Extensions of the Borsuk-Ulam theorem

J. Korean Math. Soc. 34 (1997), No. 3, pp. 599-608

EXTENSIONS OF THE BORSUK-ULAM THEOREM

IN-SOOK KIM

ABSTRACT. In this paper we give a generalization of the well-known Borsuk-Ulam theorem and its extensions to countably many prod- ucts of spheres.

1. Introduction

Borsuk's antipodal theorem which was conjectured by Ulam has equivalent formulations in great variety. In the present paper it is nat- ural to investigate the Borsuk-Ulam theorem on more general condi- tions, by replacing odd map by equivariant map with respect to group actions on the spheres.

This observation is motivated by the almost periodicity of orbits as a generalization of the periodicity. In contrast to actions of compact Lie group, a new result on Z-actions is of considerable practical im- portance. furthermore, our result is extended to finitely as well as countably many products of spheres, where the idea lies in the approx- imation theorem for almost periodic functions, see [4]. In particular, we show that this extension holds for the case of rationally indepen- dent numbers. We know that there is a close relationship between Borsuk-Ulam theorem and index theory for group actions, see [7], [10].

Therefore, we here provide an appropriate framework for an index the- ory. This kind of application is one of the main reasons to study a Z-version of the Borsuk-Ulam theorem.

2. Preliminaries

From elementary observations we will see that the approach to the

Received January 7, 1997.

1991 Mathematics Subject Classification: 58D19, 58GlO, 11J72.

Key words and phrases: Borsuk-Ulam theorem, equivariant, rationally indepen- dent, index theory.

Borsuk-Ulam theorem can be extended to other group actions, for in- stance Zq- or Z-actions.

To present these facts, we introduce the concept of equivariance.

DEFINITION 1. Let ^O'.j be a real number and let k j , lj be natural numbers for j = 1"" ,n and a := (e27ria1,.'. ,e27rian). A map h = (hI,··· ,hn ): S2k1-1 x ... ^X S2kn-1 -+ s2h^-1 x ... ^X S21n-1 will be called equivariant with respect to

a,

if

for every x = (Xl,·'· ,Xn ) ^E S2k1-1 X ••• X S2kn-1. In this case we write h: (S2k1-1 x ... X S2kn-1,a) -+ (S21 1-1 x··· X S21ⁿ-1,a).

The classical Borsuk-Ulam theorem says that given a continuous map h: sn -+lRn, there exists anX E sn such that hex) = he-x). It was well-known that this result is equivalent to the following theorem.

THEOREM 2. For every n, mEN with n

>

m there is no continuous oddmap h: sn -+srn.

For the proof we refer to Bredon [3].

The following result which we will need later can be deduced from the so-called Borsuk-Ulam theorem.

THEOREM 3. Let0'.E lR'\Z. For every k, lE N with k

>

I there is no continuous equivariant map (S2k-1, e27ria) ---7(S21-1, e27ria ).

Proof For 0'. E Q \Z the theorem is well-known, see [5]. Now let

0'. E lR \Q. Suppose that there is a continuous equivariant map h : (S2k-1, e27ria ) ---7 (S21-1, e27ria ) for k, I E N. Since {e27rina: n E Z}

= SI, there exists a sequence(nj) of integers such that

.lim

e27rinja

=

J-+ex>

-1. As h is continuous and equivariant, we have that for every z E

S2k-1

Therefore, it follows from Theorem 2 that k _~I.

h(-z) = h(.!im e27rinjaz)

J-+ex>

.!im e21rinjah(z)

J ...ex>

-h(z).

o

Thus Theorem 3 is a generalization of the Borsuk-Ulam theorem.

The basic idea is to use the fact that {e27rina :n E Z} = SI if0'. is an irrational number. It will be shown in Theorem 10 more generally.

Extensions of the Borsuk-Ulam theorem 601

3. Extensions to finitely many products

By modifying Theorem 3, we can now obtain the following results on finitely many products of spheres. For j = 1,'" ,n, let kj , lj be natural numbers, let

J:= {j

E {I"" ,

n}:

kj

>

lj}

i= 0,

and let Pj, qj be nonzero integers such that _~ is in reduced form and qj

i=

^l.

