Topological degree and the theorem of Borsuk for general covariant mappings with applications

Nonlmeor Anulysis. Theory, Printed in Great Britain.

Methods

& Applications,

Vol.

16, No. 5, pp. 399-420,

0362-546X/91 $3.00+ .Oil C 1991 Pergamon Press plc

1991.

TOPOLOGICAL DEGREE AND THE THEOREM OF BORSUK FOR GENERAL COVARIANT MAPPINGS WITH APPLICATIONS PATRICK J. RAEXER Institute

for Computational

Mathematics and Applications, Pittsburgh, Pittsburgh,

Department of Mathematics PA 15260, U.S.A.

and Statistics,

University

of

(Received 1 September 1989; received for publication 30 July 1990) Key words and phrases: Topological maximal

degree,

transversality,

group

representation,

covariant

action,

torus.

1. INTRODUCTION

AND

PRELIMINARIES

IN TERMS of topological degree, the celebrated theorem of Borsuk [3] reads as follows: given a bounded open subset a of R” symmetric with respect to the origin and such that 0 E Q the degree d(f, C&O) of any odd mapping f E Co@; R”) such that 0 $ f(iFKl), is odd. The original result by Borsuk deals with the case when Sz is a ball and is not expressed in terms of topological degree, but the above statement is equivalent, only more general regarding the geometry of 52. An alternate way of saying that a mapping is odd is to say that it is covariant under the action of the group Z, = (1, - 1) represented by the operators Z and -Z in GL(R”)-the group of linear isomorphisms of R”. On the other hand, a subset of R” is symmetric with respect to the origin if and only if it is invariant under (I and) - I. Finally, an integer is odd if and only if it is not congruent to 0 mod 2, and 2 = 1Z21, the order of Z2. The theorem of Borsuk can then be rephrased as follows: let G = Z2 and let R denote the representation (I, - 4 of G in GL(R”). If C2 is an open subset of R” invariant under R with 0 E Szand if f E C?“(Ct; R”) is covariant under R with 0 $ f(Kl), then d(f, Cl,0) # OmodlGI. Instead of d(f, Q 0)# 0 modlG[, one could have said that d(f,Sz, 0)= 1 modlGl, or that d(f,Cl, 0)= - 1 mod1 GI: all these statements are equivalent since ICI = 2. Among other things, we shall here prove results of this nature when G = Z, is replaced by an arbitrary compact Lie group, possibly finite, and f is a covariant mapping. A number of authors have investigated the problem of extending the theorems of Borsuk or Borsuk-Ulam in varieties of forms, especially during the past decade or so. Relevant contributions can be found in [5,7,8, 12, 13, 15, 18, 19,21,23], among others. However, most efforts have been devoted to special configurations (balls, spheres) or focused on the case when the representations are the same (or equivalent, a trivial variant) in the source and target spaces. Restriction to balls or spheres allows one to use both the machinery and concepts of algebraic topology, which are not available for completely general configurations. On the other hand, assuming that only one group action is involved is a significant simplification. In either of these assumptions, sometimes both, sharp formulas for the degree or even the fixed point index have been obtained. A distinctive feature of this paper is that the results we prove are valid for invariant open subsets of arbitrary shape while the representations R and S of G involved in the source and 399

400

P. J. RABIER

target spaces, respectively, may be different. Some conclusions do not even require making any assumption about S. The price we pay for such a generality is that some formulas we give for the degree are not sharp: essentially, they consist of statements guaranteeing that the degree is nonzero (for finite groups, we get a bonus: the degree is a unit modulo ICI). However, as is well known, nothing more is needed in many applications. When the representations are equivalent, our conclusions are more precise but some of them also follow from a theorem of Nussbaum [19]. Still, our approach yields a much less technical proof (the details of which are not given in [19] for the general case). Expanded comments about the relationship of this work to others in the literature are given in Section 5. Another distinctive feature of this paper is that the approach taken here leads to a principle for proving existence of solutions to covariant nonlinear equations that preserve some symmetry. To corroborate the value of this principle, we have used it to prove a new theorem for bifurcation problems involving symmetries. This theorem is of a spirit rather different from those to be found in the related literature. To complete with introductory comments, we point out that our exposition is self-contained as far as degree, assuming knowledge of only the basic notions. Aside from elementary properties of finite groups, major ingredients are transversality theory and standard Lie groups theory. No use is made of any argument from algebraic topology. The remainder of this section is devoted to some preliminaries regarding notation as well as basic definitions and results to be used throughout this paper. In what follows, G will always denote a compact Lie group, possibly finite, multiplication being the group operation. Accordingly, 1 will be the neutral element of G. If G is finite, ICI stands for the order of G. By representation of G in GL(R”), we mean a continuous group homomorphism R : y E G -+ R, E GL(R”). Note that R, = Z (1 = neutral element of G). Two representations R and S of G are said to be equivalent if there is an element A E GL(R”) such that S, = A-‘R,A (i.e. AS, = R,A),

G. The representations R and S are said to be compatible if det R, = det S,, V y E G, and not compatible otherwise. Obviously, equivalent representations are compatible. A subset XC R” is said to be invariant under the representation R, or R-invariant, if R,X c X, v y E G. This is equivalent to R,X = X since RY’ = R,-1 from R being a group homomorphism. Balls B,(O) (or B,(O)) with center 0 and radius r > 0, relative to the usual euclidian structure of R”, are invariant under every orthogonal representation of any group. Given a R-invariant subset X c R”, a mapping f:X -+ R” is said to be (R, S)-covariant if f(R,x) = S,f(x), v y E

d(f, Cl, 0) = 0 if R and S are not compatible.

Theorem of Borsuk

for covariant

If G is finite and X C R” is R-invariant, let covariant part off, denoted by f
lX-‘_f(x)

=

&

c

401

mappings

f:X + R” be any mapping.

The

(R, S)-

(1.1)

S,'f(R,x)E R”.

-fcG There is no difficulty in checking that the mapping J is indeed (R, S)-covariant and that f”= if f is (R, S)-covariant. Finally, recall that the representation R is said to be faithful if it is one-to-one, i.e. R, # vy~G,y#l. 2. THE

MAIN

THEOREM

FOR

FINITE

GROUPS

AND

SOME

OF

ITS

f I,

CONSEQUENCES

One possible proof of Borsuk’s theorem begins with showing that, given a bounded open subset fi of R” symmetric with respect to the origin with 0 $ S2, the degree d(f, Q, 0)of any odd mapping f E C?'(ai; R”) with 0 $ f (6X2), is even. The general form of this result is given below. 2.1. Let G be a finite group and let R and S be two representations of G in GL(R”) such that R is faithful. On the other hand, let Q be a bounded open subset of R” invariant under R and such that R,x # x for every x E Q and every y E G, y # 1 (in particular, 0 $ a). Finally, let f E e"(~i; R”) be (R,S)-covariant and such that 0 $ f(dQ). Then THEOREM

d(f,L2,0) = 0 modlG]. Proof.First, observe that faithfulness of R is necessary, for otherwise Sz = 0 is the only open subset satisfying the required hypotheses. From now on, we assume C2 # 0. The case G = (1) being trivial, suppose ]G] 2 2. Also, d(f, Cl, 0)= 0 if R and S are not compatible (see Section l), so that one may assume that R and S are compatible. With the usual perturbation argument, it suffices to consider the case when f E CT'(CJ; R”). Note that covariance may be preserved by replacing the smooth perturbation by its (R,S)-covariant part. Suppose first that 0 is a regular value off, sothat d(f,Cl, 0)=

c

sgn detf’(x).

x K-l(o) From (R,S)-covariance, f-'(O)is R-invariant. Also, for x E Q, the points R,x E Q are all distinct. Indeed, R,x = R,,ximplies R,x = x with 6 = y-iy’, so 6 = 1 by hypothesis. Using once again (R,S)-covariance, one finds f'(R,x) = S,f'(x)R;'. As sgn det R, = sgn det S, by compatibility of R and S, it follows that sgn detf’(R,x) = sgn detf’(x), v y E G. All these remarks show that the above sum can be rearranged as a sum of terms all equal to 3~1GJ, so that d(f,Q 0)= OmodlG/. It remains to examine the case when 0 is not a regular value off. Instead of trying to devise a tricky construction to reinstate regularity in some (R,S)-covariant perturbation off, we use the following claim. CLAIM. The subset of em(fi; R”) of those mappings g such that 0 is a regular value of the (R,S)covariant part g of g is dense for the Whitney Co topology. We shall refer to [lo] for the definition of the Whitney (3’ topology and all other notions of transversality theory to be used next.

402

P.

