of a Morse function is part of the so-called Morse inequalities (in fact this is the unique equality among them). Those inequalities are actually formulated in terms of the homology groups of M and escape our context. (7.6) The Gauss-Bonnet Formula. As we explained earlier (111.2.6, p. lOS), this is the computation of the integral curvature

for a compact hypersurface M C lRm +1 of even dimension m with Gauss mapping v. There we saw how the first equality comes from degree theory by integration, and what was left was the computation deg(v) = !X(M). But this follows from the Poincart3-Hopf Index Theorem, V.7.2, p.219, which for hypersurfaces of even dimension has a better form. Let us sketch the argument. First notice that as M is a hypersurface of dimension m, aN consists of two disjoint copies M+ and M_ of M: for every x E M there are exactly two points x+ and x_ at a distance V€ from x, and the signs can be chosen by the conditions

{

x+ = x + V€v(x), x_ = x - V€v(x),

hence 1J(x+) hence 1J(x_)

= v(x), = -v(x).

Thus p induces two diffeomorphisms M+, M_ --+ M with inverses described just above. In particular, the two derivatives of those inverses are

arbitrarily close to Id for

E

small enough.

223

7. The Poincare-HopI Index Theorem

All of this readily implies that p preserves orientations on M+, and it does not on M_. Finally, write

where 0- is the antipodal diffeomorphism of §m. Since the diffeomorphism 0- has degree +1 if m is odd and -1 if m is even (III.1.6(1), p. 99), we get deg('T]) =deg('T]IM+) + deg('T]IJ\,L) = deg(v 0 PIM+) + deg(o- 0 v

0

pIM_)

= deg(v)( +1) + (±1) deg(v)( -1) =

{o

2deg(v)

if m is odd, if m is even.

As we have already explained, deg('T]) = X(M), and for m odd we again get that X(M) = 0 and we get nothing about deg(v). But for m even we obtain X(M) = 2 deg(v), and the Gauss-Bonnet Formula follows.

Exercises and problems Number 1. Let §m C

]Rm+l

be a sphere of even dimension. Use the tangent vector field

to compute the Euler characteristic X(§m).

Number 2. Prove the following product formula for the Euler characterisitic

X(M x N) = X(M) . X(N). Number 3. Show that the torus T C ]R3 generated by the circle y = 0, (x - 2? +Z2 = 1 around the axis x = y = 0 is diffeomorphic to the product §l x §l C ]R4, and deduce that

v.

224

The HopE Theorems

its Euler characteristic is O. Confirm this in three other ways: (1) Obtain a tangent vector field on T without zeros, and apply the Poincare-Hopf Index Theorem. (2) Compute the degree of the Gauss mapping 1/ : T -+ §2, and apply the GaussBonnet Theorem. (3) Exhibit a Morse function on T, and use the Morse Inequalities. Notice that (1) and (3) can also be done for latter is not a hypersurface.

§1

x

§1

C lR\ but (2) cannot, as the

Number 4. Prove, using Euler characteristics, that a sphere of even dimension cannot be homeomorphic to a product of two other spheres. Number 5. Let M C lR 3 be a compact smooth surface, and let f : M -+ lR be a Morse function. Show that f has at least two critical points, and if there are no more, then M has genus 0 (hence M is diffeomorphic to the sphere §2). Number 6. Compute the Euler characteristic of the rotation group M using the Morse function h in Problem Number 5 of V.6.

= 80(3) C lR3X3

Number 7. Let M C lR 3 be the cornered sphere X4 + y4 + Z4 = 1. Check the GaussBonnet Theorem (i) through explicit computation of the integral curvature and (ii) computing the degree of the Gauss mapping.

Names of mathematicians cited Aleksandrov, Pavel Sergeevich (1896-1982), 32 Amann, Herbert, 39 Bernstein, Sergei Natanovich (1880-1968),34 Betti, Enrico (1823-1892), 13 Bohl, Piers (1865-1921), 13 Brouwer, Luitzen Egbertus Jan (1881-1996),14 Brown, Arthur Barton, 35 Caccioppoli, Renato (1904-1959), 34 Cauchy, Augustin Louis (1789-1857),2 Cech, Eduard (1893-1960), 30 Deimling, Klaus, 40 Dyck, Walter Franz Anton (1856-1934), 31 Dylawerski, Grzegorz, 46 Elworthy, K. David, 43 Fuhrer, Lutz, 38 Fuller, F. Brock, 45 Gauss, Karl-Friedrich (1777-1855), 2 Geba, Kazimierz, 43 Hadamard, Jacques Salomon (1865-1963), 14 Heinz, Erhard, 38 Hermite, Charles (1822-1901), 7 Hopf, Heinz (1894-1971), 28 Hurewicz, Witold (1904-1956), 30 Ize, Jorge, 43

Jacobi, Carl Gustav Jacob (1804-1851),7 Jodel, Jerzy, 46 Jordan, Camille (1838-1922), 15 Kronecker, Leopold (1823-1891), 7 Leray, Jean (1906-1998), 33 Liouville, Joseph (1809-1882), 2 Marzantowicz, Waclaw, 46 Massabo, Ivar, 43 Miranda, Carlo, 28 Nagumo, Mitio, 35 Nirenberg, Louis, 43 Ostrowski, Alexander (1893-1986), 2 Picard, Charles Emile (1856-1941), 12 Poincare, Jules Henri (1854-1912), 11

Pontryagin, Lev Semenovich, 43 de Rham, Georges (1903-1990),37 Riemann, Friedrich Bernhard Georg (1826-1866), 13 Romero Ruiz del Portal, Francisco, 45 Rothe, E., 29 Rybicki, Slawomir, 48 Sard, Arthur (1909-1980), 35 Schauder, Juliusz Pawel (1899-1943),33 Schmidt, Erhard (1876-1959), 28

-

225

226

Schonfiies, Arthur Moritz (1853-1928),15 Schwartz, Laurent (1915-2002), 38 Smale, Stephen, 42 Sturm, Jacques Charles Franr;ois (1803-1855),2 Sylvester, James Joseph (1814-1897), 7 Thorn, Rene (1923-2002), 42 Tromba, Anthony J., 43 Veblen, Oswald (1880-1960), 15 Vignoli, Alfonso, 43 Weierstrass, Karl Theodor Wilhelm (1815-1897), 10 Weiss, Stanley A., 39

