Let K denote, as in § i, the field
K = Q,[{(JI + [f~11 [L is an odd root of i}) and recall that for x e F, a finite extension of K, we set
Tr(^)= I-F'KI^^^'
We begin the proof of Theorem 2 with the following lemma. The results of § 3 are used in this section only in the proof of this lemma.
Lemma 9. — For each real representation 6 of the cyclic group Z^, n == 2° 6, with b odd, there is a representation ^ of Z^ defined over K with character ^ == 2a+lTr(^Q) and go+iQ _p p ^ ^ _[_ p^ p ^ regular representation of Z^.
Addendum to Lemma 9. — If n is a power 0/2, n== 2°', the a + i ^ ^ conclusion of the lemma can be replaced by a.
We argue by induction on n. If n 4= o (mod 4) or n = 4 it is well known [CR], [Se] that every real representation of G is linearly equivalent to a representation defined over K and so there is nothing to prove. Assume now that n == 2ab, a>_ 2 and a =t= 2 or b > i. The lemma is true, from the inductive hypothesis, for representations 6 in the image of T^ : R(ZJ -> R(ZJ, where T^ : Z^ -> Z^, m<n. Thus we need only consider the irreducible representations
a,: Z,-^SO,
given by
27T; . 27T7
cos — sin —
/,. ^ 72
<^W =
J . 271:; 27T7
— Sin —— COS ——
n n
for o<j<2a~lb, j prime to n = 2ab, and h the generator of Z^. But from Propo-sition 7 of § 3, 86^. + 8_ i ^40^+40^-1 & _j +8-1 where 8_i is the nontrivial one-dimensional representation. Then 8cr, + p ^ y + p, where y = 4crJ + 4^-1 &--? ^d Tr(x8oj) = ^(Xy)- However, y is induced from the representation 4^. of Z^o-i^.
3^.9
122 S Y L V A I N E . C A P P E L L A N D J U L I U S L . S H A N E S O N
Now, as observed for example by Schultz, for 6 and T] representations of H, a subgroup of the finite group G, and Ind(9) and Ind(6) denoting the induced repre-sentations of G, then O^/T] implies Ind(6) ^Ind^). This is obvious if we regard Ind(6) as the H-equivariant functions from G to the underlying vector space of the representation 9.
Now, by the inductive hypothesis on n, the representations (T, of Z^-i;, satisfy the conclusion of the lemma. Hence so do the representations a- ofZg^.
The proof of the addendum to the lemma is the same induction but starting with the fact that every real representation ofZ^, a<^ is easily seen to be defined over the field K.
Proof of Theorem 2. — First we show that T == (^^O + w^) is topologically equivalent to the realification of a representation 9 defined over K with ^ =:Tr(x)•
Recall that from the Brauer theory (as in [Se]), we may write, letting <pc denote the complexification of,
Qc=a-b
where a and b are complex representations ofG which are sums of representations induced from i-dimensional complex representations of subgroups of G. Thus we may write
6c + b = a
and thus the equation of real representations
26 + (3 == a,
where p = b + &, a == a + a, u = the complex conjugate of u. Hence,
(1) ^ O + ^ P + P G - ^ a + p ^ .
Now as i-dimensional complex representations factor through cyclic groups, we may apply the above Lemma 9 to get
(2) ^OC+PG-PI+PG (3) ^(B+PG-^+PG
where 9, is the realification of a representation defined over K, with ^ == 2f l ^ + lTr(Y ),
^==2< 7 + lTr(7p). Substituting (2) and (3) in (i) gives
(4) (^e+^+PG^Pi+pG)
with Tr applied to the characters of both sides giving the same function. We may write 92=(dim(p2)pQ--(p3, 93 some representation defined over K. Adding 93 to both sides of (4) gives the result.
THE TOPOLOGICAL RATIONALITY OF LINEAR REPRESENTATIONS 123
Now we prove Part A of Theorem 2. From what we have just shown, for y, = 2g+2Qi + mpQ, m •== max(wQ , m^), we have y^y^, <p^ defined over K with Tr(%^) == ^p.. But then <p^ and 93 are representations over K with the same character.
