B U L L E T I N O F T H E P O L IS H A C A D E M Y O F S C IE N C E S

M A T H E M A T I C S Vol. 48, No. 4, 2000

ALGEBRAIC TOPOLOGY

Borsuk-Ulam Type Theorems on Product Spaces Iby

Zdzisław DZEDZEJ, Adam IDZIK and Marek IZYDOREK

Presented by C. BESSAGA on January 27, 2000

S u m m ary . The ideal-valued G-index theory is used to prove Borsuk-Ulam and Bourgin- Yang type theorems on the product of spheres. A generalization of a theorem of Zhong is given.

1. In trodu ction . Let Sn denote the unit sphere in the Euclidean space Rn+1. The famous Borsuk-Ulam theorem states that for every continuous function / : Sn —► Rn there exists a point x £ Sn such that f ( x ) = f ( —x) (see [1]). It can be formulated also in the equivalent form:

T h e o r e m 1 .1 . Let f : Sn —> Rn be an odd function, i.e. f ( —x) = —f ( x ) for every x £ Sn. Then the set / -1 (0) is nonempty.

One of the most important generalizations of it is the Bourgin-Yang theorem (see [2, 6]):

T h e o r e m 1 .2 . Let f : Sn —► R k be an odd function. Then the covering dimension dim / -1 (0) ^ n — k.

In 1991 Zhong [7] extended the Borsuk-Ulam theorem to functions de­fined on the product of two spheres.

T h e o r e m 1 .3 . Suppose that f = ( / i , / 2 ) : Sn x Sm —> R n x Rm is a continuous function satisfying:

— f i ( ~ x , y ) = - f i ( x , y ) , f i ( x , - y ) = f i ( x , y ) for every (x , y ) £ Sn x S m;— f 2 { ~ x , y ) = f 2 ( x , y ) , f 2 ( x , - y ) = - f i { x , y ) for every (x , y ) £ Sn x S m.

Then there exists a point (x , y ) £ Sn X Sm such that f ( x , y ) = 0.

1991 MS Classification: 55M20, 47H04, 52A20.Key words: Borsuk-Ulam theorem, free G-space, G-index. Supported by KBN Grant 2 P03A 017 11.

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3 8 0 Z. Dzedzej, A. Idzik, M. Izydorek

It is easily seen that / is equivariant under the suitable action of the group Z2 X Z2 on R n + 1 X R m+1.

In this paper we present a natural generalization of the theorem of Zhong to the product of q spheres with the natural free action of the group (Z2)9, q £ N. We also generalize Theorem 1.2 to that case. As the main tool we use the ideal-valued G-index defined by Fadell and Husseini in [5]. Our proof is different from that of Zhong and gives a more general result.

2. G-spaces and the G-index. Throughout the paper we will use the Cech cohomology with coefficients in Z2, the group of integers mod 2. This particular cohomology theory is chosen because it is defined for paracompact spaces and has the continuity property, i.e.

H*(X, Z2 ) = fim H* ( Xn, Z2),

if X = flnGAT^n, X n+1 C X n and X n C Y for n € N, are paracompact subsets of a given paracompact space Y , X is closed in Y, and for each neighbourhood X C U there exists n such that X n C U.

Let G be the direct sum of q copies of the group Z2, G = (Z2)9, for some q £ N. Assume that G acts freely on a paracompact Hausdorff space X , i.e. for x G X and g G G, gx = x implies g = 0 in G. We call X a free G-space.

It is well known that any free G-space X admits an equivariant function h : X —► EG into a contractible free G-space EG (see [4]); any two such functions are equivariantly homotopic (see [4, Theorem 1.8.12 and Theo­rem 1.6.14]). The function h induces a function h : X —>■ BG on the orbit spaces X := X/G and BG := EG/G , which is unique up to homotopy. Consequently, we have the unique ring homomorphism

h* : H*(BG,Z2) ^ H*(X,Z2).