Names

Historical references We list below the references mentioned in Chapter I. We also include three basic papers on the history of degree theory: [Siegberg 1980a], [Siegberg 1980bJ, and [Mawhin 1999J. [Adams 1969] J.F. Adams: Lectures on Lie Groups. New York-Amsterdam: W.A. Benjamin, Inc., 1969. [Aleksandrov-Hopf 1935] P. Aleksandrov, H. Hopf: Topologie. Berlin: Verlag von Julius Springer, 1935. [Amann-Weiss 1973] H. Amann, S.A. Weiss: On the uniqueness of topological degree. Math. Z. 130 (1973), 39-54. [Banach 1922] S. Banach: Sur les operations dans les ensembles abstraites et leurs applications aux equations integrales. Fundamenta Math. 3 (1922), 133181. [Bohl1904] P. Bohl: Uber die Bewegung eines mechanischen Systems in der Nahe einer Gleichgerwichtslage. J. Reine Angew. Math., 127 (1904), 179-276. [Brouwer 1912a] L.E.J. Brouwer: Uber abbildung von Mannigfaltigkeiten. Mat. Ann. 71 (1912), 97-115. [Brouwer 1912b] L.E.J. Brouwer: On looping coefficients. Konink. Nederl. Akad. von Wet. Proc. 15 (1912), 113-122. [Brouwer 1912c] L.E.J. Brouwer: Continuous one-one transformations of surfaces in themselves. CNAG Proc. 15 (1912), 352-360. [Brouwer 1976] L.E.J. Brouwer: Collected works, vol. II. Geometry, analysis, topology and mechanics. H. Freudenthal (ed.). Amsterdam-Oxford: North Holland, 1976. [Brown 1935] A. B. Brown: Functional dependence. Trans. Amer. Math. Soc. 38 (1935), 379-394. [Caccioppoli 1936] R. Caccioppoli: Sulle corrispondenze funzionalli inverse diramata: teoria generale e applicazioni ad alcune equazioni funzionali non lineari e al problema di Plateau, I & II. Rend. Accad. Naz. Linzei 24 (1936), 258-263 & 416-421.

-

227

228

Historical references

[Cauchy 1837aJ A.-L. Cauchy: Calcul des indices des fonctions. J. Ecole Poly technique XV (1837), 176-226. [Cauchy 1837bJ A.-L. Cauchy: Extrait d'une lettre sur une memoire publie it Thrin, Ie 16 Juin 1833, et relatif aux racines des equations simultanees. Comptes Rendus Acad. Bc. Paris IV (1837), 672-675. [Cauchy 1855J A.-L. Cauchy: Sur les compteurs logarithmiques. Comptes Rendus Acad. Bc. Paris XL (1855), 1009-1016. [Cech 1932J E. Cech: Hoherdimensionale Homotopiegruppen. Verhandlungen der Internationalen Mathematiken Kongresses, Zurich (1932), 203. [Dancer 1983J E.N. Dancer: On the existence of zeros of perturbed operators. Nonlinear Anal. 7 (1983), 717-727. [Dancer 1985J E.N. Dancer: A new degree for §l-invariant gradient mappings and applications. Ann. Inst. Henri Poincare, Anal. Non-lineaire 2 (1985), 329370. [Dancer et al. 2005J E.N. Dancer, K. Geba, S.M. Rybicki: Classification of homotopy classes of equivariant gradient maps. Fundamenta Math. 185 (2005), 1-18. [Deimling 1985J K. Deimling: Non-linear functional analysis. Berlin: SpringerVerlag, 1985. [Dyck 1888J W.F.A. Dyck: Beitrage zur Analysis situs I. Aufsatz ein- und zweidimensionale Mannigfaltigkeiten. Math. Ann. 32 (1888), 457-512. [Dyck 1890J W.F.A. Dyck: Beitrage zur Analysis situs II. Aufsatz Mannigfaltigkeiten von n-dimensionen. Math. Ann. 34 (1888), 273-316. [Dylawerski et al. 1991J G. Dylawerski, K. Geba, J. Jodel, W. Marzantowicz: An §l-equivariant degree and the Fuller index. Ann. Pol. Math. 52 (3) (1991), 243-280. [Elworthy-Tromba 1970aJ K.D. Elworthy, A.J. Tromba: Differential structures and Fredholm maps on Banach manifolds. In Global Analysis (Berkeley, 1968) 45-94, Proc. Symp. Pure Math. 15. Providence, R.I.: American Mathematical Society, 1970. [Elworthy-Tromba 1970bJ K.D. Elworthy, A.J. Tromba: Degree theory on Banach manifolds. In Nonlinear Functional Analysis (Chicago, 1968) 75-94, Proc. Symp. Pure Math. 18, part 1. Providence, R.I.: American Mathematical Society, 1970. [Fuhrer 1971J L. Fuhrer: Theorie des Abbildungsgrades in endlich-dimensionalen Raumen. Dissertation, Freie Universitat Berlin, 1971. [Fuhrer 1972J L. Fuhrer: Ein elementarer analytischer Beweis zur Eindeutigkeit des Abbildungsgrades in jRn. Math. Nachr. 54 (1972), 259-267. [Fuller 1967J B. Fuller: An index of fixed point type for periodic orbits. American J. Math. 89 (1967), 133-148.