J. RABIER

The proof is easily completed from the claim and the following remarks: for x E a, set E(X) = dist(x, 8Q). Then, U = (g E e”(Q R”); /g(x) - f(x)] < E(X), vx E Sl) is open in the Whitney Co topology. This follows from continuity of E (as is easily checked, E is Lipschitz continuous). Moreover, f E U since E > 0 on a, hence U # 0, and every g E U can be continuously extended to an element of e”(O; R”) by setting g = f on %2 (note E = 0 on aa). Using the claim, pick g E U such that 0 is a regular value of g. As g = f on K? andf is (R, S)covariant, one has g = f on &2. Thus, d(f, Cl,0) = d(g, 0, 0) = 0 modlG] as was seen before. We now prove the claim through a simple application of Mather’s multijet transversality theorem. Clearly, it suffices to show that for every 0 I r I n - 1 the subset of (?“(a; R”) of those mappings g such that either g(x) # 0 or g(x) = 0 and rank g’(x) # r for every x E Q is residual in the Whitney Cm topology. Set /GI = p L 2 and number the elements of G from 1 to p, beginning with y, = 1, say R and S are given by the families G = 1~1 = 1, ~2, . . . . y,]. The representations

R = {R, = I, . . . . R,],

s = (S, = I, . ..) S,)

where Ri (respectively Si) stands for Ryi (respectively ST,). As already observed, RiX # RjX, 1 I i, j I p, i #j, vx (x, R 2x, . . . , Rpx) E &” where @p’ = ((Xi )

E Cl.

Hence,

given x E Q, one has

1 Ii,jSp,i#j).

. . . , X,)E@;Xi#Xj,

Let then 0 I r 5 n - 1 be fixed and suppose g(x) = 0, rank g’(x) = r for some x E Q. This means (see (1.1)) .E$=ISklg(Rkx) = 0 and rank Cfl=,S,‘g’(R,x)R, = r. Put in other terms, the p-fold l-jet jdg(x, R,x, . . . . Rpx) = (x, . . . . RPx, g(x), . . . . g(R,x), g’(x), . . . . g’(R,x)) intersects the subset W of a@) x (R”)p x (.S(Rn))p defined by W = W, x W2 x W, and (x 1, a.., xp) E W, e x, =

Rkx,,1 I k I p

P (t

1,

. . ..r.>

(A i, . . ..A.)

E

w,

*

c k=l

E W, w rank

&AZ,

i

=

0

SklAkRk

= r.

k=l

Obviously, WI is a submanifold of &2@’with dim W, = n, W, is a subspace of (R”)p with dimension np - n (W, is the null-space of a surjective linear mapping; recall S, = I) and W, is a submanifold of (6:(R”))P with codimension (n - r)’ (note that W, is the inverse image of the submanifold I/ c $(R”) of operators with rank r through a surjective linear mapping; that V is a manifold with codim V = (n - r)2 is well known). A simple count reveals that W has codimension np + (n - r)2 in Q2@’x (R”)p x (.AZ(R”))~ and hence codim W > np = dim Q”‘) since 0 5 r I n - 1. From the multijet transversality theorem (see [lo]), the mappings g E CY’(Q R”) such that jdg intersects W transversely form a intersection residual set. On the other hand, codim W > dim no’) implies that transversal n amounts to nonintersection, and we are done.

Remark 2.1. It is not clear whether standard

results

about

G-transversality

the proof of theorem 2.1 could be shortened by using (as for instance in [2] or [9]). The problem is that

Theorem

of Borsuk

for covariant

mappings

403

coincide with transversality general (if G # 11)) and the definition whether it does in the hypotheses theorem 2.1. Note when G = Z, and R = S = {Z, -Zj that the hypotheses about a in theorem 2.1 do reduce to saying that Sz is symmetric with respect to the origin and 0 $ Q. In the remainder of this section, we shall prove some corollaries to theorem 2.1, including a first generalization of Borsuk’s theorem. We now introduce a useful notation: given a representation R of some finite group G in GL(R”), we shall set C, = (x E R”; iny E G, y # 1, R,x = x)

(2.1)

and, given two open subsets Sz and o of R” CR(O, cl) = c,

In particular, C, = C,(0)

R”).

n (sz -

0).

(2.2)

Note that CR(w, Q) is closed, hence compact if Sz is bounded.

Corollary 2.1. Let G, R and S be as in theorem 2.1. Next, let w c Sl C R” be two bounded open subsets of R” invariant under R and let f E e”(O; R”) be (R, S)-covariant. Suppose rhat 0 $ f(Xl U &a U C&o, a)). Then d(f, Q, 0) = d(f, a, 0) modiGi. Proof. By excision of do, one has d(f, i-l, 0) = d(f, Q - ao, 0). Since Sz - ao is the disjoint union o U (Q - a), one has d(f, Sz, 0) = d(f, o, 0) + d(f, Q - c3,O). The desired result is then equivalent to d(f, fi - Q, 0) = OmodlGl.

(2.3)

As 0 $ f (CR(o, Q)) by hypothesis and from compactness of CR(o, a), the excision property yields d(f, Q - Q, 0) = d(f, a, 0), where we have set Q = (Q - a) - CR(W, Q) = n -’ (ii, u CR(O, n)).

(2.4)

The set C, is invariant under R. Indeed, if x E C,, y. E G, y. # 1, and Rrox = x, then = RrRrox = RYrox, v y E G. Using yyo = yi y with y1 = yyoy-’ and since yr # 1 from y. # 1, one finds R,x E C, . As o and Q are invariant under R, the same is true of Q and a. Because G is a group, Rinvariance is unaffected by replacing a set by its complement in R”. The operations of taking union or intersection of R-invariant sets do not affect invariance either, and it follows from all this that CR(o, a), ii, U CR(o, Q) and, finally, fi in (2.4), are invariant under R. If x E fi, then R,x # x, tl y E G, y # 1. In order to make theorem 2.1 available with fi replacing a, thus proving (2.3), it remains to show that 0 $ f (aa). Given a compact set K c R”, one has a(Q - k) c K U t&2. Taking K = 0 U CR(o, a) yields ad c &2 U Q U CR(w, a). But no point of o is in the closure of d so that afi c aQ U do U C&w, Q). The condition 0 $ f(ab) thus follows from 0 @f(S2 U do U C&O, cl)). n

R,x

Recall that the representation R of the group G is said to be free if C, = (01, namely vx E R” - (0). For instance, with G = Z, = ( - 1, 11, the b’y~G, y#l, representation R, = I, R_, = -Z is free. With n = 2 and G = Zk represented by R as the

R,xfx,

404

P. J.RABIER

group generated by the rotation of angle 2n/k, one finds that R is free for every k. It should be noted that existence of a free representation R when lG[ 2 3 requires that n be even: first, it is clear that a nonfaithful representation cannot be free irrespective of n being even or odd. Next, if n is odd, either 1 or - 1 is an eigenvalue of R, for every y E G. Hence, 1 is an eigenvalue of Rt = Ryz. If R is free, it follows that Rt = I, v y E G, so that the eigenvalues of R, are f 1. Thus, R, = Z or R, = -I, tl y E G, since R is free. But this is possible only if [Cl = 2 because R is faithful. If R is free, CR(o, a) = @ if 0 E o and corollary 2.1 is available if 0 ef(X2 U &IO).As an application of this remark, we obtain the following property of invariance of the degree, modulo 1GI . 2.2. Let G, R and S be as in theorem 2.1 and suppose in addition that R is free. Let fi2, and Q2, be two bounded open subsets of R” invariant under R such that 0 E fi2, n Q. Finally, let f~ C”(Qz,; R”), g E C3°(Qi,; R”) be (R, S)-covariant and suppose 0 @f(%2,), 0 $ g(X$). Then COROLLARY

W,&,O)

= d(g,%,O)modIGI.

Proof. Choose r > 0 small enough, so that the ball B,(O) is contained in Q, fl Q, and let h E (?‘(a, ; R”) E > 0 be so small that w, = e&2i c B,(O) c fi2, fl k&. Define the function through h(x) = f(x/c). It is trivial that o1 is invariant under R, that h is (R, S)-covariant with 0 $ h(aw,) and that d(h, oi ,0) = d(f, a,, 0) (use the product formula for the degree). As o1 n X& = 0, one may extend h as a continuous function on 0, such that h = g on aa,. Moreover, replacing h by its (R, S)-covariant part (see Section 1) one may assume that h is (R, S)-covariant and, still, h = g on an,. As 0 $ h(aC&) = g(iQ), one has 0 $ h(&o, U &A,). Also, 0 E oi C 0,. From the remark preceding the corollary, d(h, ol, 0) = 4h, Q, 0) mod/GI, while d(h, Q, 0) = d(g, C&, 0) since h = g on X2, and the proof is complete. n The practical value of corollary 2.2 is obvious: to calculate d(f, Q2,, 0) mod[Gl, it suffices to calculate d(g, &, 0) mod/G1 where a2 and g can be chosen once and for all. As a particular case of corollary 2.2, let us note the following corollary. COROLLARY 2.3. Let G, R and S be as in theorem 2.1 and suppose in addition that R is free and S is equivalent to R (so that S is free, too) say S, = A -‘R,A, v y E G. Let Q be a bounded open subset of R” invariant under R such that 0 E fi and letf E (?‘(a; R”) be (R, S)-covariant. Then, if 0 ef(as2) one has

d(f, fi, 0) = sgn(detA) More particularly,

mod/Gl.

if R = S above d(f, Q 0) = 1 modlG1.

