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Topological and combinatoric methods for studying sums of squares Yiu, Paul Yu-Hung 1985

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TOPOLOGICAL AND COMBINATORIC METHODS FOR STUDYING SUMS OF SQUARES By PAUL YU-HUNG YIU B.A.(Honours), Univers i ty of Hong Kong, 1975 M . P h i l . , Univers i ty of Hong Kong, 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF MATHEMATICS) We accept t h i s thes is as conforming to the required standard THE UNIVERSITY OF BRITISH May 1985 © Paul Yu-hung Y iu , COLUMBIA 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission f o r extensive copying of t h i s thesis f o r scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of MATHEMATICS The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date • S e p t e m b e r 1 8 . 1 9 8 5 i i S u p e r v i s o r : Professor Kee Yuen Lam ABSTRACT We study sums of squares formulae from the p e r s p e c t i v e of normed b i l i n e a r maps and t h e i r Hopf c o n s t r u c t i o n s . We begin with the geometric p r o p e r t i e s of q u a d r a t i c forms between e u c l i d e a n spheres. Let F: S m >• S n be a q u a d r a t i c form. For every po i n t q i n the image, the in v e r s e image F 1 ( q ) i s the i n t e r s e c t i o n of S m w i t h a l i n e a r subspace W , whose dimension can be determined q' e a s i l y . In f a c t , f o r every k ^ m+1 w i t h nonempty Y^ = {q £ S n: dim Wg = k}, the r e s t r i c t i o n F 1(Y^) > Y k i s a great ( k - 1 ) -sphere bundle. The q u a d r a t i c form F i s the Hopf c o n s t r u c t i o n of a normed b i l i n e a r map i f and only i f i t admits a p a i r of "poles" ±p such that dim W + dim W = m+1. In t h i s case, the in v e r s e p -p images of p o i n t s on a "meridian", save p o s s i b l y the p o l e s , are mutually i s o c l i n i c . Furthermore, the c o l l e c t i o n of a l l poles forms a great sphere of r e l a t i v e l y low dimension. We a l s o prove that the c l a s s i c a l Hopf f i b r a t i o n s are the only nonconstant q u a d r a t i c forms which are harmonic morphisms i n the sense that the composite with every r e a l valued harmonic f u n c t i o n i s again harmonic. Hidden i n a q u a d r a t i c form F: S m >• S n are nonsingular b i l i n e a r maps x R m s- R n, one f o r each p o i n t i n the image, a l l r e p r e s e n t i n g the homotopy c l a s s of F, which l i e s i n Im J . Moreover, every hidden nonsingular b i l i n e a r map can be homotoped to a normed b i l i n e a r map. The e x i s t e n c e of one sums of squares formula, t h e r e f o r e , a n t i c i p a t e s others which cannot be obtained simply by s e t t i n g some of the indeterminates to zero. These geometric and topolog ica l propert ies of quadratic forms are then used, together with homotopy theory resul ts in the l i t e r a t u r e , to deduce that cer ta in sums of squares formulae cannot e x i s t , notably of types [12,12,20] and [16,16,24] . We also 25 23 prove that there is no nonconstant quadratic form S > S Sums of squares formulae with integer c o e f f i c i e n t s are equivalent to " in te rca la te matrices of co lors with appropriate s igns" . This combinatorial nature enables us to es tab l i sh a stronger nonexistence r e s u l t : no sums of squares formula of type [16,16,28] can ex is t i f only integer c o e f f i c i e n t s are permitted. We also c l a s s i f y in tegra l [10,10,16] formulae, and show that they a l l represent ±2v £ -n^. With the a id of the KO theory of rea l pro ject ive spaces, we determine, for given 6 ^ 5 and s , the greatest possible r for which there ex is ts an [r ,s ,s+6] formula. An e x p l i c i t so lut ion of the c l a s s i c a l Hurwitz-Radon matrix equations is a lso recorded. iv TABLE OF CONTENTS INTRODUCTION -. . 