Hence <p^ == 93 and thus Yi ^ 9i = ?2 ^ Ta*
To prove Part B of Theorem 2 it suffices to show, under the hypothesis on odd cyclic subgroups of G, that if 63^63 then Tr(^Q) = Tr(^ ). It suffices to check this on cyclic subgroups of G; so set G = = Z ^ . We may also inductively assume the result for all proper subgroups of Z^. Thus we may assume that
(5) Tr(^)(/)=Tr(^)(/)
for fe Z^ and not a generator. We must only show the same formula forf a generator.
The case n odd is covered by the hypothesis. Assume n is even. Now we may decompose any real representation of Z^, 6^ = o^ + Pi 5 where o^ is a sum of representations of Zy, which factor through various Z^, m < n and m dividing n, and ^ is a sum of 2-dimensional / 27y . 2nj\
I cos —— sin — \cos — sin real representations ^ which send the generator h of Z^ to n n
. 27y 27y -sin— cos— /
n n J where j is prime to n. By considering the points of the representation space of 6^ which are fixed by the action of some nontrivial subgroup ofZ^, we see that, as e^^Og, we have a^^ocg.
Hence, from our inductive hypothesis for m < n, (6) Tr(^)=Tr(^)
it remains only to check that for n even and^a generator of Z^, Tr(^p ) (jf) == Tr(^n) (V) • When TZ = 2 (mod 4)5 writing n == 2^, A odd, we have
%0,(/)=-%C,,((^+I)/)
and hence,
(7) Xp,(/)= =-X3,(^+I)/) ^ 1 . 2 .
As (k + i)f is not a generator of Z,, from (5), Tr(^((A + i)/)) = Tr(^((^ + i)/)) and hence, from (6), Tr(^((& + i)/)) == Tr(^(^ + i)/). Substituting this in (7) gives that Tr(^)(/) = Tr(^)(/). _ _
When n=.Q (mod 4) we show that Tr(/p ) (/) == o = Tr(^p (/)), / a generator
331
124 S Y L V A I N E . C A P P E L L A N D J U L I U S L . S H A N E S O N
ofZ^. Set /= kh, h the canonical generator ofZ^ and k an integer prime to n. Then P, was a sum of the Op z = i, 2, and
Tr^JW^Tr^cos2 7^) 7 \ w ;
= Tr([ji 4- i^)? ^ a primitive n-th root of unity,
= ' . S ^, ,=[K((i+i;):K],
^ l^j<n j prime to n I
= - . 0 = 0 , c
as ^ is divisible by 4.
The addendum i to Theorem 2 is proved similarly to Theorem 2, but using further the fact that any irreducible complex representation of G, a supersolvable group, is induced from a i-dimensional representation of a subgroup. The proof of addendum 2 is similar, but using the fact (left as an exercise) that ifG has in addition a central 2-Sylow subgroup, every irreducible real representation is induced by a representation, of some subgroup, which factors through a cyclic group.
Proofs of Theorems 3 and 1. — Theorem 3 (and (3.2)) is immediate from Theorem 2.
Corollary (3.1) then follows by using standard representation-theoretic methods of computing the number of irreducible representations over a field (see [CR], [Se]), in this case applied to K. Corollary (1.2) can be obtained from Corollary (3.1) using the elementary fact that, in a 2-group, (ii) of (3.1) implies that for each ^eG, g^ is conjugate to g or g~1 for all odd p. To obtain Theorem i from these results, note that by standard representation theory, for G a 2-group, RQ(G) -> RK:(^) ls an isomorphism and a rational representation is determined by the dimensions of the fixed points of the cyclic subgroups. Moreover, the coefficient 2g+2 in Theorem 2 is improved to 2g+l
in Theorem i by using the addendum to Lemma 9 of this section.
Next we prove the results on compact Lie groups.
Proof of Theorem J. — It is easy to see that conditions (i) and (2) imply that 63^62. Just decompose the G representations 6^ as 6^ =6?°+6^, i = = i , 2 . Then condition (i) implies that 6^° ^6^° and (2) implies that Q[ and 62, as they have the same characters, are linearly equivalent [Ad] and, a fortiori, topologically equivalent.