Proof. Suppose first that R = S. As g = Z is (R, R)-covariant and 0 E Q, corollary 2.2 with Q2, = fi2, = Q yields d(f, Q, 0) = 1 modlGI. If now R and S are equivalent, with S, = A -‘R,A, then the mapping Af is (R, R)-covariant and the conclusion follows from the first part of the proof and the product formula for the degree. H

405

Theorem of Borsuk for covariant mappings

Remark 2.2. If Q and fare as in corollary 2.3, then f(0) covariance off and the fact that R and S are free. 3. A RECIPROCITY

= 0. This is trivial

from

(R, S)-

FORMULA

We shall next give a generalization of corollary 2.3 in the case when the representations R and S are free but not necessarily equivalent. This will follow from an interesting “duality” between (R, S)- and (S, R)-covariance. THEOREM 3.1 (reciprocity

formula). Let G, R and S be as in theorem 2.1 and suppose in addition that R and S are free. Let 02, and C& be two bounded open subsets of R” such that 0 E 52, n 52,. Suppose also that Q2, is invariant under R and 0, is invariant under S. Finally, let f E C?‘(L=& ; R”) be (R, S)-covariant and g E e”(@; R”) be (S, R)-covariant and suppose that 0 ef@Q,), 0 $ g@Q). Then d(f,Q,O)d(g,%,O)

= 1modlGi.

Proof. Upon possibly changing the norms in R” in two different ways, one may assume that R and S are orthogonal. Now, consider the function F E C”(Q, x &; R” x R”) defined by F(x, y) = (f(x), g(y)). As a(Q, x Q,) = (asZ, x &) U (ai, x X&), one has (0, 0) $ F(W2, x Cl,)). Also, for y E G, (R,x, S,y) E fi2, x C&, v (x, y) E Cl, x C& and F(R,x, S,y) = (f(R,x), g(S,,y)) = &f(x), $g(y)). In other words, Q2, x Sz, is R-invariant and F is (R, s)-covariant where R and S are the representations of G in GL(R” x R”) given by

It is trivial that E and s are free since the same is true of both R and S. Moreover, are equivalent, for s7 = A -‘I?_$ where A=A-‘=

0

z

z

0

a and 9

(> *

Observe in passing that det A = ( - 1)“. Since (- 1)” = 1 modlGl when IGI I 2 or when n is even (the only two cases when the hypotheses of the corollary make sense) one finds from corollary 2.3 that d(F, Q2, x Cl,, (0,O)) = 1 modlG1. On the other hand, it is trivial that d(F, Q2, x 4,

(O,O)) = d(f, Q2,, 0) d(g, Qz, 0).

n

Remark 3.1. Among the by-products of theorem 3.1, one finds the degree modulo covariant mappings from that of (R, S)-covariant ones. We also need the following

technical

LEMMA 3.1. Let R and S be representations

is free and orthogonal. 0 @ g(dB,(O)).

Then,

1GI of (S, R)-

lemma.

of a finite group G in GL(R”) and suppose that R there is a (R, S)-covariant mapping g E C?‘(B,(O); R”) such that

406

P. J.RABIER

Proof. Since the result is obvious if G = (11, we assume IGI L 2. As R is free, one has R,x # R,tx, v y, y’ E G, y # y’, vx E R” - (0). With [GI = p 2 2 and in the notation of the proof of theorem 2.1, this means (x, R,x, . . . , R,,x) E (R” - lo))@‘, v x E R” - (0).

If h E (?Y(R” - (0); R”) is such that there is x E a&(O) with 6(x) = 0, then the p-fold O-jet @z(x, . . . , R,x) = (x, . . . , Rpx, h(x), . . . , h(R,x)) intersects the subset WC (R” - lo))@) x (R”)p defined by W = W, x W, where (x 1, **a,XP) E w, # xk = &Xl ,

lsksp,

IXll = 1,

P

c

(t ,,..., &,)Ewz”

&‘<,=o.

k=l

Obviously, WI is a (n - 1)-dimensional manifold and W, is as in the proof of theorem 2.1, namely a (np - n)-dimensional space. It follows that codim W = np + 1 > np = dim(R” - lo))@‘. Therefore, the multijet transversality theorem ensures that $h does not intersect W for h in a residual subset of C”(R” - 10); R”) relative to the Whitney em topology. This proves existence of h E C”(R” - (0); R”) such that i;(x) # 0, vx E d&(O). Since the representation R is orthogonal, both B,(O) and aB,(O) are R-invariant. With h as above, set g(0) = 0,

g(x) =

I-m-4xl)

Then, g E (?‘(a; R”) and g is (R, S)-covariant n g = h”on M,(O), so that 0 $ g(M,(O)).

if x E B,(O) - (0).

(as R is orthogonal,

it is norm-preserving).

Also,

The conclusion of lemma 3.1 is trivial when there is a (R, S)-covariant linear isomorphism. But this means that S is equivalent to R, a case already covered by corollary 2.3. We are now in a position to prove the generalization of corollary 2.3 mentioned earlier. 3.2. Let G, R and S be as in theorem 2.1 and suppose in addition that R and S are free. Let Q be a bounded open subset of R” invariant under R such that 0 E Q and let f E CT”(L2i; R”) be (R, S)-covariant. Then, if 0 $ f(&2), d(f, Cl, 0) is a unit modulo IGI (in particular, d(f, L2,O) # 0 if G # (1)) and d(f, Cl, 0) mod[Gl is independent of fi and f. THEOREM

Proof.

free, that This tion

Once again, assume that R and S are orthogonal with no loss of generality. As S is it follows from lemma 3.1 that there is a (S, R)-covariant mapping g E e’(B,(O); R”) such 0 $ g(X?,(O)). Thus, using theorem 3.1, one finds d(f, Cl, 0) d(g, B,(O), 0) = 1 modlG1. shows that d(f, Cl, 0) is a unit modulo IGl, hence d(f, Cl, 0) # 0 if G # (1). The last asserfollows, e.g. from corollary 2.2. n 4.THE

CASEOF

COMPACT

LIE GROUPSOF

POSITIVE

DIMENSION

We now turn to the case when G is a compact Lie group. The results previously established for finite groups will be essential in our treatment, but it will no longer be necessary to confine attention to free representations. We shall say that the representation R of the (arbitrary) group G in GL(R”) is fixed point free if x = 0 is the only element of R” such that R,x = x for every y E G. Obviously, a free

Theorem of Borsuk for covariant

mappings

407

representation is fixed point free, but the latter notion is much weaker (for instance, it does not even imply that the representation is faithful). For nontrivial compact Lie groups, a notion midway between free and fixed point free representations is as follows: given a representation R of a group G in GL(R”) and x E R”, the isotropy subgroup of x relative to R is the set G,(R) = [y E G; R,x = x). If now G is a compact Lie group with dim G 2 1 it is well known that G contains a maximal torus with dimension 11 and that any two maximal tori of G are conjugate (see e.g. [l] or [3]). We shall say that x E R” is a quasi-fixed point (QFP) of R if G,(R) contains a maximal torus of G. Every fixed point of R is a QFP; in particular, 0 is a QFP. We shall say that the representation R is quasi-fixed point free (QFP-free) if no x # 0 is a QFP of R. Equivalently, R is QFP-free if no maximal torus of G is contained in any isotropy subgroup G,(R), x E R” - (0). Since isotropy subgroups are closed, they are compact Lie groups and hence (recall that the rank of a Lie group is the dimension of its maximal tori) R is QFP-free

e rank G,(R) < rank G,

vx E R” - 10).