The space EG can be identified with the q-fold Cartesian product of the infinite dimensional spheres S°°:

EG = S°° x . . . x S°°

with a free action of G defined by

1 % q) ~ (2?1 * • • • * X ki • • • -,X q) ,

where gk are fixed generators of G, k = 1 , . . . , q . The orbit space BG is the Cartesian product of q-copies of the infinite dimensional real projective space P°°:

BG = P°° X . . . X P°°.It is well known that , Zf) is the polynomial ring Zo[x], where x

corresponds to the generator of H 1 ( P °° , Z2 ) — Z2 . By the Kiinneth formula

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Borsuk- Ułam Type Theorems on Product Spaces I 3 8 1

we obtainH * ( B G , Z i ) = Z 2[xu . . . , x q\,

the ring of polynomials of q variables. The elements x\, . . ,x q correspond to generators g1, .. . , g q of H 1 (BG, Z2) = ( Z2)q.

Now, let us recall the Fadell^and Husseini definition of the ideal-valued G-index, / G(X ), for a G-space X (see [5]) formulated for the particular case when A” is a free (Z2)9-space.

D e f i n i t i o n 2 .1 . The G-index of a free G-space X is the ideal I G( X ) = ker h* in the ring H* { BG , Z2) = Z2 [x\, x 2, ..., x q\.

Most of the properties of the G-index are immediate consequences of the definition. In particular, we have: _

— Monotonicity: if G acts freely on X and Y , and f : X -+ Y is an equivariant map, then I G(Y ) C I G(X) .

— Dimension: if dim X < m, then x j1 . . . x q G I G( X) whenever t\ + ... -f ^ m, where dim denotes the covering dimension.

An important special case of the above is:— Nontriviality: if I G( X ) ^ Z2 [ x i , . . ,x q], then J / 0.Let G act freely on X and let A C X be a compact G —space. Since the

Cech cohomology theory has the continuity property and ring Z2 [xi, . . ,x q] is Noetherian, we obtain: _

— Continuity: There is an open neighbourhood U of A in X , which is a G-space, such that I G(U) = I G(A).

The concept of the G-index was introduced by Yang [6] for G = Z2 and next extended to other more general settings by several authors, notably to actions of compact Lie groups by Fadell and Husseini [5].

3. Borsuk-Ulam type theorems. Let us fix a sequence of natural numbers ii\, n2,..., nq. For k = 1, . . . , q consider a subspace of EG

M k = S°° x . . . x S°° x Snk -1 x S°° x . . . x 5°°which is a G-space itself. Clearly,

M k = P°° x . . . x P°° x P nk -1 x P°° x . . . x P°°.

It is well known that the cohomology ring H*( Pm, Z2) is equal to the truncated polynomial ring Z2 [x] l {xm+l), m ^ 0. By the Kiinneth formula we obtain

H* ( Mk, Z 2) = Z2 [xu . . . , x q]/(xnkk), where by (a, ...,&) we denote the ideal generated by elements a, ...,&.

L e m m a 3 .1 . If M — IJ L i M k, then x ^ x ^ 2 . . . x qQ G I G{M).

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3 8 2 Z. Dzedzej, A. Idzik, M. Izydorek

P r o o f . Since a G-mapping h is unique up to G-homotopy, we can choose h =7 , the natural inclusion. Consider the diagram

M k EG

v v

M k — BG

and observe that the induced homomorphism

i* : Z2 [x1 , . . . ,x q\ -> Z2 [x1 , . . . , xq]/(xlk)

maps x k onto x k, k = 1,..., <7. Since /* is the ring homomorphism, we obtain

= = o.Therefore x^k is an element of I G( Mk).

Put M := \Jqk z = 1 M k and consider the long exact sequence of the pair = 1,

. . . H nk (M , M k) A H nk (M) - Ł H nk ( Mk) - > . . .

Let h : M —*• EG be the natural inclusion function and let h* : Z2 [ x i , . . . , x q] —> H * ( M ) be the corresponding ring homomorphism. Now, ik(h*(x^k)) = 0 because x^k € I G( Mk). Thus, there is an element a k £ H nk( M , M k) such that j k(ak) = h*(x^k).

From the following commutative diagram

H ni( M , M i ) ® . . . ® H n* ( M , M q) —=U- #« ,+ . . .+« , (M , M ) = 0

r

H n' { M ) ® . . . ® H n*(M) -------- y--------^ ^na + ...+n,(M )

where U denotes the cup-product (see [3]), we conclude

a i Ua 2U. . .Ua, = 0, ( i i 0 - . . .® a 9) = h'(x™1) ® . . .®h*(xq*),

and finally

0 = H O ) = h * ( x F) U h*(xJ2) U . . . U h*(x? ) = h 'ix ’l 1 •. . . • * ) .