THEOREM 4. Let aj

=

~ be in reduced form for j

=

1,'" ,n.

Suppose that there is a jo E

J

such that the numberqjo does not divide the least common multipleof {qll'" , qjo-l, qjo+!,'" , qn}. Then there is nocontinuous equivariant map

h: (S2k¹-1

x···

^X S2kⁿ

-l,a)

^---? (S2^h-1

x···

^X S21ⁿ

-I,a),

Proof. Suppose that there is a continuous equivariant map h. Let c:= lcm{ql,'" ,qjo-l,qjo+l,'" ,qn}. Hence, for every Z = (Zl,""

zn) E S2k¹-1 X ... X S2kⁿ

-1,

we obtain

h(

Zl,'" ,Zjo-be27l"ico<'JOzjo,Zjo+I,'" ,Zn)

h(

e27l"ica¹Zl ... e27l"icO<joz· ... e²7l"icO<nZ )

" 30" n

= (e27l"icO<l hI (z), ... , e27l"icO<jo h jo (z), ... , e²7l"icO<nh n (z)) (hI (z), ... , h jo - l (z), e27l"icO<jo hjo (z), h jo+! (z), ... , hn(z)).

Let aj E S2kr l be fixed for each j E {I"" ,n} \

{io}.

Define

h:

S2kjo-1 ---* S21jo-1 by

for x E S2kjo -l. Then

h

is continuous and equivariant, because

27l"ico<' h ( )

e JO jo al,'" , ajo-l, x, ajo+b ... , an e27l"icO<jo

h(

x).

for every x E S2kjo -l. Since qjo does not divide c, by Theorem 3, this

contradicts the fact that jo E

J.

0

We now turn our attention to the case when al,' .. ,an are taken as rational and irrational numbers together.

THEOREM

5.

Suppose O'.il' jl E

{l,···

,n}, is the only one irra- tional number andO'.j = _~, j E

{l, ... ,n}\ Vd,

arerational numbers in reduced form.

(1) Hjl E

.1,

then there is no continuous equivariant map

(2) Hjl (j.

.1

andifthe numberqjo does not divide the least com- mon multiple of{qI, ... ,qn} \ {qjo, qj1} withjo E

.1,

then the same conclusion holds.

Proof. For simplicity, we may suppose that 0'.1 is an irrational num- ber and 0'.2, • .. ,O'.n are rational numbers. There are two cases to con- sider.

Case 1 : 1 E

.1;

that is k1

>

11 . Assume that there is a continuous equivariant map h. Since ^0'.1 E ~\Q, there exists a sequence

(nk)

of integers such that

lim e27rinkQ2···qn0l1 = e27ri0l1.

k->(X)

Furthermore, e27rinkQ2"'QnOlj = 1 for j = 2,· .. ,n and for all kEN.

Since h is continuous and equivariant, it follows that

for all z

=

(ZI,··· ,zn) E S2k1^-1 X ... X S2kⁿ-l. Hence the func- tion hI : S2k1-1 - 7 S2l1-1, X ~ hI(x, a2,· .. ,an), for fixed aj E S2kj-l, j = 2,···

,n,

is continuous and equivariant. By Theorem 3, this contradicts the fact that k₁

>

1_{1 .} Thus the proof of the statement (1) is complete.

Case 2: 1

(j..1.

Without loss of generality we may suppose that 2E

.1;

that is k2

>

1_{2 .} Assume that there is a continuous equivariant map h.

Let c be the least common multiple of {q3' ... ,qn} such that q2 does not divide c. Hence there exists a sequence

(nk)

of integers such that

and lim ^e27rimk0l1 = l.

k->(X)

It follows that

Extensions of the Borsuk-Ulam theorem 603

and e2-rrimkOtj

=

1 for j

=

3, ... ,n.

Therefore, for all Z= (ZI, ... , zn) E S2k^{1-1 X ... X} s2k^n-1, we have

h(

ZI,e^2-rricOt2Z2,Z3,··· ,Zn⁾

which, as in the proof of Theorem 4, leads to a contradiction to k₂

> b.

This proves the statement (2). 0

Theorem 5 can be formulated in more general situation as follows.