Thus, a QFP-free representation is one for which the isotropy subgroups are not “too large”. In particular, it is obvious that a representation R such that G,(R) is finite for every x # 0 is QFP-free, and hence the latter notion generalizes the former, which has sometimes been used in the literature in connection with degree calculations (see Section 5). Also, if G = 7’is a torus, a representation is QFP-free if and only if it is fixed point free. For instance, the representation R of SO(2) = (z_E C: IzI = 1) in C = R* corresponding to multiplication by .zk, k E Z, is QFP-free for all k # 0 but is free only if k = + 1. The standard representation of SO(3) in GL(R3) is fixed point free, but every x E R3 is a QFP. We now give a useful equivalent characterization of QFP-free representations that better relates them to free ones. PROPOSITION 4.1. Let G be a compact Lie group with dim G 1 1. The representation R of G in GL(R”) is QFP-free if and only if there is y E G” (the identity component of G) such that R,x # x, V x E R” - (0). Moreover, this must hold whenever y is a generator of a maximal torus of G.

Proof. Suppose first that there is y E G” such that R,x # x, Vx E R” - 10). As is well known, y is contained in a maximal torus T of G and hence no isotropy subgroup G,(R), x E R” - (0), contains T. If T’ is another maximal torus in G, there is g E G” such that T’ = g-‘Tg. If T’ were contained in G,(R) for some x E R” - (0), then T would be contained in G,,(R) with y = R,x # 0, a contradiction. Thus, R is QFP-free. Suppose now that y E G” is a generator of a maximal torus T of G and there is x E R” - (OJ such that R,x = x. Clearly, one has R,kx = x, v k E Z, and hence R,x = x, v 6 E T by denseness of (yk) and continuity. Thus, T c G,(R) and R is not QFP-free. n

The next proposition

establishes the property making the connection with finite groups.

PROPOSITION 4.2. Let G be a compact Lie group with dim G 2 1 and let R” be finitely many QFP-free representations of G in GL(R”). Then, for every large enough prime p, there is a subgroup Gp of G with G,, = Z, such that the representation of Gp deduced from R’O is free for every index 1.

P. J. RABIER

408

Proof. From proposition 4.1, R$)x f x, V x E R" - f0)or , equivalently, v x E R”, 1x1= 1, provided that y E G” is a generator of some maximal torus T. By compactness of the unit sphere and continuity, one has R$Ox # x, tl x E R”, 1x1= 1, v I, v 6 E N,, where N, is a small enough neighborhood of y in T. Now, identify T = (SG(2))k (k = rank G 2 1) and set y = (e’ol, . . . , e’ok) with 0 I Oj < 2n, 1 I j I k. Given any E > 0 and any large enough integer p > 0, one can find integers 0 I qj < p such that ((Oj/27r) - (qj/p)( 5 E. For E small enough, 6 = (e2aiql’p, . .., e2?riqk’p) lies in the neighborhood N, and hence Rax # x, Vx E R”, 1x1= 1. Obviously, 6p = 1 (=(l, 1, . . . . 1)) and, ifp is a prime, 8‘ # 1 for 1 I k < p. This shows that the subgroup Gp C T generated by 6 is isomorphic to Zp . Hence, every element 6’ E G,, 6’ # 1, is also a generator of Gp, so that Ri?x # x, v x E R”, 1x1= 1, V 6’ E Gp, 6’ # 1. In other words, the representation of G,, deduced from R@ is free. n V I,

Remark 4.1. From Section 2).

proposition

We are now in a position Lie groups.

4.2 no QFP-free

to prove that a stronger

representation

version

exists in odd dimension

of corollary

(see

2.2 is true for compact

THEOREM 4.1.

Let G be a compact Lie group with dim G 2 1 and let R and S be two representations of G in GL(R”), n even, and suppose that R is QFP-free. Let a1 and CJ2 be two bounded open subsets of R” invariant under R such that 0 E C2r 17 & . Finally, let f E C?“(QZ,; R”) and g E C?‘(C&; R”) be (R, S)-covariant and suppose 0 $ f(&2,), 0 $ g(dCl,). Then d(f,%,O)

= d(g,G,

0).

of Gp Proof. Use proposition 4.2 to find Gp c G with Gp = Z, such that the representation deduced from R is free as soon asp is a large enough prime. From corollary 2.2 with Gp replacing G, one finds d(f, Q, , 0) = d(g, C&, 0) modp. But this holds for arbitrarily large prime p, and the conclusion follows by taking p > Id(f, Q2,, 0) - d(g, !&, 0)l. n Remark 4.2. From remark 4.1, it is clear that the hypothesis used explicitly) is indispensable in theorem 4.1.

that n is even (which has not been

THEOREM 4.2. Let G, R and S be as in theorem 4.1 and suppose in addition that R and S are equivalent, say S, = A-*+4, v y E G. Let fi be a bounded open subset of R”, n even, invariant under R and such that 0 E C& and let f E C?‘(!% R”) be (R, S)-covariant. Then, if 0 $ f(dQ), one has d(f, C2,O) = sgn(det A). More particularly, if R = S d(f, c&O) = 1.

Proof. Repeating the proof of theorem 4.1 but using corollary 2.3 instead yields d(f, Q, 0) = sgn(det A) modp for p arbitrarily large prime. Any P > b(f,

leads to the desired

result.

n

Q2, 0) -

sgn(detA)1

of corollary

2.2

Theorem of Borsuk for covariant mappings

409

Once again, n must be even, andf(0) = 0 is implicit in theorem 4.2. When R and S are not necessarily equivalent, one still has the following theorem. THEOREM 4.3. Let G be a compact Lie group with dim G L 1 and let R and S be two QFP-free representations of G in GL(R”), n even. Let 0 be a bounded open subset of R” invariant under R such that 0 E a and let f E e”(Oi; R”) be (R, S)-covariant. Then, if 0 ef(%2) one has

d(f, Q2,0) f 0. Moreover,

d(f, fi, 0) is independent

of a and f verifying the above hypotheses.

Proof. Use proposition 4.2 to find Gp c G with Gp = Z, such that the representations of Gp deduced from R and S are free as soon as p is a large enough prime. Theorem 3.2 with Gp replacing G thus yields d(f, S& 0) # 0 modp, hence d(f, Q, 0) # 0. The last assertion in theorem 4.3 follows from theorem 4.1. W

The appropriate version of the reciprocity formula (theorem 3.1) in the context of compact Lie groups is as follows. THEOREM 4.4.Let G be a compact Lie group with dim G L 1 and let R and S be two QFP-

free representations of G in GL(R”), n even. Let Q2,and 0, be two bounded open subsets of R” with 0 E 9, fl Q22and suppose that Sz, is invariant under R and fi2 is invariant under S. Finally, letf E eO(G; R”) be (R, S)-covariant and g E co@&; R”) be (S, R)-covariant and suppose that 0 ef(a&), 0 6 g(X&). Then d(f,

sz,, 0)

= d(g, c&, 0) = +

1.

Proof. Arguing just as in the proof of theorem 4.3 and using the reciprocity formula in theorem 3.1 yields d(f, Q2,, 0) d(g, Q,, 0) = 1 modp, for p an arbitrarily large prime number. Taking p > I&f, Q, 0) d(g, Q, 0) - 1) one finds d(f, Sz, , 0) d(g, Sz, , 0) = 1, hence d(f,c&,O) = d(g,c&,O) =+1. n

Comparing theorems 4.3 and 4.4, one should not infer that d(f,Cl,0) is always f 1 in theorem 4.3. Indeed, it is important to notice that existence of f e C?‘(q; R”) and g E CO(G; R”) being (R, S)- and (S, R)-covariant, respectively, and such that 0 B f(Kl,), 0 $ g(&&) is an assumption in theorem 4.4, and this assumption may not be satisfied (i.e. there is no analog of lemma 3.1 for representations of compact Lie groups such as R and S in theorem 4.4). An equivalent formulation of this remark is as in the following theorem. Let G be a compact Lie Group with dim G L 1 and let R and S be two QFP-free representations of G in GL(R”), n even. Let a, and fi2, be two bounded open subsets of R” with 0 E Sz, fl 52, and suppose that Sz, is invariant under R and Q2, is invariant under S. Finally, let f E C?‘(q; R”) be (R, S)-covariant and such that 0 $ f(dQ,), and suppose that d(f,C12,, 0) # k 1. Then, for every mapping g E e”(G; R”) which is (S, R)-covariant, there is x E S& such that f(x) = 0. THEOREM 4.5.