Hence, 6^ and 63 are topologically equivalent.
Next we show the converse, that 61^63 implies (i) and (2). It is easy to see that a topological equivalence of 6^ to 63 carries the subspace 6^° to 6^°. Condition (2) is thus immediate. Now decompose, as above, 6^ =6?° + 6o 1 = 1 , 2 . Clearly
(6^)GO = o. We will complete the argument by showing that Q[==%.
THE TOPOLOGICAL RATIONALITY OF LINEAR REPRESENTATIONS 125
Consider the difference of characters Xoi—Xe,- For each geG, Q^(g) and Q^g) are orthogonal matrices. By an elementary argument of [DeR 2] (cf. [KR]) as Q^(g) and Q^{g) are furthermore topologically equivalent, those eigenvalues of Q^(g) and of Q^(g) which are not roots of unity are the same, counted with multiplicity. In particular, for each g e G, ^{g) — Xe,QO has value a sum of roots of unity. Moreover, X e , — Z e . = ( X e 2 — X e s ) + ( X e i - X e « ) . Q?^6?0. ^d clearly, as 6?1 may be regarded as a representation of the finite group G/Go, the eigenvalues of (6?°)(^) are all roots of unity, for g e G. Hence ^(g) — %e;0?) ls a sum of roots of unity, for g e G. But the function /^ — ^ thus maps continuously the group G, regarded as a space, to the set of algebraic numbers. Hence, a == /^ — ^ is constant on connected components of G; it is, of course, zero on the component of the identity element of G, as dim 61= dim 63. Thus, a can be regarded as defined by a class function G/Go-^-R which vanishes on the identity element and has equal values on inverse elements.
Writing such a class function of the finite group G/GQ in terms of characters of repre-sentations, we see that (61—62) has the character of an element of R(G/Go)®R. But as an element of R(G)®R is determined by its character [Ad],
(61-63) e!mage(R(G/Go)®R -> R(G)®R).
But this image is detected under the map R(G) -> R(G/Go) sending 6->6GO. As (61)°° - (62)°° = 0 - 0 = 0 , we see that Q[ = 63.
Proof of Corollaries (5.2) to (5.5). — All except (5.2) and (5.4) are immediate.
from Theorem 5 and standard facts about compact Lie groups. To prove (5.4), use additionally the fact that for representations of dimension less than 6 of a finite group, topological equivalence implies linear equivalence [CS 9].
To prove (5.2), using a maximal torus, reduce to the case where G is a torus.
In that case, as g is hyper-regular it follows that 6 (A, ) and r\(h, ) are topologically equivalent for each heG respectively. Then use an argument similar to that of the proof of Theorem 5.
Proof of Theorem 4. — First of all it is easy to see that a representation which is topologically equivalent to a bounded representation is bounded; bounded represen-tations are characterized by the o-vector having a neighborhood whose image under the action of G is a subset of a compact set. Now, standard methods in representation theory show that bounded representations are linearly equivalent to orthogonal repre-sentations. As such orthogonal representations behave very much as in the compact case, the proof proceeds along much the same lines as that of Theorem 5.
Proof of Theorem 6. — We first show that when G has no element of order 4^ with k>_ 2, Q^ = Qy in R(G). It clearly suffices to prove this for G a cyclic group Z^ with n ^ o (mod 4) or n == 4. Arguing by induction, we may, by considering first the factors of n, assume that ^ = ^ on any proper subgroup ofZ^. Now, by results
233
ia6 S Y L V A I N E . C A P P E L L A N D J U L I U S L . S H A N E S O N
of [0] the fixed points of a periodic map / on a (mod 2) homology sphere has Euler characteristic the Lefschetz number off. Hence, when the fixed point set of an element of Z^ is discrete, there are o or 2 of them. If o, there is nothing to prove. If 2 and n is odd, as a mod 2 homology sphere is, in particular, a rational homology sphere, we are essentially in the case discussed in [Sa], using the methods of [AS], [AB], [M], When n== 2k, k odd, or n == 4, let V be the fixed point set of ZgCZ^. By Smith theory, V is a (mod 2) homology sphere. Letting o^ (resp., <Xy) denote the representation of G on the tangent space of V at x (resp.,^) we may write 6^ = a^ + Pu5 u == x ov Y) where o^ is a representation ofZga^ factoring through Z^a-i^ and (3y is a sum of repre-sentations GJ sending the generator of Z^ to
/ 27y . 27y\
/ cos — sin — \ n n
. 27T7 27T7
\ — sin — cos —- /
\ n n ]
for j odd.