THEOREM 6. Let Uj be a real number for j

=

1,··· ,n. Suppose that for some jo E

:r,

there exists a sequence

(mk)

ofintegers such that

for j = jo

for jE{l,···,n}\{jo}·

Then there is no continuous map

h: S2k¹-1 x ... ^X S2kn-1 __ S2h-1 X .•. X S2lⁿ-l such that h is equivariant with respect to

a.

Proof. Without loss of generality we may suppose that jo = l.

Assume that there is a continuous equivariant map h. Then, for all Z

=

(Zl'··· ,zn) E S2k¹-1 x··· ^X S2kn-1, we obtain

h(e2-rriOtl ZI, Z2,···, zn)

= h ( lim e^2-rrimkOtlZl,· .. , lim e2-rrimkOtⁿzn)

k-->oo k-->oo

_ klim (e2-rrimkOt lh1(z), ... , e27rimkO!nhn(z))

-->00

= (e 2-rriO!lh1(z), h2(z),···, hn(z)).

As in the proof of Theorem 5, a similar argument leads to a contadic-

tion, and the proof is complete. 0

REMARK. In fact, there is no sequence (mk) of integers such that lim e27rimkQ!1 = e 27riO<l and lim e 27i-imk Q!2 1

k ...oo k ...oo

if aI, a2 E lR \

Q

have the property Clal

+

^{C2a2 E}

Q

for some

Cl, C2 E

Q\ {O}.

We are now in position to study when there exists a sequence to satisfy the condition of Theorem 6. First, Remark leads us to the following definition.

DEFINITION 7. The n real numbersaI, . .. ,an will be called ratio- nally independentif any relation of the formclal

+ ..

·+enan = 0 with rational numbers Cl, . . . ,en implies that Cl

= ... =

^en

=

O. Countably many real numbers aI,··· ,an,"· will be called rationally indepen- dent if any finitely many numbers which belong to {an :n E N} are rationally independent.

Next, the following resultisbased on the main theorem of Kronecker.

LEMMA 8. Letal,··· ,an,!3l ,··· ,(3n be real numbers. Ifthe num- bers 1,aI, .. ' ,an arerationally independent, then the system of the n inequalities

I

ajXn+l - Xj - (3j

I <

^{E forj =}

1,···

,n has solutionsXl,··· , Xn+l E Z for anyE

> o.

A

proof of this lemma can be found in Perron [9].

Using Lemma 8, we can prove a result which gives a sufficient con- dition for the existence of a sequence stated in Theorem 6. It is also useful for a classification of the type of orbits, see [7].

THEOREM 9. Let 1,al,··· ,an be rationally independent. Then, for every(Sb··· ,sn) E (Sl)n, there exists a sequence(mk) ofintegers such that

lim e27rimkQ!j

=

^Sj

k ...oo for j

=

1, ... ,n.

Extensions of the Borsuk-Ulam theorem 605 Proof. Let (Sl,"',Sn) E (Sl)n. Then there is aj3j E [0,1] with

Sj = e27ri{3j for j = 1,'" ,n. For every positive integer k there exists an Ek

>

0such that for all

x

ER with

Ix -

j3j

I <

^Ek mod 1 we have

for j = 1, ... ,n.

By Lemma8, there are integers mk, Ykl,··· , Ykn such that for j = 1, ... ,n .

For j = 1,··· ,n, since

I

e27rimkCtj - e27ri{3j

I <

^{2-k ,} we conclude that

lim e27rimkCtj k--oo

- e 27ri{3j - s·

- - J'

o

4. An extension to countably many products

The following result is an analogue of Theorem6for countably many products of spheres.

THEOREM 10. Let Q.j be a real number and kj , lj natural numbers for everyj E N. Suppose that for somejo E {j EN: kj

>

lj}, there exists a sequence

(nk)

of integers such that

for j = jo for j EN\{jo} . Then there is no continuous map

h:

IT

^{S2kj -l - - - t}

IT

^S21j-l

jEN jEN

which is equivariant with respect to 0:

=

(e27riCtl, . .• ,e21riCtj , .•• ).