410

P. J.RABIER

Note. From theorem 4.1, the hypotheses listed above depend only on the representations R and S.

are independent

of a,

and hence they

The hypotheses of theorem 4.5 are never satisfied when R = S (see theorem 4.2). Theorem 4.5 is meaningful, e.g. in the following simple case: let n = 2 and R2 = C, and take G = SO(2) = (eie; 0 I 0 < 277). Consider the representations of SG(2) in CL(C) given by R = (e’o; 0 I 0 < 2n],

S = lezio; 0 5 0 < 2nJ

the action being simply multiplication of complex numbers. Obviously, R is free and the isotropy subgroup G,,(S) for y E C - 101 is (1, - 1). Thus, both R and S are QFP-free. The mapping z E K(O) c C -+ z2 is (R, S)-covariant and d($, B,(O), 0) = 2. Hence, there is no (S, R)-covariant continuous function on q(O), that does not vanish on S?,(O). In fact, g = 0 is the only (S, R)-covariant mapping as is obvious writing g(e2iez) = eieg(z) and taking 0 = 7~. But this is specific to the case n = 2 with R and S as above. This example shows that lemma 3.1 is not true when G is a compact Lie group. But, when compared to theorems 4.4 and 4.5, our final result raises the question whether lemma 3.1 remains valid when both R and S are free. 4.6. Let G be a compact Lie group with dim G 2 1 and let R and S be two free representations of G in GL(R”), n even. Let s2 be a bounded open subset of R” invariant under R such that 0 E Sz and let f E e”(O; R”) be (R, S)-covariant. Then, if 0 $ f(%2), one has THEOREM

d(f,QO) Moreover, Note.

d(f, Cl, 0) is independent

Unlike

in theorem

= 21.

of 0 and f satisfying

the above hypotheses.

4.2, R and S need not be equivalent

here.

Proof. Let C c G be a subgroup with E = SO(2) (existence of C follows from existence of a maximal torus T C G with dim T r 1). Obviously, the representations of C deduced from R and S are free and hence one may suppose G = SO(2) in the first place. If so, for every integer p 2 1, the representations Z, deduced from R and S are free, too. From theorem 3.2 with Z, replacing G one finds d(f, 0,O) # 0 modp for every p > 1, so that d(f, i2,O) is a multiple of no positive integer other than 1, hence the conclusion. The last statement follows from theorem 4.1. n 5,COMPARISONS

ANDCOMMENTS

This short section is devoted to a few comments, most of which are intended to put the results of Sections 2-4 in the perspective of other works in the literature. For brevity, we have limited ourselves to discussing the features of those works that strictly relate to ours. It goes without saying that many of the contributions mentioned below have interesting aspects in other, often closely related, directions. 1. With G = Z, in theorem 3.2 or corollary 2.3, one recovers Borsuk’s theorem. Indeed, it is easily seen that R = S = {I, -I) is the only pair of representations satisfying the required hypotheses. Theorem 4.3 should be considered as the appropriate generalization to compact Lie groups, with complements given in theorems 4.2, 4.4 and 4.6 in particular cases.

Theorem

of Borsuk

for covariant

mappings

411

2. All the results in Sections 2 and 3 in the case of a finite group G are obviously valid when g is replaced by one of its subgroups. 3. If G is cyclic and &2i = Sz, (repectively n) is a ball, corollary 2.2 (respectively corollary 2.3) may also be obtained directly from results of Krasnosel’skii [12] (reproduced in Krasnosel’skii and Zabreiko [ 131) about periodic maps on spheres. The method in [ 12, 131 heavily relies on the special structure of spheres and cannot be extended to arbitrary bounded open subsets of R". We point out that even the case when Sz, and fi, (respectively n) are homeomorphic to a ball cannot be reduced to that of the ball because covariance of mappings is not preserved without stringent additional hypotheses on the homeomorphism. The assumption that G is cyclic cannot be eliminated either because it is ensuring periodicity. At the best, we note that using a standard theorem of the theory of finite groups known as Cauchy’s theorem (see e.g. [14, Volume I, p. 1591) and using (2) above, one may handle the case when ICI is a product of distinct primes (and Sz, = Sz, (respectively a) is a ball). With the same restriction that G is cyclic and Sz, is a ball, the conclusion d(f, fi2,, 0) # 0 in theorem 3.2 also follows from [13, theorem 8.6, p. 261 on periodic maps (quoted with no proof). But, in [13], no mention is made of some reciprocity formula. 4. In a fairly recent article, Dold [7] proves that given a finite dimensional paracompact space X invariant under the free action of a finite group, every covariant mapping g E e”(X; x) is essential, namely is not homotopic to a constant map. If a is a ball (and in other cases, but not all) and X = 80, it follows from Hopf’s theorem that d(f, SJ, 0) # 0 for every continuous extensionfof g to Sz. This is a weak form of corollary 2.3 (note that the results of Sections 2-4 are not affected if covariance is required of only the restriction to the boundary of the mappings involved). 5. Fade11 [8] shows that Dold’s conclusion remains valid for general compact Lie groups. In some cases, this yields a weak form of theorem 4.6 (showing only d(f, a, 0) # 0). He also proves a theorem more complete than theorem 4.3 ([8, theorem 5.11) when Sz is a ball and the isotropy subgroups G,(R) and G,,(S) are finite for X, y E R" - (0)(sothat R and S are QFPfree). Fadell’s approach relies entirely on methods of algebraic topology, and the same is true of the somewhat less general works by Liulevicius [15] and Necochea [ 181. 6. If G is a finite group, R a faithful representation of G in GL(R") and f E C?‘(Q; R") is a (R, R)-covariant mapping with no fixed point in a&2 (and 0 is R-invariant) such that f(a)is compact, Nussbaum [ 191 gives the formula Mf,

Q) = k,(f,

CR fl Sz) mod/Cl

(5.1)

where iRn, ic, are fixed point indices and C, is as in (2.1). Replacing f by I - f,one finds corollary 2.3 at once (since C, = (0) and hence icR = 1). Also, (5.1) can be used to prove the main theorem 2.1 when R = S (or, more generally, when R and S are equivalent). Indeed, C, n a = 0 so that ic, = 0. Nussbaum’s results are valid for finite groups of homeomorphisms over general finite dimensional ANRs (not only linear homeomorphisms, although this is the most important case in practice). However, in [19], a complete proof is given only in the periodic case. The proof of the general result ([19, theorem 5 ‘I) is only sketched and involves, among other technicalities, an admittedly complicated construction of a suitable retraction. The proof of corollary 2.3 given here is then significantly simpler than the one based on (5.1) (the difficulties encountered to prove (5.1) are not greatly decreased by confining attention to linear homeomorphisms).

412

P. .I. RABIER

7. In [5], Dancer derives a formula (properly corrected in [6]) relating the degree of a covariant mapping to the degree of the same mapping restricted to the fixed point space of the group action. Dancer’s result is an alternate formulation of a theorem by Rubinsztein [21] and is valid for general compact Lie groups, including finite ones, but limited to balls and to the case R = S (or, more generally, R and S equivalent). When C2 is a ball, corollary 2.3 and theorem 4.2 both easily follow from Dancer’s formula. 8. The work of Wang [23] allows for invariant configurations more general than Dancer’s but, again, assumes R = S and hence yields only corollary 2.3 and theorem 4.2. 6,APPLICATIONSWITHFREEORQFP-FREEREPRESENTATIONS

We shall here describe applications of Borsuk’s section.

a few applications of the results in Sections 2-4, in line with traditional theorem. Other applications (of corollary 2.1) will be given in the next

6.1. Let G be a finite group, G # (11, and let R and S be two free representations G in GL(R”) (n even if IGI 2 3). Let f E C?‘(R”; R”) be (R, S)-covariant and such limlf(x)l = 00 as 1x1 -+ 00. Then,f(R”) = R”. THEOREM

of that

Proof. With no loss of generality, suppose that R is orthogonal. Pick y E R” and consider h(t, x) = f(x) - ty. From the growth property of f, one has 0 $ h([O, l] x M,.(O)) for r > 0 large enough. Hence d(f, B,(O), y) = d(h(1, *), B,(O), 0) = d(h(0, e), B,(O), 0) = n d(f, B,(O), 0) # 0 from theorem 3.2. If IGI is odd, G has no subgroup with two elements and theorem 6.1 does not follow from the case G = Z,. Also, theorem 6.1 is obviously false if no covariance is required (i.e. G = (1)). The version of theorem 6.1 when G is a compact Lie group, obtained through the same proof except for using theorem 4.3 instead of theorem 3.2, reads as follows. THEOREM 6.2. Let G be a compact Lie group with dim G L 1 and let R and S be two QFP-free representations of G in GL(R”), n even. Let f E C?‘(R”; R”) be (R, S)-covariant and such that limIf(x)I = ~0 as x 4 0~). Then, f(R”) = R”.