Historical references

229

[Gauss 1813] K.-F. Gauss: Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata. Gottingische gelehrt Anzeigen, April 5 (1813), 545-552. [Gauss 1833] K.-F. Gauss: Zur mathematischen Theorie der electrodynamische Wirkungen. In Werke, 5. G6ttingen: K6niglichen Gesellschaft der Wissenchaften zu G6ttingen (1867),601-626. [Geba et al. 1986] K. Geba, I. MassabO, A. Vignoli: Generalized topological degree and bifurcation. In Non-linear Functional Analysis and Applications (Proc. NATO Adv. Study Inst., Maratea, Italy, 1985) 55-73, Mathematical and Physical Sciences 173. Dordrecht: Reidel, 1986. [Geba et al. 1990] K. Geba, I. MassabO, A. Vignoli: On the Euler characteristic of equivariant gradient vector fields. Boll. Un. Mat. Ital. A(7) 4 (1990), 243-251. [Hadamard 1910] J.-S. Hadamard: Sur quelques applications de l'indice de Kronecker. In [Tannery 1910], 437-477. [Heinz 1959] E. Heinz: An elementary analytic theory of the degree of mapping in n-dimensional space. J. Math. and Mech. 8 (1959), 231-247. [Hopf 1925] H. Hopf: Uber die Curvatura integra geschlossener Hyperfliichen. Math. Ann. 95 (1925), 340-367. [Hopf 1926a] H. Hopf: Abbildungsklassen n-dimensionaler Mannigfaltigkeiten. Math. Ann. 96 (1926), 209-224. [Hopf 1926b] H. Hopf: Vectorfelder in n-dimensionalen Mannigfaltigkeiten. Math. Ann. 96 (1926), 225-250. [Hopf 1931] H. Hopf: Uber die Abbildungen der dreidimensionalen Sphiire auf die Kugelfliiche. Math. Ann. 104 (1931), 637-665. [Hopf 1933] H. Hopf: Die Klassen der Abbildungen der n-dimensionalen Polyeder auf die n-dimensionale Sphiire. Comment. Math. Helvetici 5 (1933),39-54. [Hopf 1935] H. Hopf: Uber die Abbildungen von Sphiiren auf Sphiiren niedrigerer Dimension. Fundamenta Math. 25 (1935),427-440. [Hopf 1966] H. Hopf: Ein Abschnitt aus der Entwicklung der Topologie. Jber. Deutsch. Math.- Verein. 68 (1966), 96-106. [Hurewicz 1935a] W. Hurewicz: Beitriige zur Topologie der Deformationen I. H6herdimensionalen Homotopiegruppen. Proc. Akad. Wetenschappen 38 (1935), 112-119. [Hurewicz 1935b] W. Hurewicz: Beitriige zur Topologie der Deformationen II. Homotopie und Homologiegruppen. Proc. Akad. Wetenschappen 38 (1935), 521-528. [Hurewicz 1936a] W. Hurewicz: Beitriige zur Topologie der Deformationen III. Klassen und Homologietypen von Abbildungen. Proc. Akad. Wetenschappen 39 (1936), 117-126.

230

Historical references

[Hurewicz 1936b] W. Hurewicz: Beitrage zur Topologie der Deformationen IV. Aspharische Raume. Proc. Akad. Wetenschappen 39 (1936), 215-224. [Ize 1981] J. Ize: Introduction to bifurcation theory. In Differential Equations (Sao Paulo, 1981), 145-203. Lecture Notes in Math. 957. Berlin-New York: Springer-Verlag, 1982. [Ize et al. 1986] J. Ize, I. Massabo, A. Vignoli: Global results on continuation and bifurcation for equivariant maps. In Non-linear Functional Analysis and Applications (Maratea, Italy, 1985), 75-111. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 173. Dordrecht: Reidel, 1986. [Ize et al. 1989] J. Ize, I. Massabo, A. Vignoli: Degree theory for equivariant maps, I. Trans. Amer. Math. Soc. 315 (2) (1989), 433-510. [Ize et al. 1992] J. Ize, I. Massabo, A. Vignoli: Degree theory for equivariant maps, II: The general §l-actions. Memoirs Amer. Math. Soc. 481 (1992). [Iz€-Vignoli 2003] J. Ize, A. Vignoli: Equivariant Degree Theory. de Gruyter Series in Nonlinear Analysis and Applications 8. Berlin: Walter de Gruyter & Co., 2003. [Jordan 1893] C. Jordan: Cours d'analyse de l'Ecole Poly technique, f. 1893. [Kronecker 1869a] L. Kronecker: Uber Systeme von Functionen mehrerer Variabeln, I. Monatsberichte koniglich Preuss. Akad. Wissens. Berlin, March 4 (1869), 159-193. [Kronecker 1869b] L. Kronecker: Uber Systeme von Functionen mehrerer Variabeln, II. Monatsberichte koniglich Preuss. Akad. Wissens. Berlin, August 5 (1869), 688-698. [Leray-Schauder 1933] J. Leray, J.P. Schauder: Topologie et equations fonctionnelles. Comptes Rendus Acad. Sc. Paris 1197 (1933), 115-117. [Leray-Schauder 1934] J. Leray, J.P. Schauder: Topologie et equations fonctionnelles. Ann. Sc. Ecole Normale Sup. Paris 51 (1934),45-78. [Liouville-Sturm 1837] J. Liouville, J.-Ch.-F. Sturm: Note sur un tMoreme de M. Cauchy relatif aux racines des equations simultanees. Comptes rendus Acad. Sc. Paris 40 (1937), 720-724. [Mawhin 1999] J. Mawhin: Leray-Schauder degree: A half century of extensions and applications. Topol. Methods Nonlinear Anal. 14 (1999), 195-228. [Miranda 1940] C. Miranda: Un'osservazione su un teorema di Brouwer. Boll. Un. Mat. ftal. (2) 3 (1940), 5-7. [Nagumo 1951a] M. Nagumo: A theory of degree of mapping based on infinitesimal analysis. American J. Math. 73 (1951),485-496. [Nagumo 1951b] M. Nagumo: Degree of mapping in convex linear topological spaces. American J. Math. 73 (1951), 497-511.