Now by induction o^==ay. To see that jB.,==Py, note that for 7 2 = 2 (mod 4) or for n = 4, any sum of representations of the form (T, is determined by its restriction to Z^g, whence the result.
Now to complete the proof of the theorem, we show that in general Tr(^e ) = Tr(/Q ) and apply Theorem 2. Again, we may assume G == Z^, n == 2^, b odd, and argue by induction on n. The cases a = o or i have already been considered.
Also, by induction, we may assume Tr(^)(/) = = T r ( ^ ) ( / ) f o r / not a generator ofZ^. Define V as above as the fixed point set of the subgroup ZgCZ^; by Smith theory, this is a mod 2-homology sphere and again write 6y == o^ + Pi^ u == x or jy, o^ the representation of G on the tangent space of V at u. Again by induction, Tr(^ ) == Tr(^ ). So we need only check that
(i) Tr(x3j(/)=Tr(x3;(/)
for/any generator of Z^, n == 2^, a>_ 2 and (^ or (By a sum of representations of the form (T. defined as above. But, the same kind of calculation as that made at the end of the proof of Theorem 2 shows that Tr(^.)(/) == o, f a generator of Z^; thus both sides of (i) vanish identically f o r / a generator.
BIBLIOGRAPHY
[A] V. I. ARNOLD, Ordinary Differential Equations, Cambridge, MIT Press, 1973.
[Ad] J. F. ADAMS, Lectures on Lie Groups, New York, Benjamin, Inc., 1969.
[AB] M. ATIYAH and R. BOTT, A Lefschetz fixed point formula for elliptic complexes: II. Applications, Ann.
of Math., 88 (1968), 451-491.
THE TOPOLOGICAL RATIONALITY OF LINEAR REPRESENTATIONS 127 [AS] M. ATIYAH and I. SINGER, The index of elliptic operators. III, Ann. of Math., 87 (1968), 546-604.
[Bre i] G. BREDON, Representations at fixed points of smooth actions of compact groups, Ann. of Math., 89 (i969)» 512-532.
[Bre 2] — Introduction to compact transformation groups. New York, Academic Press, 1972.
[CKP] C. CAMACHO, N. KUIPER, J. PALIS, The topology of holomorphic flows with singularity, Publ. Math.
I.H.E.S., 48 (i979), 5-33.
[CR] C. CURTIS and I. REINER, Representation theory of finite groups and associative algebras. New York, Interscience-Wiley, 1962.
[CS i] S. E. CAPPELL andj. L. SHANESON, Pseudo-free group actions I, Proc. of the Aarhus Topology Conference of 1978, Springer Lecture Notes in Math., 763 (1979), 395-447.
[CS 2] — Linear Algebra and Topology, Bull. Amer. Math. Soc., New Series, 1 (1979), 685-687.
[CS 3] — Nonlinear similarity of matrices. Bull. Amer. Math. Soc., New Series, 1 (1979), 899-902.
[CS4] — Nonlinear similarity, Ann. of Math., 113 (1981), 315-355.
[CS 5] — Fixed points of periodic smooth maps, Proc. National Acad. of Sci. USA, 77 (1980), 5052-5054.
[CS 6] — Fixed points of periodic differentiable maps (to appear).
[CS 7] — The geometry of linear representations of groups and subgroups (to appear in Amer. Jour. of Math.).
[CS 8] — Matrices, topology, and elementary number theory (to appear).
[CS9] — Nonlinear similarity and linear similarity are the same in dimensions less than 6 (to appear).
[CS 10] — Class numbers and periodic smooth maps (to appear).
[DeR i] G. de RHAM, Sur les nouveaux invariants topologiques de M. Reidemeister, International Conference of Topology (Moscow, 1935), Recueil Mathematique, Moscow, 1936, t. I (43), 737-743.