Other applications deal with the problem of determining whether an equation of the form f(x) = Ag(x) has a solution (A, x) with x lying on the boundary of some open set a. It is easy to obtain many results of this type with covariant f (not g) and invariant fi, the argument being always that f(x) = lg(x) with A = 0 if there is x E an such that f(x) = 0 whereas d(f, Q, 0) # d(g, Cl, 0) if 0 $ f(Xl) from the choice of g and properties of d(f, Cl, 0) established earlier. It is clear how corollary 2.3 as well as theorems 3.2, 4.2, 4.3, 4.4 or 4.6 can be used for this purpose, depending on G and the hypotheses made about the representations. We shall pass on the rather obvious generalizations of the Borsuk-Ulam theorem when G is a finite group (use corollary 2.3) or a compact Lie group (use theorem 4.3 or 4.6), but we mention the following theorem. THEOREM

equivalent

Lie group with dim G 2 1 and let R and S be two QFP-free of G in GL(R”), n even. Let fi be a bounded open subset of R”

6.3. Let G be a compact

representations

Theorem

symmetric f E e’(R”;

of Borsuk

for covariant

mappings

413

with respect to the origin (not necessarily invariant under R) such that 0 E Q and let R”) be (R, S)-covariant. Then, there is x E aa such that f( -x) = -f(x).

Proof. First, assume that Q is also invariant under R and note that (R, Qcovariance off implies (R, S)-covariance of g with g(x) = f( -x), hence that of h with h(x) = f(x) + f( -x). As h is even, the degree d(h, a, 0) is even if 0 $ h(aQ). On the other hand, it follows from theorem 4.2 that d(h, Q, 0) = + 1, a contradiction. Therefore, there must be x E a&2 such that h(x) = 0, namely f( - x) = -f(x). If now Q is only symmetric with respect to the origin, set d = U, f G R,Q. Clearly, d is open, R-invariant and is symmetric with respect to the origin since the same is true of every subset R,SZ. Also, d is bounded: to see this, just recall that R is equivalent to an orthogonal representation, so that IR,I is uniformly bounded for y E G. From the first part of the proof, there is x E afi such that f( -x) = -f(x) and hence f( - R,x) = -f(R,x) for every y E G. To complete the proof, it suffices to show that R,x E 22 for at least one 6 E G. Since x E ad, there is a sequence x, E d such that lim k+co xk = x. For each k, let yk E G be such that x, E R,,i2. Because G is compact, there is y E G and a subsequence, still denoted by Yk, such that limk,, Yk = y. Let yk E Q be such that xk = R,,yk . By COIltiIlUity Of )’ --f R,, one finds that limk,,yk = R;‘x = y. Obviously, y E fi and, in fact, y $ Q for otherwise x = R,y E R,Q C d, a contradiction. Thus, y E aQ and the conclusion follows with 6 = y-l since y = R;‘x = RT-lx. n The same proof as above easily shows that theorem 6.3 remains valid when R and S are free but not necessarily equivalent upon using theorem 4.6 instead of theorem 4.2. But this result has no interest because, in this case, it is easily seen that every (R, S)-covariant mapping must be odd. In the hypotheses of theorem 6.3, a (R, S)-covariant mapping f need not be odd if n > 2 (n even). For instance, with G = W(2) = (e’o; 0 I 0 < 27~1, n = 4 and R = S where R ?, y = e”, identifies with (zl , z2) E C2 = R4 -+ (e’oz, , eiz0z2) E C2 one finds that the mapping f(z, , z2)= (0, z:) is (R, R)-covariant and even. Remark 6.1. If G is a finite group and the representations R and S are free, theorem 6.3 is either trivial (if ICI is even) or false (if ICI is odd). Indeed, if ICI is even, then G has a subgroup with two elements (i.e. Z,) through Cauchy’s theorem mentioned in Section 5. Since the only free representation of Z, in GL(R”) is (I, -I), every (R, S)-covariant mapping is odd. On the other hand, take G = Z,, n = 2 and R = (1, e2rrr’3, e4ni’3 ), with action being simply multiplication of complex numbers. Then, R is free, the function f(z) = zw2is even and (R, R)-covariant, but 0 $ f (w(0)). A classical theorem of Hopf and Rueff ascertains that a system of n( 2 1) continuous mappings f,,1 5 I I n, in n + 1 complex variables z = (z, , . . . , z,+ ,) and satisfying fr(& 19 ..*, 5zn+1) = rq’f,(zl ) ...,Zn+l),

1 I I I n

for every < E C with 151 = 1, vanishes on the boundary of every ball in C”+’ provided that the q/s are nonzero integers (in [ll], Hopf and Rueff proved this theorem when q1 = ... = q,,). A more general result follows from a straightforward application of theorem 4.3.

P. J. RABIER

414

THEOREM 6.4. Letfi E C’(C”“; C"),1 I 1 I n, n > 1, be a family of functions for which there are nonzero integers* pk, 1 I k 5 n + 1 and qr, 1 I I I n such that, with z = (z, , . . . , z”+~), .mPIZ,

9 . . . . rpn+‘z,+l)

= 5qJfj(z,, . . . . Zn+,), 1 I I I n

for every < E C, 151 = 1. Let Sz be a bounded z E 8Q such that fi(z) = .Se= f,(z)= 0.

open subset of Cn+r with 0 E 0. Then,

(6.1) there is

Proof. Let G = SO(2) = (e”; 0 I 0 < 27~) and consider the two representations of G i:, c,5(Cn+‘) c GL(R2"+2)given by R = (diag(eiPle, . . . , eiPn+le)), and S = (diag(eiqla, . . . , e n , e’o)). Because pk # 0, (respectively qr # 0) the isotropy subgroups G,(R) (respectively G,(S)) are finite for z E C”+’ - 10). Hence, both R and S are QFP-free. Define f E (Z?“(C”fl; Cntl) through f(z) = cfi(z) , . .., f,(z), 0). It is trivial to check that condition (6.1) ensures (R, S)-covariance off. Suppose now that fi is invariant under R. If 0 $ f(&X), then d(f, Sz, 0) # 0 from theorem 4.3. On the other hand, f takes values in a proper subspace of Cn+‘, so that d(f, a, 0) = 0, a contradiction. This shows that 0 E f(Xl), namely there is z E aa such that fi(z) = ... = f,,(z) = 0. If Q is not invariant under R, the trick used in the proof of Theorem 6.3 allows one to reduce n the problem to the previous case. In their proof when a is a ball and pk = 1, 1 I k I n + 1 (and q1 = ... = q,J Hopf and Rueff argue through explicit calculation of the degree. This can still be done here if pk = 1, 1 5 k I n + 1 (see remark 6.2 below). With n = 1, a simple example when one cannot take p1 = p2 = 1 isf(zl , zz) = z1 + z:. However, f(t2zl, Ez2) = t2f(z,, q), so thatp, = 2, p2 = - 1 and q1 = 2 works. When Sz is a ball, theorem 6.4 also follows from theorem 5.1 in Fade11 [S] already mentioned of Section 5. For arbitrary 0 containing the origin, theorem 6.4 seems to be new: the validity of theorem 4.3 for arbitrary invariant sets is crucial to this generalization. Remark 6.2. The usefulness of the results about mapping-independence of the degree in Sections 2 and 4 is well exemplified as follows: let f = Cfr),=l,n E e’(C”; C”), n 2 1, be such that fi(
. . . , e”); 0 5 0 < 27r),

R is free and fi is R-invariant.

S = (diag(eiqle,

. . . . eiqne); 0 I 0 < 2nJ.

Thus,

if 0 $ f(aQ), it follows from theorem 4.1 that satisfying the same hypotheses asf. Choosing = .z? if q/ < 0, a g(z 1, .**, z,) = (Wl(Zl)Y . . .9 v,(z,)) where vr(zl) = zpI if q/ > 0 and I, straightforward calculation provides d(g, B,(O), 0) = q1 -. . qn . Hence, d(f, L2,O) = q1 . . . qn. When Q is a ball, this is the argument used by Hopf and Rueff. d(f, 0, 0) = d(g, B,(O), 0) where g is any mapping

Finally, recall that one of the most popular applications of Borsuk’s theorem is the result saying that if n is a bounded open subset of R" symmetric with respect to the origin with 0 E &2, *Not necessarily

positive.