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[Nirenberg 1971] L. Nirenberg: An application of generalized degree to a class of nonlinear problems. In T'roisieme Colloque sur l'Analyse Fonctionnelle (Liege, 1970), 57-74. Louvain: Vander, 1971. [Picard 1891] Ch.-E. Picard: Sur Ie nombre des racines communes it plusieurs equations simultanees. Comptes Rendus Acad. Sc. Paris 113 (1891), 256358. [Picard 1891/1905] Ch.-E. Picard: TraiU d'Analyse, I and II. Paris: GauthierVillars, 1891 and 1893/1905. [Poincare 1883] J.H. Poincare: Sur certains solutions particulieres du probleme des trois corps. Comptes Rendus Acad. Sc. Paris 97 (1883), 251-252. [Poincare 1885a] J .H. Poincare: Sur les courbes definies par les equations differentielles. J. Math. Pures Appl. 1 (1885), 167-244. [Poincare 1895b] J.H. Poincare: Analysis Situs. J. Ecole Poly technique (2) 1 (1895), 1-123. [Poincare 1886] J.H. Poincare: Sur les courbes definies par une equation differentielle. J. Math. Pures Appl. 2 (1886), 151-217. [Poincare 1899] J.H. Poincare: Complement it l'Analysis Situs. Rend. Circ. Mat. Palermo 13 (1899), 285-343. [Poincare 1900] J.H. Poincare: Complement it l'Analysis Situs. Proc. London Math. Soc. 32 (1900), 277-308. [Pontryagin 1955] L.S. Pontryagin: Smooth manifolds and their application in homotopy theory. Amer. Math. Soc. Transl. (2) 11 (1959), 1-114, from Trudy Mat. Inst. im Steklov 45 (1955). [de Rham 1955] G. de Rham: VarieUs difJerentiables. Paris: Hermann, 1955. [Rothe 1936] E. Rothe: Uber Abbildungsklassen von Kugeln des Hilbertschen Raumes. Compos. Math. 4 (1936), 294-307. [Ruiz del Portal 1991] F.R. Ruiz del Portal: Teoria del grado topol6gico generalizado y aplicaciones. Dissertation. Madrid: Universidad Complutense, 1991. [Ruiz del Portal 1992] F.R. Ruiz del Portal: On the additivity property of the generalized degree. Math. Japonica 37 (1992), 657-664. [Rybicki 1994] S. Rybicki: A degree for §l-equivariant orthogonal maps and its applications to bifurcation theory. Nonlinear Anal. 23 (1994),83-102. [Sard 1942] A. Sard: The measure of critical points of differentiable maps. Bull. Amer. Math. Soc. 48 (1942), 883-897. [Schonflies 1902] A.M. Schonflies: Uber einen grundlegenden Satz der Analysis Situs. Gott. Nachr. Math. Phys. Kl. (1902), 185-192. [Siegberg 1980a] H. W. Siegberg: Brouwer degree: History and numerical computation. In Numerical solutions of highly nonlinear problems, W. Forster (ed.), 389-411. Amsterdam: North-Holland, 1980.

232

Historical references

[Siegberg 1980bJ H. W. Siegberg: Some historical remarks concerning degree theory. American Math. Monthly 88 (1981), 125-139. [Smale 1965J S. Smale: An infinite dimensional version of Sard's Theorem. American J. Math. 87 (1965), 861-866. [Sylvester 1853J J.J. Sylvester: On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm's functions, and that of greatest algebraic common measure. Philos. Trans. Roy. Soc. London 143 (1853), 407-562. [Tannery 1910J J. Tannery: Introduction Ii la theorie des fonctions d'une variable, t. 2, 2ieme edition. Paris: Hermann, 1910. [Thom 1954J R. Thom: Quelques proprif~tes globales des varietes differentiables. Comment. Math. Helvetici 28 (1954), 17-86. [Veblen 1905J O. Veblen: Theory of plane curves in non-metrical analysis situs. Trans. Amer. Math. Soc. 6 (1905), 83-98.

Bibliography For prerequisites, we recommend the quite ad hoc texts [4], [10], [16], and [18]. On the other hand, there is a wealth of literature on degree theory and related topics; we suggest as further reading the following books: [1] J. Cronin: Fixed points and topological degree in nonlinear analysis. Mathematical Surveys 11. Providence, R.I.: American Mathematical Society, 1964. [2] K. Deimling: Nonlinear Functional Analysis. Berlin: Springer-Verlag, 1985. [3] A. Dold: Teoria de punto fijo (I, II, III). Mexico: Monografias del Instituto de Matematica, 1986. [4] J.M. Gamboa, J.M. Ruiz: Iniciacion al estudio de las variedades diferenciales (2a edicion revisada). Madrid: Sanz y Torres, 2006. [5] A. Granas, J. Dugunji: Fixed Point Theory. Springer Monographs in Mathematics. New York: Springer-Verlag, 2003. [6] V. Guillemin, A. Pollack: Differential Topology. Englewood Cliffs, N.J.: Prentice Hall, Inc.,1974. [7] M.W. Hirsch: Differential Topology. Graduate Texts in Mathematics 33. Springer-Verlag, New York-Heidelberg: Springer-Verlag, 1976. [8] M.A. Krasnosel'skii: Topological methods in the theory of nonlinear integral equations (translated from Russian). New York: Pergamon Press, 1964. [9] W. Krawcewicz, J. Wu: Theory of degrees with applications to bifurcations and differential equations. New York: John Wiley & Sons, 1997. [10] S. Lang: Differential manifolds. Berlin: Springer-Verlag, 1988.

-

233

Bibliography

234

[11] E.L. Lima: IntrodUf;ao Ii Topologia Diferencial. Rio de Janeiro: Instituto de Matematica Pura e Aplicada, 1961. [12] N.G. Lloyd: Degree Theory. Cambridge Tracts in Mathematics 73. Cambridge: Cambridge University Press, 1978.

[13] 1. Madsen, J. Tornehave: From calculus to cohomology: de Rham cohomology and characteristic classes. Cambridge, New York, Melbourne: Cambridge University Press, 1997. [14] J. Milnor: Topology from the differentiable viewpoint. Princeton Land-

marks in Mathematics. Princeton, N.J.: Princeton University Press, 1997. [15] L. Nirenberg: Topics in nonlinear functional analysis (with a chapter

by E. Zehnder and notes by R.A. Artino). Courant Lecture Notes in Mathematics 6. New York: Courant Institute of Mathematical Sciences, American Mathematical Society, 1974. [16] E. Outerelo, J .M. Ruiz: Topologia Diferencial. Madrid: AddisonWesley, 1998. [17] P.H. Rabinowitz: Theorie du Degre Topologique et applications Ii des problemes aux limites non lineaires (redige par H. Berestycki). Lec-

ture Notes Analyse Numerique Fonctionelle. Paris: Universite Paris VI, 1975. [18] M. Spivak: Calculus on manifolds: A modern approach to classical theorems of advanced calculus. Boulder: Westview Press, 1971.

Symbols = f(x)

y

zn

+

+ ... + an =

alZ n - l

°

J:~(j)

Z(Z) = X(X, y)

+ iY(x, y)

IKI --7 ILl

1

9 = 'ljJ 0 go cp-l

2

d(j)

3

F(t, x) =

3

§+

28

:

24

25

IM~~l=g=g~1I

27

f 211"i r

Z'{z)d Z{z) z

3

f(Kl' K 2 )

28

N

JYl Yo

(.d)

3

X(M)

30

4


34

d(
34

x-F(x)=O

34

dist

36

d(j,G,a)

36

1

=

XI

Xo

~8 = /-Ll -/-L2 w=

aF ax

aF ay

£l.