[DeR 2] — Reidemeister's torsion invariant and rotation of S^*, International Conf. on Differential Analysis, Bombay Colloquium of 1964, Oxford Univ. Press, 1964, 27-36.
[DeR 3] G. de RHAM, S. MAUMARY, and M. A. KERVAIRE, Torsion et type simple d'homotopie. Lecture Notes in Math., 48, Springer-Verlag, 1967.
[I] Ju. S. IL'IA§ENKO, Global and local aspects of the theory of complex differential equations, Proc. Int.
Cong. of Math., Helsinki, 1978, 821-826.
[Ka] K. KAWAKUBO, Weyl group actions and equivariant homotopy equivalence, Proc. of Amer. Math. Soc., 80 (1980), 172-176.
[Kla] B. KLARES, Classification topologique des n-tuples de champs de vecteurs holomorphes commutatifs sur Pn+i(C), These, Universite de Strasbourg, 1980, partie II.
[K i] N. H. KUIPER, The topology of a solution of a differential equation on R^ Proc. Int. Congress on Manifolds, Tokyo, 1973, 195-203.
[K 2] — La topologie des singularites hyperboliques des actions de R2, Asterisque, 59-60 (1978), 131-150.
[K 3] — The topology of linear C^-flows on C^ to appear in the Proceedings of a Conference on Dynamical Systems, Rio de Janeiro, July 1981.
[KR] N. KUIPER and J. W. ROBBIN, Topological classification of linear endomorphisms, Inuenfiones Math., 19 (i973), 83-io6.
[L] S. LANG, Algebra, Addison-Wesley, 1965.
[LW] C. LEE and A. WASSERMAN, On the groups JO (G), Memoirs Amer. Math. Soc., Vol. 2, Issue i. No. 159, Amer. Math. Soc., Providence, R. I., 1975.
[M] J. MILNOR, Whitehead torsion. Bull. Amer. Math. Soc., 72 (1966), 358-426.
[0] R. OLIVER, Fixed point sets of group actions on finite acyclic complexes. Comment. Math. Helv., 50 (1975), I55-I77.
[Pe i] T. PETRIE, The Atiyah-Singer invariant, the Wall groups L^( , i), and the function ^ + i)/^— i), Ann. of Math., 92 (1970), 174-187.
[Pe 2] — G-Surgery I-A survey. Springer Lecture Notes, 664 (1978), 197-233.
[^ 3] — Three Theorems on Transformation Groups, Springer Lecture Notes, 763 (1979), 549-572.
[P] H. POINCARE, Sur les courbes definies par les equations differentielles, (Euvres de H. Poincare, Vol. I, Paris, Gauthier-Villars (1926).
[R] M. G. ROTHENBURG, Torsion invariants and finite transformation groups, Symposia in Pure Math., XXXII, part i, 267-312, Amer. Math. Soc., 1978.
335
S Y L V A I N E . C A P P E L L A N D J U L I U S L . S H A N E S O N
C. SANCHEZ, Actions of groups of odd order on compact, orientable manifolds, Proc. Amer. Math. Soc., 54 (i976)> 445-448.
R. SGHULTZ, On the topological classification of linear representations. Topology, 16 (1977), 263-270.
J.-P. SERRE, Representations lineaires des groupes jinis, Paris, Hermann (1967).
J. L. SHANESON, Wall's surgery obstruction groups for Z X G, Ann. of Math., 90 (1969), 296-334.
P. A. SMITH, New results and old problems in finite transformation groups. Bull. Amer. Math. Soc., 66 (1960), 401-415.
P. TRACZYK, On the G-homotopy equivalence of spheres of representations. Math. ^eitschrift, 161 (1978), 257-261.
C. T. C. WALL, Surgery on compact manifolds, New York, Academic Press, 1970.
— Classification of Hermitian forms, VI, group rings, Ann. of Math., 103 (1976), i-8o.
Gourant Institute of Mathematical Sciences 251 Mercer Street
New York, N.Y. 10012 Department of Mathematics Rutgers University
New Brunswick, N.J. 08903
Manuscrit re^u Ie 2 novembre 19799 Revise Ie 2 mars 1981.