Theorem

of Borsuk

for covariant

415

mappings

and if f E Co@; R”) is such that 0 $ f(%2) and f(x) and f( -x) do not point in the same direction for x E aa, then d(f, Sz, 0) is nonzero. This is because f is homotopic to its odd part. The generalization of this is quite simple: let G be a finite group, G # (l), and let R and S be two free representations of G in GL(R”). Suppose that M is invariant under R with 0 E Sz. Let f E (!?‘(a; R”) be such that 0 $ f(%2). Certainly, f is homotopic to its (R, S)-covariant part f” if f(x) # -If(x) for 13 I 0 and x E aa. Note that this also implies 0 $ f(a(asz). Denoting by (*, *) the inner product of R”, this condition is satisfied if S is orthogonal* and Y;d (f(R,x), Alternatively,

S&x))

> 0,

&f(x))

< 0,

(6.2)

one may assume c (f&x), +V=G

VX E aa

(6.3)

for then f(x) # Af(x) for A 2 0 and x E a&X, and f is homotopic to -f: In both cases, d(f, Cl, 0) # 0 follows from theorem 3.2, and condition (6.2) is vacuous iff is (R, S)-covariant. When G = Z, and R = S = (I, -I), (6.2) reads (f(x),f( -x)) < If(x)\‘, vx E X2. Exchanging the roles of x and -x, one finds (f(x),f( -x)) < min(If(x)12, ]f( -x)12) 5 If(x)1 ]f( -x)1, i.e. that f(x) and f(-x) do not point in the same direction. Condition (6.3) makes no sense when G = Z, and R = S = [Z, -I), but does in general. Again, if G # Z,, it is necessary that n be even for either criterion (6.2) or (6.3) to be available, but then (6.2) and (6.3) combined as

c

(f&x),

S,fW) z 0,

vx

E

an

(6.4)

provide a large variety of new conditions since there is complete freedom of choice for G, R and S. The example of (R, S)-covariant mappings shows that these new criteria can be used to handle cases when the mapping f is even. If now G is a compact Lie group with dim G 2 1, and R and S are QFP-free (so that n is even) the analog of (6.4) reads (assuming again S is orthogonal)

G

(f W,x), S,f(-9) dy f 0,

vxEai2

(6.5)

where dy is Haar measure. The conditions (6.4) or (6.5) ensuring that d(f, Q 0) # 0 can be used, e.g. for proving existence of periodic solutions to ordinary differential equations in R”, n even. We shall refer to Lloyd [16] for various existence theorems explicitly involving a condition of the form d(f, Cl, 0) # 0. Boundary value problems are also considered in [16], under the same kind of condition. 7. APPLICATION

WITH

ARBITRARY

REPRESENTATIONS

If one assumes only that the representation R is faithful, there is no longer any restriction regarding the dimension of the space R”, but the set CR introduced in (2.1) need not reduce to the origin. If so, there is a variety of situations depending not only on the group G and the *Otherwise,

the inner product

must be changed

appropriately.

416

P.

J. RABIER

representation R but also on the mappings involved. Still, a general result of special significance is that much less than homotopy is needed for constancy of the degree modulo ICI. This is the content of theorem 7.1 below, generalizing [13, theorem 8.9, p. 291 where fi is a ball and G is cyclic. THEOREM 7.1. Let G be a finite group and let R and S be two representations of G in GL(R”) such that R is faithful. Let Sz be a convex bounded open subset of R” invariant under R with 0 E 0. Finally, let f E C’(fi; R”), g E C?‘(i’li;R”) be (R, S)-covariant and such that 0 ef(aQ); 0 $ g(%2). Suppose also that f and g are semi-homotopic in the sense that there is a mapping H E e’([O, l] x (C, n aa); R”) such that H(t, *) is (R, S)-covariant for t E [0, 11, 0 $ H(]O, 11 x (CR n dQN and II(O, *) = ficRnaa, W,

-) = gicRnan. Then

W, Q, 0) = 4g, Q, 0) mdGI. If G above is not finite but a compact d(f,

Lie group

with dim G 2 1, then

Q2,0) = d(g, Q2,0).

Proof. With no loss of generality, suppose that R is orthogonal. As Q is convex and 0 E 0 the half line (tx; t 2 1) intersects XJ at exactly one point p(x) for every x E Sz - (0). It is easily seen that p is continuous. Extend H to ((0) x aQ) U ((I) x Xl) U [0, l] x (C, n aQ) by setting is continuous and hence H can be further H(O,*) =flaa, H(l,*) = gl,,. This extension extended as a continuous mapping on [0, l] x an. Replacing H(t, *) by its (R, S)-covariant part, one may assume that H(t, *) is (R, S)-covariant. Every x E Q - M/2 can be written in the form x = (1 + r(x))p(x)/2 with 0 I T(X) I 1. Moreover, T(X) = 0 (respectively 1) if and only if x E a(Q/2) = KU2 (respectively an). As r(x) = 2(x, p(x)>/lp(x)12 - 1, the mapping r is continuous. For x E Q, set ifxEQ - a/2 H(r(x), P(X)) 4(x) = if x E Q/2. i f (2x)

If x E a(fi2/2), then r(x) = 0 and p(x) = 2x, so that q5 is continuous. Also, 4(x) # 0 for x E a(fi/2) since 0 ef(aQ). As 4(x) = g(x) for x E X& one finds $(x) # 0 for x E aa. Finally, 4(x) # 0 for x E (Q - Q/2) n C, since 4(x) = H(r(x), p(x)) and p(x) E C, n an whenever x E c, - (0) (recall that C, is a union of linear spaces). Thus, 4(x) = 0 would contradict 0 $ H([O, I] x (C, fJ &J)). Finally, it is easily seen that p is (R, R)-covariant and that, by orthogonality of R, z is R-invariant. This shows that 4 is (R, S)-covariant. Applying corollary 2.1 yields d(& Cl, 0) = d(q5, Q/2,0) modlGl. As 4 = g on aa, the lefthand side is d(g, Cl, 0). Next, as 4(x) = f(2x) for x E a(Q/2), the product formula shows that d($, M/2,0) = d(f, ~2, 0) and the conclusion follows. If G is a compact Lie group with dim G 2 1, replace G by Z, c SO(2) C G with arbitrarily large p to get d(f, Q 0) = d(g, GO). n If R is free, theorem 7.1 gives again corollary 2.2 when fi2, = a2, = Q and a is convex. From the proof, convexity is not essential in theorem 7.1 but we do not know whether it is valid for arbitrary R-invariant bounded open subsets a. To use theorem 7.1 to calculate d(f, Sz, 0), it suffices to find g such that f and g are semi-homotropic and d(g, Cl, 0) is known. For instance, if R = S, g(x) = x is (R, R)-covariant and f is semi-homotopic to g provided that f(x) # - Ax, VA > 0, vx E C, n aQ (take H(t,x) = (1 - t)f(x) + tx).

Theorem of Borsuk for covariant mappings COROLLARY 7.1. Suppose R = S in theorem 7.1 and suppose v x E CR n X2. Then, if G is finite, one has d(f, Sz, 0) = 1 mod)G].

group

417

f(x) # - Ax, V A > 0, If G is a compact Lie

with dim G L 1, one has d(f,Q 0) = 1.

Of course, there is a trivial variant of corollary 7.1 when R and S are equivalent. Observe also that the weaker statement d(f,Cl,0) # 0 if G # 11) can be obtained from the case when G is cyclic by replacing G by any cyclic subgroup. Thus, if a is a ball, d(f,Cl,0) # 0 follows from [13]. The stronger conclusion d(f, Q,O) = 1 modIG] is useful, e.g. for the following application: if n is odd and ICI 2 3, there is A # 0 and x E CR fl afi such that f(x) = Ax. Indeed, otherwise, corollary 7.1 can be used with both f and -f,whence 1 = (- 1)” mod]G], which is absurd if n is odd and (GI 2 3. This result is not true if G = Z2. If G contains an element of order 2 3, one may replace G by the cyclic subgroup generated by this element but this approach fails to work, e.g. with G = Z, x Z2, even when a is a ball. To complete with, we shall expand on an immediate consequence of corollary 2.1 which is of some importance regarding existence of solutions to nonlinear equations. With S thus being another representation of G in GL(R”) (not necessarily faithful) and o c Q c R” bounded open subsets invariant under R, let f E CT’(Q; R”) be (R, S)-covariant and such that 0 $ f(Kl U ao). Another way to read corollary 2.1 is to say that if d(f, co, 0) # d(f, Q, 0) mod(G(, then there must be x E CR(w, a) such that f(x) = 0 (see (2.2) for the definition of CR(o, a)). In other words, a discrepancy modulo \Gj between the degree of f relative to the sets o and Q automatically implies existence of y E G, y # 1, and x E Q - 6~ such that f(x) = 0 and R,x = x. Of course, without involving covariance, a discrepancy between the degrees implies existence of a solution x E Q - Q to f(x) = 0. The point is that if this discrepancy persists modulo ICI, one gets existence of a solution preserving some “symmetry”. This approach for proving existence of solutions preserving some symmetry is competing against a perhaps more natural one, which is as follows: fix y E G and define the space Xy = (x E R”; R,x = x) c R”. Because f is (R, S)-covariant,

one finds f(F)

(7.1)

c Yy where

Yy = (y E R”; S,y = y].