1 CHAPTER ZERO: BACKGROUND INFORMATION 8 A. History up to Hurwitz-Radon 8 B. De f in i t i ons and prel iminary resu l ts 14 CHAPTER ONE: QUADRATIC FORMS BETWEEN SPHERES 2 2 1. Geometric propert ies of quadratic forms between spheres..22 2. P a r t i t i o n of a quadratic form into sphere bundles 28 3. When i s a.quadrat ic form a Hopf form? 32 4. Quadratic forms with more than one pair of poles 38 CHAPTER TWO: HOMOTOPY CLASSES OF SPHERES REPRESENTED BY QUADRAT IC FORMS 4 2 5. Nonsingular and normed b i l i n e a r maps hidden in a quadratic form 42 6. Retr iev ing hidden normed b i l i n e a r maps 48 7. A l l quadratic forms between spheres represent Image of J elements 56 8. Some homotopy theorems relevant to the study of sums of squares formulae 60 CHAPTER THREE: NONEXISTENCE OF SUMS OF SQUARES FORMULAE 64 9. Nonexistence of ce r ta in nonconstant quadratic forms 64 10. A lower bound of (2 a +l)*2 D 70 11. Nonexistence of [12,12,20] and [16,16,24] formulae 72 CHAPTER FOUR: SUMS OF SQUARES FORMULAE WITH INTEGER COEFFICIENTS 75 12. Intercalate matrices and the H o p f - S t i e f e l condit ion 75 13. Determination of r* s for r < 9 83 14. Nonexistence of [16,16,28] formulae with integer c o e f f i c i e n t s 87 15. Structure of [10,10,16] formulae with integer c o e f f i c i e n t s 92 CHAPTER FIVE: SUMS OF SQUARES FORMULAE NEAR THE HURWITZ-RADON RANGE 101 16. An e x p l i c i t so lu t ion of the Hurwitz-Radon equations 101 17. Determination of p(s + 5,s) for 6 < 5 106 REFERENCES 112 V ACKNOWLEDGMENTS I f ind myself short of words in expressing my sincere thanks, f e l t long and deeply, to my mentor Professor Kee Yuen Lam, for h is unceasing, genuine concern for my growth as a person. His love and ins ights of mathematics enriched my mathematical l i f e ; h is teaching and research i n i t i a t e d my interest in the problem of "sums of squares". This work could not have been completed without h is longsuffer ing and i n s p i r i n g guidance throughout the past few years. Thanks to my other teachers at UBC, from whom I benefitted much; espec ia l l y to Professor Erhardt Luft for h is encouragements on many timely occasions, and in words and deeds, reminding me of the necessity of hard working. Professor Roy Westwick has lent me many valuable hours for a report on th i s work, and I am thankful to him for point ing out an error in the f i r s t draf t of th i s t h e s i s . My f r iends Mr. Tim Lee and Dr. Wolfgang Holzmann have given me computer terminal t u t o r i a l s that helped me overcome machine-phobia enough to f i n i s h typing t h i s t h e s i s . For f i n a n c i a l supports, thanks are due to the Mathematics Department for a Teaching Ass is tantsh ip and to the NSERC of Canada for summer research grants. My appreciat ion to my beloved wife Betty for accepting my emotional up-and-downs during my study, merr i ly providing a lovely home, constantly upholding me with prayers, and above a l l , for seeing the value of my work beyond immediate app l i cat ions to th i s advancing technological wor ld . F i n a l l y , grat i tude to our f r iends , too many to name here, who, in the past few years, supported us in many ways, and shared our sorrows and joys, struggles and b less ings . 