£l.

ax

5

ay

2;i fr z~a w(j(r), O) = 2;i ff{r) ~

5 6

fIRm
w(j(r), O)

6

d(y(x),

w(j(r) , 0) = Lkak

6

fn
38

E(k, f)

8

M(W)

41

A(k, f)

8

Mo(n)

41

#E(k, f) - #A(k, f)

8

an= n\n

41

X(Fo, F l ,· .. ,Fn)

8

f, foe,

H#E(k,f) - #A(k,f))

8

d*(j, U,O)

44

11

d*(j, U)

44

11

p : §l

GL(V)

46

17

21, a = (ark::o

46

20

Deg(j, n)

46

20

Deg(j, n) = E(d*(j, n))

47

w(r, a) =

V() Z

=

Lk w(r, ak)ak

X-XC)

= Ilx-xoll 3

fFo=O(V,v)dS w=Nl -N2

75" = {x E]Rn : Ilxll a75" = §n-l •

K,IKI,IKI

~ 1}

23

§l

*a

=1

n, z)

f*

--7

38 38

43

47

-

235

236

Symbols

Ilull < r(x)}

Deg(f,il) = (a r ) 8kf 8X'l··· 8x'k 'P = ('Pl, ... , 'Pp)

47

{(x, u) E 11M:

49

H(t, x), Ht(x), Ht

75

54

[M,N]

75

x = (Xl, ... ,xm)

54

7rk(N)

=

[§k, N]

75

'l/J-l 0 'P

54

7rk(M)

=

[M, §k]

75

dimx(M), dim(M) Un M = f- 1(0)

54

(8 fi ) 8xj

55

h(U n M) = V n (l~m X {O})

55

lHIm : A ~ 0

56

8M

56 Mo U Ml

=

57 58

Tx M

..iLl 8x, x -

~( 8x, x

)

58 58

gradx(f)

dxf(u)

=

(gradx(f), u)

T(x,y)(M x N) dxf : TxM =

76 85 85

(M = {(x: x E M}

8([0,1] x M)

dag

55

g(x)11 < c(x)

Ilf(x) (m

--7

=

TxM x TyN

TyN

(d b'l/J)-1

0

dxf 0 da'P

58 58 59 59

1" (0) = do l' (1 )

60

dxf(u)

60

=

(f 0 1')'(0)

JR.lP'm,Cr

61

TM VM

61

Cf , Rf

62

Rf=N\f(Cf )

62

Txf-l(a)

62

61

=

ker(dxf)

O(n)

66

OM

66

11M

71

y - p(y) 1- Tp(y)M

71

dist(y,M)

=

Ily-p(y)11

71

[8~1 Ix'···' 8~rn IJ (MxN

=

(x

M

86

--7

JR. m+1

87 91

8(

95

signx(f)

EXEf-l(a)

95

deg(f) deg(h x ... x fr) T m = §1 x ... X §1 7r :

§m

102 102

:mpm

--7

103

r~(M)

104

d : r~(M)

--7 r~+I(M)

=

104 104

d(a 1\,8)

dw

85 86

((M,(N)

=

signx(f) II :

72

104

0, w = da

H~(M,JR.)

104

H~(M,Z)

104

H;'(JR.m,JR.)

= JR.

104

Hm(JR.m,JR.)

=

0

104

fMda

104

0

=

fM : H;'(M, JR.)

--7

JR.

104

h = E"!' .=1 !!..& 8x,

105

w t-+ f*w

105 deg(f) -IN w

107

grad (f) / II grad(f) II

108

fM f*w II

=

=

OM = det(lI, .)

108

237

Symbols

= deg(g)

dxv : TxM -+ Tx M

108

deg(17(g))

K(x) = det(dxv)

109

f(x) # -g(x)

186

Il,=JMKrlM

109

7r m (M)

193

rlsm

109

V*rlsm = KrlM

109

X(M)

109

=Z 7rm(§m) = 7rm(§m) = Z deg(f· g) = deg(f) + deg(g) 7rm(x) = Z

109

§2m-l -+ §m

200

Hl(§l x §l,JR) = JR2

110

H(f 0 g) = H(f) deg(g)

200

deg(f)

111

7r3(§2)

205

Il,

=

~vol(§m)X(M)

176

193 195 198

exp: JR -+ §l

113

h = - f + (f,g)g : §m -+ §m

113

=Z 7r7(§4) = Z EB Z4 EB Z3 7r15(§S) = Z EB Zs EB Z3 EB Z5 H(g 0 f) = deg(g)2H(f)

117

m ~.~ ~ -- I: i=l tax;

206

p, -p,cP,iP

119

206

cPa,b f-l(a) x f-l(b) -+ §2m-2

119

e=(6,···'~m) dz~ : TzM -+ TzM

119

det(dz~)

207

H(f) = deg(cPa,b) £(f-l(a), f-l(b))

119

indz(~)

Hm(§2m-l, JR) = 0

123

fl(f) = JS2m-l a /\ da

123

deg2(f)

124

w2(M,p)

124

W2(p)

127

Ilfll

137

d(f, D, a)

138

w(f, a) = dU, D, a)

156

f(x) = ±f(-x)

160

~#±~ IIf(x)1I IIf(-x)1I

163

f(x) _ ~ IIf(x)1I - =i=lIf(-x)1I

f

* g: §m+n+l -+ §m+n+l

deg(f)

= deg(1/J 0 f 0 cp-l)

113

121

deg(

= sign det(dz~)

nfrr Is)

Ind(~)

= I:z indz(~)

205 205 205

207 207 209 209

grad (f)

211

Gxcp

212

Jxe

212

Hx(f 0 cp)

212

Qz(f)

213

Qz(f) = (f 0 'Y)"(t)

213

indz(f)

214

Ind(~)

214

166

= I:z(-l)ind. U ) Ind(~) = I:z(-l)k ak dist(x, H) = II!II (x, a)

169

80(3)

217

s(X), s'(X)

171

Ind(~)

= deg(7J)

219

d(gof,D,a)

172

daf = IdE EBda~

219

17(g)