This proves Lemma 3.1. □

Let X = Sni X . . . X SUq be a standard G-subspace of EG.

P r o p o s i t i o n 3.2. There is no G-equivariant function f : X —*■ M .

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Borsuk- Ułam Type Theorems on Product Spaces I 3 8 3

P r o o f . On the contrary, suppose that there exists / : X M . It induces a function on orbit spaces f : X —► M , where X = P ni x . . . X P Uq. Since H * ( P m, Zf) = Z 2 [z]/(zTn+1), we obtain from the Kiinneth formula

H \ X , Z 2) = z 2[x1,. . . ,x ,] / (x ? -+1, . . . ,x ^ +1).

Moreover I G( X ) = 1, x '^+1), which is a consequence of the factthat the ring homomorphism

t* : Z2 [xu . . . ,xq] -+ Z2 [xi, . . . ,x q\/(xi1+1, ...,Xgq+1)

maps x k onto x k, k = 1, . . . , q. Therefore l*{x” 1 . . . x q9 ) ^ 0.Consider the diagram

X M e g

T T TX — f-+- M — ^ BG.

By Lemma 3.1

/ * * • ( * ? ' . . . i j * ) = /* (0 ) = 0.On the other hand we have the following diagram

X — EG

X — 5 G

where 7 and t are inclusions. Since 7 and h o / are G-homotopic, the corre­sponding ring homomorphisms coincide:

l* = ( h o / ) * = / * o h * .This contradicts our previous calculations. □

Let R Ul X . . . x Rnq be a representation of G — (Z2)g with the action given by

9 k ( % l 5 •••» X k, ••••> % q) — ( i C i , . . . , X k , ••••) X q)

where gk are generators of G as before, k = 1, . . . , q.

T heorem 3.3^7/ / : 5 ni x S 712 x . . . x SHq —► Rni x . . . x RUq is a G-function, then / -1 (0) 7̂ 0.

P r o o f . Suppose that / _1 (0) = 0. Denote by irk : Rni x . . . X RUq —► 72n* the standard projection and let f k = irk o f .

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3 8 4 Z. Dzedzej, A. Idzik, M. Izydorek

The sets A k = /^_1(0) are G-spaces and by our assumption fj?=1 =We can choose 6 > 0 such that

f| Ns(Ai) = 01 = 1

and all ^-neighbourhoods Ns(Ak) are G-sets.For k = 1 , . . . , q define Xk ' SUl X . . . X SUq R by the formula:

Xk(x) = dist(a;, SUl X . . . x Sn* \ Ns( Ak)).

They are all G-invariant functions, because G acts by isometries.Now, denote by pk the composition

S "1 x . . . x Sn° — R n ' + 1 x . . . x R n - + 1 -> fln*+1

where the first arrow is the inclusion, and the second is the projection. Define gk{x) — Xk(%) • Pk(x ) and consider the function

F : Sni x . . . x SUq —► R 2 n i + 1 x . . . x R 2rlq+\F { x ) = (F1 ( x ) ,F 2 ( x ) , . . . , F q(x)) = (( /i(a :) ,f lf i(a :)) ;.. . ;( /9(a:),flr9(x ))).

One can easily check the following properties:— F is a G-function,— F f 1 (0) = 0 for every k = 1 , . . . , q,— F ( S ni x . . . x Snq) C ( J L i R2ni+1 x . . . x R n' x . . . x R2nq+1. The following formula

Fi(x) Fq(x) \

defines a G-function <p : .S'” 1 x . . . X Snq —> M contrary to Proposition 3.2. □

Let d i,...,d q be natural numbers and let SHl+dl X . . . X Sn<1+dq be the standard G-subspace of EG.

T heorem 3.4. If f : Sni+dl x . . . x SUq+dq R ni x . . . x R Uq is a G-function, then x dl . . . x q 9 (£ / G( / _1(0)).

P r o o f . Put A = / _1(0) and consider the diagram

A ------ ► sni+dl x . . . x SHq+dq — EG

y t yA ------► P ni+ di x . . . x P n«+d* — BG

where the left horizontal arrows are inclusions.

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Borsuk- Ułam Type Theorems on Product Spaces I 3 8 5

Assume that h*(xdl . . . Xq9) ^ = 0 G H*( A, Z 2 ). Then there exists a neighbourhood A of A such that h*(xdl . . .Xq9) ^ = 0. Denote by A the G-set p~l (N) which obviously is a neighbourhood of A.