1 INTRODUCTION T h i s t h e s i s i s devoted t o a study of the geometry and to p o l o g y b e h i n d a sums of squares formula of type [ r , s , n ] , namely, ( x 2 + ... + x j M y 2 + ... + y 2 ) = f2 + _ + f 2 f where f , , . . . , f are b i l i n e a r forms w i t h r e a l c o e f f i c i e n t s i n 1 n x ^ . . . , x r , and y l f . . . , y . Such an [ r , s , n ] f o r m u l a i s e q u i v a l e n t t o a normed b i l i n e a r map f: R r x R s > R n s a t i s f y i n g the norm c o n d i t i on |f U,y> I = |x | |y | , x G R r, y G R s. S p e c i f i c a l l y , we put f ( x , y ) = (f (x, y ) " , . . . , f ( x , y ) ) . We s h a l l , t h e r e f o r e , speak of sums of squares formulae and normed b i l i n e a r maps i n t e r c h a n g e a b l y . H u r w i t z [1923] and Radon [1922] proved t h a t t h e r e e x i s t s an [ r , n , n ] f o r m u l a i f and o n l y i f r < pin), where pin) = 8a + 2 b i f n = 2 4 a + b ( 2 c + 1 ) , 0 < b < 3. Normed b i l i n e a r maps of typ e s [ p ( n ) , n , n ] a re s a i d t o be of Hurwitz-Radon t y p e s . For g i v e n i n t e g e r s r and s, we denote by r * s ( r e s p e c t i v e l y r * z s ) the l e a s t i n t e g e r n f o r which t h e r e e x i s t s an [ r , s , n ] f o r m u l a w i t h r e a l ( r e s p e c t i v e l y i n t e g e r ) c o e f f i c i e n t s . These are i n g e n e r a l v e r y d i f f i c u l t t o d e t e r m i n e . We s h a l l improve some of 2 the e x i s t i n g lower bounds in the l i t e r a t u r e by adopting the fo l lowing perspect ive: a normed b i l i n e a r map f of type [ r , s ,n ] determines a quadratic form F: S r + S 1 > S n v ia the Hopf const ruct ion : F(x,y) = (|x| 2-|y| 2, 2 f ( x , y ) ) . The geometric and topological propert ies of th i s Hopf construction are pert inent to the existence problem of the [ r , s , n ] formula. In Chapter 1, we study general quadratic forms mapping spheres into spheres. Let F: S m > S n be one such quadratic form. We show that for every point q £ S n in the image of F, the inverse  image F 1 (q) i_s a great sphere, namely, the in tersect ion of S m  with a subspace of R m + 1 (Theorem 1.5) . These inverse images do not necessar i ly have the same dimension for d i f fe rent points in the image of F. However, i f for each k < m+1, we c o l l e c t into those points with inverse images each a great (k -1 ) - sphere , then the r e s t r i c t ion of F to F 1(Y^) > i s a sphere bundle project ion , provided, of course, that i s nonempty (Theorem 2 .4 ) . Thus, each quadratic  form between spheres is a d i s j o i n t union of great sphere bundles. Not much i s known about the structure of Y^ in general ; even to decide i f Y^ i s nonempty i s n o n t r i v i a l . Nevertheless, we s h a l l show (Theorem 4.2) that if_ there is more than one point q with greatest possible dimension of F 1(q) = r-1 , then indeed  the c o l l e c t ion Y^ i_s a great / -sphere, 1 < / < p ( r ) . These geometric propert ies of quadratic forms are c r u c i a l in proving the nonexistence of cer ta in sums of squares formulae in the subsequent chapters. 3 The r e s u l t s in Chapter 1 also lead to a solut ion of an extension of a problem of Eel ls -Lemaire [1983] on harmonic morphisms. A harmonic map between Riemannian manifolds i s one with vanishing Laplac ian . A harmonic morphism i s a harmonic map <t>: M > N universal in the fo l lowing sense: for every real valued harmonic map g: N > R, the composite qo<t>: M >• R i s always harmonic. Ee l ls -Lemaire [1983] asks: for what integers r and n does there ex is t a sums of squares formula of type [ r , r , n ] whose Hopf construct ion i s a harmonic morphism? We