176

Ind( -~)

limllxll-HOO If(x)1

= +00

= (-l)m Ind(~)

214 215

220

238

Ind(~)

Symbols

= X(M)

220

X=F-E+V

221

X = 2 - 2g

221

X(§2) = 2

221

X(M) = Lk(-l)kak

222

deg(v) = ~X(M)

222

Index Accessible points, 16 Additivity, 37, 39-41, 45, 46, 149 Admissible class of mappings, 41 Admissible extension, 44 Alexandroff compactification, 43 Amann-Weiss degree, 42 Analysis Situs, 13 Antipodal extension, 187 Antipodal images, 186 Antipodal preimages, 167 Approximation and homotopy, 78 Approximation by smooth mappings, 76 Arcs do not disconnect, 15, 180 Attractor, 207 Axiomatization of Euclidean degree, 149 Balls do not disconnect, 180 Barycentric subdivision, 25 Bolzano Theorem, 167 Borsuk Theorem, 195 Borsuk-Ulam Theorem, 163 Boundaries are not retractions, 13, 133 Boundary, 14 Boundary of a manifold, 56 Boundary of a simplicial complex, 23 Boundary Theorem, 19, 96, 101, 111, 150, 157 Brouwer Fixed Point Theorem, 13, 20, 28, 29, 134, 152, 169 Brouwer-Kronecker and Euclidean degrees, 157, 158 Brouwer-Kronecker degree, 31, 99, 111

Bump function, 51 Canonical orientation, 85 Cauchy Argument Principle, 5, 6, 160 Cauchy index of a function, 3 Cauchy index of a planar curve, 5, 11 Cauchy index with respect to a planar curve, 15 Cell, 14 Center, 208 Change of coordinates, 54 Change of variables for integrals, 38, 107 Characteristic of a regular function system, 8 Circulation, 208 Classification of compact differentiable curves, 63 Closed differential form, 104 Cobordism, 42 Cohomotopy group, 75 Coincidence index of two mappings, 29 Colevel of a space with involution,

171 Combinatorial topology, 13, 23 Completely continuous mapping, 33 Complex multiplication, 100 Complex projective space, 61 Converse Boundary Theorem, 194, 199 Convolution kernel, 144 Counterclockwise orientation, 85 Critical point, 62 Critical value, 35, 62

-

239

240

Curve, 54 Curve germs, 60, 90 Cusp, 60 Degree for mappings between spaces of different dimensions, 43 Degree of a complex polynomial, 151 Degree of a complex rational function, 190 Degree of a composite mapping, 27, 100, 111 Degree of a differentiable mapping, 114,154 Degree of a symmetry, 100 Degree of the antipodal map, 99 Degree on manifolds with boundary, 193 Deimling Axiomatization, 149 Deimling axiomatization, 40 Derivative of a differentiable mapping, 59 Diffeomorphism, 54 Diffeotopies of Euclidean spaces, 81 Diffeotopies of manifolds, 82 Diffeotopies preserve orientation, 86 Diffeotopy, 80 Differentiable function, 49 Differentiable manifold, 54 Differentiable mappings, 49 Differentiable mappings are homotopic to their derivatives, 153 Differentiable retraction, 67 Differential form, 104 Dimension of a manifold, 54 Dipole, 210 Directional derivation, 58 Divergence of a vector field, 105 Easy Sard Theorem, 63, 187 Electric field flow, 11, 160 Equivariant degree, 46 Equivariant mapping, 46 Equivariant mapping of spaces with involution, 171 Equivariant mapping of spheres, 171 Equivariant topological degree, 45 Euclidean degree, 138, 143, 145

Index

Euclidean degree is locally constant, 139, 146 Euler characteristic, 30, 109, 220 Euler formula, 221 Even and odd extension lemmas, 161 Even and odd mappings, 160 Exact form, 104 Excision, 37, 44, 46 Exhaustion by compact sets, 54 Existence of solutions, 1, 19, 27, 34, 37, 40, 44, 46, 149 Extension by zero, 53 Exterior differentiation, 104 Exterior of a compact hypersurface, 129 Exterior of a planar curve, 15 Face, 23 Fibrations of spheres, 202 Finite representation, 46 First homotopy group, 30 Fixed Point Problem, 1 Fixed Point Theorem for odd mappings, 167 Fixed points of mappings of spheres, 21, 27 Flow of a tangent vector field, 207 Flow through a surface, 11 Framed cobordism, 43 Fredholm operator, 34 Fubini Theorem, 65, 216 Fiihrer Axiomatization, 149 Fuller index, 45 Fundamental group, 30 Fundamental Theorem of Algebra, 2, 102, 150 Fiihrer axiomatization, 39 Gauss curvature, 109 Gauss mapping, 30, 108 Gauss Theorem on electric flows, 11, 160 Gauss-Bonnet Theorem, 31, 109, 222 Geba-MassabO-Vignoli degree, 43 Genus of a compact surface, 30, 221 Global equations, 55, 129

Index

Global normal field, 215 Good simplicial approximation, 25 Gradient of a smooth function, 58 Gradients on arbitrary manifolds, 211 Gramm matrix, 212 Green's Theorem, 16 Hadamard index, 36 Hadamard Integral Theorem, 17, 159 Hedge-hog Theorem for manifolds, 221 Hedge-hog Theorem for spheres, 135 Height function, 215 Heinz Integral Formula, 38, 144 Hessian of a function at a critical point, 213 Hirsch Theorem, 166 Homology groups, 222 Homomorphisms on cohomology, 105 Homotopic mappings, 75 Homotopy, 75 Homotopy and smooth homotopy, 79 Homotopy for manifolds with boundary, 193 Homotopy for non-compact manifolds, 193 Homotopy for non-orientable manifolds, 194 Homotopy group, 30, 75 Homotopy invariance, 25, 27, 37, 39-41, 44, 46, 98, 101, 107, 112, 122, 140, 142, 144, 147, 156 Homotopy of mappings into spheres, 186 Hopf fibration, 202, 205 Hopf invariant, 30, 119 Hopf invariant in odd dimension, 122 Hopf invariant of a composite mapping, 200 Hopf Theorem, 29, 191, 196 Hypersurface, 54