Proof. As in the proof of Theorem 6, a similar argument establishes

the result for countably many products. 0

We can make a modification of Theorem 9 with the aid of a diagonal process.

THEOREM 11. Let aj,!3j be real numbers for every j E N. If 1, al,··· ,G-j,··· arerationally independent, then there is a sequence

(nk)

ofintegers such that

lim ^e27rinkOlj

k-+oo for all j EN.

Proof. Let 1,aI, ... ,aj,'" be rationally independent. By Lemma 8, for every n E N, the-re is a sequence (mn,k) of integers such that

for j

=

1"" ,no

We now consider the sequence (mk,k) on the diagonal. Then, for all kEN, we have

for j = 1"" ,k

which implies that lim e27rirnk,kOlj = e^27ri/3j for all j EN. 0

k-+oo

Here we have seen that the Borsuk-Ulam theorem is preserved un- der certain conditions related to the number theory. The generalized Borsuk-Ulam theorem is fundamental in measuring complexity of al- most periodic orbits.

5. Application

Inthis section we give a definition of a Z-index induced by a home-- omorphism of a compact space. Then a Z-version of the Borsuk-Ulam theorem plays a fundamental role in an index theory for Z-actions.

Moreover, an index theory for group actions is important from the viewpoint of applications to differential equations, see [1], [2], [6] and

[8].

Given a continuous action1r :G x X ^----l-X of a topological group G on a topological space X, we denote

E(X,G)

:=

{A

eX:

A

is G-invariant}.

Extensions of the Borsuk-Ulam theorem

AG-index is a mapping

i: _~(X,G)- tNU {O,oo}

607

which has the following properties (1) i(A)

= °

if and only if A

= ^0.

(2) IfA, B E ~(X,G) and <P : A ^{- t} B is a continuous equivariant map, then i(A):::; i(B).

(3) IfA E ~(X,G) is a closed set, then there exists an open neigh- borhood U E ~(X,G) ofA such that i(A)

=

i(U).

(4) If A, B E ~(X,G) are closed sets, then i(AUB) :::; i(A) +i(B).

We consider a compact topological space X and a homeomorphism

f :

X ^{- t} X such that

(1)

{fn:

n E Z} is uniformly equicontinuous; and

(2) there is an irrational number a E [0,1] such that for every

x

E X there exists a homeomorphism <P :0

(x)

^{- t} SI with the property

<P(J(u))

=

e

²

1l"iO:<p(u)

for all

u

E

O(x)

where

O(x)

denotes the closure of

O(x)

:=

{fn(x) :

n E Z}.

Inthis framework we define the Z-index i(A) of an invariant subset A of X as the smallest integer k such that there exist an mEN and a continuous map <P :A - tS2k-l satisfying the following equivariance property

<P(J(u))

=

e

²

1l"imO:<ll(u)

for all

u

E

A.

In [7] we show that this is an index in the sense of the above definition.

References

[1] V. Benci, A geometrical index for the group SI and some applications to the study of periodic solutions of ordinary differential equations, Comm. Pure Appl. Math. 34 (1981), 393-432.

[2] H. Berestycki, J. M. Lasry, G. Mancini and B. Ruf, Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces,Comm. Pure Appl. Math.

38(1985), 253-289.

[3] G. E. Bredon, Topology and geometry, Springer, New York, 1993.

(4] C. Corduneanu,Almost periodic functions, Chelsea, New York, 1989.

[5] A. Dold, Simple proofs of some Borsuk-Ulam results, Contemp. Math. 19 (1983), 65-69.

[6] E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamil- tonian systems, Invent. Math. 45 (1978), 139-174.

[7] 1. S. Kim,Zu einer Indextheorie fur fastperiodische Aktionen, Doctoral Thesis, LMU Miinchen, 1995.

[8] J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems, Springer, New York, 1989.

[9] O. Perron, Irrationalzahlen, 4.Aufl., De Gruyter, Berlin, 1960.

[10] H. Steinlein,Spheres and symmetry: Borsuk's antipodal theorem, Topol. Meth- ods NonlinearAnal. 1 (1993), 15-33.

Department of Mathematics Sung Kyun Kwan University Suwon 440-746, Korea