Set S = Sni+dl X . . . X Sn<1+dq and P = P n 1+dl X . . . X p n9+rf«. Observe that f ( S \ A ) C RUl X . . . x R n<1 \ {0 }. Similarly to the proof of Theorem 3.3 we can define an equivariant function

F : ( S \ A ) — >M = IJ S2ni+dl x . . . x S'71* "1 x . . . x S2n*+d*i

which induces a function on orbit spaces

(S \ N ) / G - ± * B G .

By Lemma 3.1 h ( x ™ 1 . . . Xq9) = 0 and thus F*h ( x ™1 . . . Xq9) = 0, i.e.

(*) x ? . . . x ? € l a ( S \ Ń ) .

Now consider the exact sequence

f j m + ...+ nq ^ p ^ f f t n + ̂ .+ n g ^ p j j j n i + - -+ n q ^ j y y

Since I G(S ) = (:ri1+dl+1, . . . ,Xq9+dq+1), h * ( x . . . Xgq) is a nontrivial ele­ment in f f ni + ■••+",(p ) .

By (*) . . . Xgq) = 0. Thus there exists a nonzero element

a G H n' + - +n* ( P , P \ A )

such that j{ot = .. .Xq9).On the other hand we consider the element

0 £ h*(xdl .. . x d«) G H dl+" +d' (P) .

By our assumption . . . Xq9) = 0 in the exact sequence

+ 2 L > J J d i + — + dq ^ p ^ J J d i + . . . + dq ^ y

Consequently, there exists /3 G H di+—+d(> (P, A ), (3 ^ 0, such that j£(/3) = h*(xdl .. .Xq9). Now we use the commutative diagram (see [3]):

H di + - + dq( P , N ) ® H ni + - +n* ( P , P\ A ) ^ H ni+" m+n' +dl + "'+d<(P, P) = 0

3\ ®j'2 jT v

p d i + .- .+ d q ^ p ^ 0 jp n i + ...+ nq ^ p ^ ------ U____ ^ j]T ii + ...+ nq+ d 1 + ...+ dq ^ p y

Since / 5Ua = 0, then j*((3 U a) = 0. Therefore

o = Ji*(/3)Uj2*(a) = h ' ( x i ' . . . x d, ' ) U h ' ( x ? . . . X ? ) = h \ x rij + d i 1

rf.nq -j~dq x q

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3 8 6 Z. Dzedzej, A. Idzik, M. Izydorek

which is in contradiction with the fact thatI G(S) = ( x i1+dl+1, . . . ,Xqq+dq+1). □

Corollary 3.5. Let f : Sn 1+dl x . . . x Snq+dq — > R ni x . . . x RUq be a G-function. Then the covering dimension of the set / -1 (0) satisfies the inequality

dim / -1 (0) ^ d1 + . . . + dq.

P r o o f . From Theorem 3.4 it follows that

Hdl+- +dq(f~1(0),Z2) ± 0. □Acknowledgements. The authors are deeply indebted to the referee for

her/his valuable comments.

I N S T I T U T E O F M A T H E M A T I C S , G D A Ń SK U N IV E R S IT Y , W IT A S T W O S Z A 57, 80-952 GDA Ń SK , P O L A N D (Z D )

( I N S T Y T U T M A T E M A T Y K I , U N I W E R S Y T E T G D A Ń S K I) e-mail: zd zedzejO k sin et.u n iv .gda .p lI N S T I T U T E O F C O M P U T E R S C IE N C E , P O L IS H A C A D E M Y O F S C I E N C E S , O R D O N A 21, 01-237

WARSZAWA, P O L A N D (A l)( I N S T Y T U T P O D S T A W I N F O R M A T Y K I PA N ) e-m ail: ad idzikO ipipan.w aw .plD E P A R T M E N T O F M A T H E M A T I C S , T E C H N I C A L U N IV E R S IT Y O F G D A Ń S K , N A R U T O W I C Z A

1 1 /12 , 80-952 G D A Ń S K , P O L A N D (M I)(W Y D Z I A Ł M A T E M A T Y K I , P O L I T E C H N I K A G D A Ń SK A ) e-m ail: izydorek O m ifg ate .p g .gd a .p l

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