241

Identity off a set, 80 Impulsor, 207 Incoming (eingang) point, 8 Index of a center, 208 Index of a dipole, 210 Index of a function at a critical point, 214 Index of a function system on a hypersurface, 19 Index of a non-degenerate zero, 207 Index of a quadratic form, 214 Index of a saddle, 207 Index of a sink, 207 Index of a source, 207 Index of an isolated zero, 209 Integral curvature, 30, 109 Integral index, 3 Integral lines, 207 Interior of a compact hypersurface, 129 Interior of a planar curve, 15 Intermediate Value Theorem in arbitrary dimension, 12 Invariance of dimension, 28, 169 Invariance of Domain Theorem, 28, 56, 167, 169, 181 Invariance of interiors, 20, 169 Invariance of the characteristic under continuous deformations, 12 Invariant set, 46 Inverse image of a regular value, 24, 35, 62 Inverse Mapping Theorem for manifolds, 59 Inward tangent vectors at a boundary point, 90 Isolated coincidence point of two mappings, 29 Jacobian, 4, 9, 58 Join of two mappings, 113 Jordan hypersurface, 31 Jordan Separation Theorem, 6, 14, 28, 55, 88, 108, 124, 126, 158, 176, 181 Jordan sets separate the space, 180

242

Kneser-Glaeser Theorem, 63 Kronecker Existence Theorem, 9 Kronecker index, 14 Kronecker integral, 11-13 Kronecker Integral Theorem, 10, 159 Lemniscate, 171 Leray-Schauder degree, 34 Letter from Brouwer to Hadamard, 21 Level of a space with involution, 171 Lifting, 113 Linearization of antipodal extensions, 188 Link coefficient, 28, 121 Liouville-Sturm Theorem, 5 Local coordinate system, 54 Local coordinate system compatible with an orientation, 85 Local diffeomorphism, 54 Local differentiable extension, 49 Local equations, 55 Local extrema of a differentiable function, 59 Local finiteness, 50 Localization, 24, 59 Locally closed set, 55 Manifold, 23 Manifold with boundary, 56 Manifolds are homogeneous, 83 Manifolds as retractions of open Euclidean sets, 70 Mappings into spheres, 99, 183, 185 Mappings into toruses, 186 Measure zero set, 64 Mod 2 degree theory, 34, 124 Mod 2 winding number, 124 Models of a manifold, 31 Morse function, 214, 222 Morse inequalities, 221 Multiplication Formula, 172 Multiplicities of roots, 151 Multiplicities of zeros, 6 Nagumo Axiomatization, 149 Nagumo axiomatization, 37 Noether operator, 34

Index

Non-degenerate critical point of a function, 212 Non-degenerate zero of a tangent vector field, 207 Non-embedded manifolds, 31 Non-homotopic mappings with the same degree, 112 Norm of a function, 137 Normal vector field, 87, 88, 108, 218 Normal vector field compatible with an orientation, 87 Normal vector field on a tube, 219 Normality, 27, 37, 39-41, 139, 149 Normalization of antipodal extensions, 188 Opposite orientations, 85 Order of the origin with respect to a surface, 17 Orientation at two antipodal points of a sphere, 88 Orientation in a linear space, 85 Orientation of a cylinder, 91 Orientation of a manifold, 85 Orientation of hypersurfaces, 87 Orientation of inverse images, 89 Orientation of spheres, 88 Orientation of the boundary, 91 Oriented distance, 215 Orthogonal group, 66 Outgoing (ausgang) point, 8 Outward normal vector field, 88 Outward tangent vectors at a boundary point, 90 Parallelizable manifold, 210 Parametrization, 54 Partial derivative, 58 Perron-Fr6benius Theorem, 134 Poincare-Bohl Theorem, 19, 159 Poincare-Hopf Index Theorem, 21, 30, 109,219 Poincare-Miranda Theorem, 28 Polyhedron, 13 Positive atlas, 85 Positive basis, 85 Positive change of coordinates, 85

Index

Preserving and reversing orientation mappings, 86 Principal curvatures, 109 Product of manifolds, 57 Proper homotopy, 76 Proper mapping, 75 Proper subset of spheres do not disconnect Euclidean spaces, 180 Properly homotopic mappings, 76 Pseudomanifold, 23 Pull-back of a differential form, 105 Quadratic transform, 144 Radial retraction, 61 Real projective hyperplane, 67 Real projective hypersurface, 67 Real projective space, 61 Real roots of a polynomial, 2, 7 Regular function system, 7 Regular point, 62 Regular value, 62 Residual set, 63 Retraction, 67 de Rham cohomology, 37, 104 Riesz Representation Theorem, 211 Rotation group, 217 Saddle, 207 Sard Theorem, 35 Sard-Brown Theorem, 35, 63 Sch6nflies Theorem, 16, 20 Screwdriver orientation, 85 Segre embedding, 189 Semi-cone, 60 Set topology, 15 Sign of a mapping at a point, 86 Simplex, 23 Simplicial approximation, 25 Simplicial complex, 23 Simplicial homology, 13 Singular cohomology, 104 Singularity of a tangent vector field, 206, 212 Sink, 207 Smooth function, 49 Smoothness, 57, 117

243

Solid torus, 60 Source, 207 Spheres are not homeomorphic to proper subsets, 180 Splitting of an isolated zero, 208 Stereographic projection and orientation, 88 Stereographic projections, 55 Stiefel bundle, 61, 66 Stokes' Theorem, 13, 38, 104 Surface, 54 Surface in the Euclidean space, 16 Suspension, 44 Suspension of a mapping, 176, 200 Symmetric set, 161 Tangent Tangent Tangent Tangent Tangent Tangent

bundle, 61 space, 58 vector, 58 vector field, 135, 206 vector fields via flows, 221 vector fields via triangulations, 220 Tangent vector fields without zeros, 21, 27 Tietze Extension Theorem (differentiable), 52 Topological beast, 16 Topological circle, 183 Topological degree, 24 Topological degree at the origin, 5 Topological sinus, 113, 145 Torus of dimension m, 102 Torus with 9 holes, 221 Total index of a tangent vector field, 209 Total index of the solutions of an equation in a Banach space, 34 Total number of zeros, 6 Transversality, 62 Triangulation, 13, 220 Thbular retraction, 71 Unitary normal vector field, 87, 88, 108, 218 Uryshon separating function, 51