(7.2)

Introducing the open sets fizy = Q rl Xy, wy = w tl Xy and identifying Xy = Rp, Yy = R4, one may consider f as a mapping from Qzy to R*. In the case when p = q (e.g. when R and S are equivalent) a relation such as d(f, my, 0) # d(f, W, 0) would then imply existence of x E fizy - tiy (thus R,x = x) such that f(x) = 0. The advantage of the above procedure is that it allows one to find solutions preserving a given symmetry R,. Its disadvantage is of course that proving d(f, my, 0) # d(f, W, 0) requires having quite a bit of information available on f and R, (in particular, on Xy) since this relation may be true for some y E G and not for others. Also, the case q # p cannot be handled by this method and it is relevant when S is the trivial representation S = (r) (so that q = n irrespective of y E G). If so, (R, S)-covariance means f(R,x) = f(x) for y E G, x E Sz, namelyfis invariant under R. This situation is encountered, e.g. when the representation R acts through permutation of some coordinates, and f is independent of such permutations. In summary, it appears that the method based on corollary 2.1 for proving existence of solutions preserving some symmetry is available in greater generality and with less information

418

P. J.

RABIER

on f and the representations R and S. As a matter of fact, these are not even specifically involved since the group G plays a role by only its order /Cl. Naturally, corollary 2.1 yields a result less precise than the other approach (when available), some uncertainty being left about which ys in G produce a solution x such that R,x = x. But even this is true only to some extent, because corollary 2.1 applies with the group G as well as with any of its subgroups H: whenever d(f, w, 0) # d(f, C& 0) modjH], one finds y E H, y # 1 and x E CX- o such thatf(x) = 0 and R,x = x (of course, the same x may correspond to different such ys). Quite obviously, finding a discrepancy modulo ICI between d(f, o, 0) and d(f, Q, 0) depends in part on the problem under consideration and in part on one’s ability of picking o and C2 in a suitable way. Such a suitable way is easy to find in the case of problems of bifurcation that we now consider as an illustration of this method. Let F = F(,u, x) be a e’ mapping from R x R” to R” such that F(,u, 0) = 0. That Fis defined over the whole space is only assumed for convenience, and we have not tried to formulate the weakest possible hypotheses of regularity either. Suppose that F(M, *) is (R, S)-covariant and that D,F@, 0) E GL(R”) for 1~1 > 0 small enough (which implicitly requires that R and S are equivalent). Suppose also that x = 0 is an isolated solution to F(0, x) = 0 and that sgn(det D,F+, 0)) changes as fi crosses 0. Irrespective of (R, S)-covariance, the above assumptions ensure that (0,O) is a point where nontrivial solutions bifurcate. With virtually the same proof, except for making use of corollary 2.1, one finds the following theorem. THEOREM

7.2. Under the above hypotheses, and given any neighborhood U of (0,O) in R x R” and any subgroup H C G with IHI 2 3, there is y E Hand (D, x) E U, x # 0, such that R,x = x and F(,u, x) = 0. Proof. Upon possibly changing the inner product of R”, one may assume that R is orthogonal. If so, every ball B,(O) C R” is invariant under R. In addition, for r > 0 small enough, there is no solution x E a&(O) to F(0, x) = 0, and hence this is true with F&, x) = 0 for 1~1 I ,D,, small enough. It is not restrictive to assume [ --pO, ,D~] x B,(O) c U, and d(F@, *), B,(O), 0) is constant for I,D] I ,D,, because of invariance of the degree by homotopy. Given a subgroup H of G with IH( 2 3, suppose that there is no solution &, x) to F(,u, x) = 0 with iu = +pu,, x E B,(O), x # 0 and R,x = x for some y E H, y # 1. Then, with n = B,(O) and o = B,(O), 0 < E < r, one has 0 $ F( fp,, , C,(Q, w)) and corollary 2.1 yields d(F( fp,, , e), B,(O), 0) = d(F( fpO, a), B,(O), 0) modlHl if 0 $ F( +,u,, , aBE). This holds for E > 0 small enough since D,F( +pu,, 0) E GL(R”). Moreover, d(F( +pO, *), B,(O), 0) = sgn(det D,F( fpO, 0)) and the change in sign as P crosses 0 leads to - 1 = 1 mod IHI, which is absurd when IHI L 3. n

Elementary examples show that theorem 7.2 is not true if (HI = 2 (recall that ICI = 2 in corollary 2.1 is the standard theorem of Borsuk). In fact, with G = H = Z, and R = S = (I, - 4, existence of nontrivial solutions preserving the symmetry -I is nonsense. The content of theorem 7.2 is that at least one symmetry of each subgroup H of G with IHI 2 3 is preserved in some bifurcated solution provided that det D,F(u, 0) changes sign. This result does not seem to have been obtained before. It should be noted that the hypotheses of theorem 7.2 are consistent with each other. For instance, consider the case when n is odd and F(,u, x) = ,ux + Q(x) where Q(O) = 0, Q’(O) = 0.

Theorem

of Borsuk

for covariant

419

mappings

If so, D,FC,u, 0) = ,d and sgn(det D,F@, 0)) = sgnp. On the other hand, it is easy to find faithful representations R of general (finite) groups G along with (R, R)-covariant mappings @ as above such that x = 0 is an isolated solution to CD= 0. For instance, one may look for @ being a homogeneous polynomial of degree ~2 (in this respect, see e.g. Sattinger [22]). When G is a compact Lie group with dim G 2 1 and with no modification of the other assumptions (especially that sgn(det D,F@, 0)) changes), one gets a more striking result. 7.3. Under the above hypotheses and given any neighborhood U of (0,O) in R x R” and any (maximal) torus T c G, there is @, x) E U, x # 0, such that R,x = x for every y E T and F(p, x) = 0.

THEOREM

Theorem 7.3 follows immediately from standard arguments and the results of Dancer [5] or Wang [23] showing that the index of F(_LI,*) coincides with the index of FQI, *) restricted to the fixed point space of T relative to R (for p # 0) and hence must change as p crosses 0. Incidentally, this implies that the fixed point space of Trelative to R is not 10). Theorem 7.3 can also be deduced from theorem 7.2 using ideas similar to those in the proof of lemma 4.2 but this method is longer. Remark 7.1. Despite theorem 7.3 is a trivial consequence of [5] or [23], it has apparently not been noticed before. The statement for finite groups in theorem 7.2 does not follow from [5] or [23], although part of it does. More precisely, if H c G is ap-subgroup (i.e. IHI = p* where p is a prime) and using for instance theorem 2 in [5], one finds that the index of F(,u, *) coincides module p (not p”) with the index of F(,u, *) restricted to the fixed point space of H relative to R. If p > 2, one has - 1 # 1 modp and bifurcation of H-symmetric solutions follows. This result is more precise than theorem 7.2 if cx L 2 and, through Cauchy’s theorem, it can be used to cover the case when IHI is divisible by some prime p L 3. But the case when His a 2-group, e.g. Z, or Z, x Z, , cannot be handled with this approach. The choice H = Z2 x Zz is especially important regarding applications. Remark 7.2. Obviously, the representations R and S cannot be QFP-free in theorem 7.3. In fact, one can prove that if L E GL(R”) is (R, S)-covariant with R and S equivalent and QFPfree, then sgn det L is independent of L (hence det D,F(p, 0) could not change sign if R and S were QFP-free in theorem 7.2). We emphasize again that change of sgn(det D,F(p, 0)) is essential for the validity of theorems 7.2 and 7.3: there are many physical examples in which S0(2)-symmetry is not preserved when bifurcation occurs at eigenvalues with even multiplicity. For instance, Hopf bifurcation can be viewed as S0(2)-symmetry breaking bifurcation. A second example can be found in [20] and a third one is the buckling of circular plates. It should be clear how theorems 7.2 and 7.3 can be used to derive corresponding statements for problems of bifurcation in Banach spaces, via Lyapunov-Schmidt reduction. The only technicality is that the latter may affect faithfulness of the representation. In this case, the group G must be replaced by a quotient group. Details are left to the reader. Acknowledgement-This

work was in part supported

under

Air Force contract

AFOSR-88-0262.

420

P. J. RABIER REFERENCES

1. 2. 3. 4.

ADAMS J. F., Lectures on Lie Groups. Benjamin, Reading, MA (1969). BIERSTONE E., General position of equivariant maps, Trans. Am. math. Sot. 234, 447-466 (1977). BORSUCK K., Drei S&e iiber die n-dimensionale euklidische Sphtire, Fund. Math. 20, 177-190 (1933). BROKER T. & TOM DIECK T., Representations of Compact Lie Groups, Grundlehr. math. Wiss. 98. Springer, Berlin (1985). DANCER E. N., Symmetries, degree, homotopy indices and asymptotic homogeneous problems, Nonlinear-Analysis

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