244

Variation of the argument, 14 Vector product, 87 Volume element, 108 Weingarten endomorphism, 108 Whitney Embedding Theorems, 56 Winding number, 5, 156, 163, 166 Winding number is locally constant, 156 Zero of a tangent vector field, 206

Index

Titles in This Series 108 107 106 105 104 103 102 101 100 99 98 97 96 95 94 93 92 91 90 89 88 87

86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71

Enrique Outerelo and Jesus M. Ruiz, Mapping degree theory, 2009 Jeffrey M. Lee, Manifolds and differential geometry, 2009 Robert J. Daverman and Gerard A. Venema, Embeddings in manifolds, 2009 Giovanni Leoni, A first course in Sobolev spaces, 2009 Paolo Alufli, Algebra: Chapter 0, 2009 Branko Griinbaum, Configurations of points and lines, 2009 Mark A. Pinsky, Introduction to Fourier analysis and wavelets, 2009 Ward Cheney and Will Light, A course in approximation theory, 2009 I. Martin Isaacs, Algebra: A graduate course, 2009 Gerald Teschl, Mathematical methods in quantum mechanics: With applications to Schrodinger operators, 2009 Alexander I. Bobenko and Yuri B. Suris, Discrete differential geometry: Integrable structure, 2008 David C. Ullrich, Complex made simple, 2008 N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, 2008 Leon A. Takhtajan, Quantum mechanics for mathematicians, 2008 James E. Humphreys, Representations of semisimple Lie algebras in the BGG category 0,2008 Peter W. Michor, Topics in differential geometry, 2008 I. Martin Isaacs, Finite group theory, 2008 Louis Halle Rowen, Graduate algebra: Noncommutative view, 2008 Larry J. Gerstein, Basic quadratic forms, 2008 Anthony Bonato, A course on the web graph, 2008 Nathanial P. Brown and Narutaka Ozawa, CO-algebras and finite-dimensional approximations, 2008 Srikanth B. Iyengar, Graham J. Leuschke, Anton Leykin, Claudia Miller, Ezra Miller, Anurag K. Singh, and Uli Walther, Twenty-four hours of local cohomology, 2007 Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations, 2007 John M. Alongi and Gail S. Nelson, Recurrence and topology, 2007 Charalambos D. Aliprantis and Rabee Tourky, Cones and duality, 2007 Wolfgang Ebeling, Functions of several complex variables and their singularities (translated by Philip G. Spain), 2007 Serge Alinhac and Patrick Gerard, Pseudo-differential operators and the Nash-Moser theorem (translated by Stephen S. Wilson), 2007 V. V. Prasolov, Elements of homology theory, 2007 Davar Khoshnevisan, Probability, 2007 William Stein, Modular forms, a computational approach (with an appendix by Paul E. Gunnells), 2007 Harry Dym, Linear algebra in action, 2007 Bennett Chow, Peng Lu, and Lei Ni, Hamilton's Ricci flow, 2006 Michael E. Taylor, Measure theory and integration, 2006 Peter D. Miller, Applied asymptotic analysis, 2006 V. V. Prasolov, Elements of combinatorial and differential topology, 2006 Louis Halle Rowen, Graduate algebra: Commutative view, 2006 R. J. Williams, Introduction the the mathematics of finance, 2006 S. P. Novikov and I. A. Taimanov, Modern geometric structures and fields, 2006

TITLES IN THIS SERIES

70 Sean Dineen, Probability theory in finance, 2005 69 Sebastian Montiel and Antonio Ros, Curves and surfaces, 2005 68 Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, 2005 67 T.Y. Lam, Introduction to quadratic forms over fields, 2004 66 Yuli Eidelman, Vitali Milman, and Antonis Tsolomitis, Functional analysis, An introduction, 2004 65 S. Ramanan, Global calculus, 2004 64 A. A. Kirillov, Lectures on the orbit method, 2004 63 Steven Dale Cutkosky, Resolution of singularities, 2004 62 T. W. Korner, A companion to analysis: A second first and first second course in analysis, 2004 61 Thomas A. Ivey and J. M. Landsberg, Cartan for beginners: Differential geometry via moving frames and exterior differential systems, 2003 60 Alberto Candel and Lawrence Conlon, Foliations II, 2003 59 Steven H. Weintraub, Representation theory of finite groups: algebra and arithmetic, 2003 58 Cedric Villani, Topics in optimal transportation, 2003 57 Robert Plato, Concise numerical mathematics, 2003 56 E. B. Vinberg, A course in algebra, 2003 55 C. Herbert Clemens, A scrapbook of complex curve theory, second edition, 2003 54 Alexander Barvinok, A course in convexity, 2002 53 Henryk Iwaniec, Spectral methods of automorphic forms, 2002 52 llka Agricola and Thomas Friedrich, Global analysis: Differential forms in analysis, geometry and physics, 2002 51 Y. A. Abramovich and C. D. Aliprantis, Problems in operator theory, 2002 50 Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, 2002 49 John R. Harper, Secondary cohomology operations, 2002 48 Y. Eliashberg and N. Mishachev, Introduction to the h-principle, 2002 47 A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and quantum computation, 2002 46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, 2002 45 Inder K. Rana, An introduction to measure and integration, second edition, 2002 44 Jim Agler and John E. MCCarthy, Pick interpolation and Hilbert function spaces, 2002 43 N. V. Krylov, Introduction to the theory of random processes, 2002 42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002 41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002 40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable, third edition, 2006 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/.

This textbook treats the classical parts of mapping degree theory, with a detailed account of its history traced back to the first half of the 18th century. After a historical first chapter, the remaining four chapters develop the mathematics. An effort is made to use only elementary methods, resulting in a self-contained presentation. Even so, the book arrives at some truly outstanding theorems: the classification of homotopy classes for spheres and the Poincare-Hopf Index Theorem, as well as the proofs of the original formulations by Cauchy, Poincare, and others. Although the mapping degree theory you will discover in this book is a classical subject, the treatment is refreshing for its simple and direct style. The straightforward exposition is accented by the appearance of several uncommon topics: tubular neighborhoods without metrics, differences between class I and class 2 mappings, Jordan Separation with neither compactness nor cohomology, explicit constructions of homotopy classes of spheres, and the direct computation of the Hopf invariant of the first Hopf fibration . The book is suitable for a one-semester graduate course. There are 180 exercises and problems of different scope and difficulty.

_ ~

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ISBN 978-0-8218-4915-6

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