prove that a quadrat ic form between spheres is a harmon ic morphi sm i f  and only i f . i t is . isometric to one of the c l a s s i c a l Hopf  f i b r a t i o n s (Theorem 4 .5 ) . A quadrat ic form F: S m > S n _is_ not necessar i ly the Hopf construct ion of a normed bi 1 inear map. This i s the case j_f and  only i f F admits a pair of poles , namely, a pair of antipodal  points in S n the dimensions of whose inverse images add up to m-1 (Proposit ion 3 .5 ) . We say that an [ r , s ,n ] formula represents a homotopy c lass o £ 7 r m ( S n ) i f the homotopy c lass of i t s Hopf construct ion i s a. Sums of squares formulae of the Hurwitz-Radon types represent elements, in fact generators, of the image of the c l a s s i c a l J-homomorphism. The f i r s t example of a sums of squares formula other than these types i s a [10,10,16] formula ant ic ipated by Kirkman [1848], and wri t ten down e x p l i c i t l y by Lam [1967]. Lam [1977a] showed that t h i s formula represents ±2v, where v i s a generator of the stable 3-stem of the homotopy groups of spheres. Hefter [1982] proved that every [ r , s , n ] formula represents elements in the image of J . On the other hand, Lam [1984] discovered that hidden inside a sums of squares formula, 4 there are nonsingular b i l i n e a r maps, one for each point in the image of i t s Hopf const ruct ion . We show that t h i s in fact i s true for a rb i t ra ry quadratic forms between spheres (Proposit ion 5 .1 ) . Furthermore, every such nonsingular b i l i n e a r map represents  the same homotopy c lass of the quadrat ic form (Corol lary 5 .4 ) . Every hidden nonsingular b i l i n e a r map can be deformed v ia  nonsingular maps to a normed bi1inear maps (Theorem 5 .5 ) . Thus, we can in fact speak of hidden normed b i l i n e a r maps. This means that the existence of one sums of squares formula ant ic ipates in a n o n t r i v i a l way other sums of.squares formulae. For example, in every [10,10,16] formula, there are hidden [4 ,16 ,16] , [8,12,16] formulae, not obtainable from the given one by r e s t r i c t i o n s . It turns out that these hidden normed bi1inear maps can be written  down very e a s i l y (Proposit ion 6 . 3 ) . This a lso gives a very simple "framed cobordism" c a l c u l a t i o n of the homotopy c lasses of spheres represented by quadratic forms. These geometric and topo log ica l resu l ts are blended together in Chapter 3 to deduce some nonexistence r e s u l t s . The proof of the nonexistence of a [12,12,20] formula (Theorem 11.1) v i r t u a l l y makes use of a l l the aforementioned propert ies of quadratic forms, and deep theorems in a lgebraic topology about the image of J (Adams [1966]), homotopy groups of S t i e f e l manifolds (Paechter [1956], Hoo-Mahowald [1965]) and pro ject ive homotopy groups of spheres (Milgram-Strutt-Zvengrowski [1977]). We a lso show that there are no [16,16,24] formulae (Theorem 11.2) nor nonconstant 25 23 quadrat ic forms S > S (Theorem 9.11) and give an upper a b bound for (2 +1)*2 (Theorem 10.3) . General nonexistence resu l ts 5 are d i f f i c u l t to come by, b a s i c a l l y because of the lack of general information on the homotopy groups of spheres and of S t i e f e l manifolds. Most of the known examples of sums of squares formulae have integer c o e f f i c i e n t s . An " i n t e g r a l " sums of squares formula of type [ r , s , n ] i s equivalent to an in te rca la te matrix of type ( r , s ,n ) signed by a (1 , -1 ) -matr ix to s a t i s f y the norm condi t ion . Roughly speaking, an in terca late matrix i s one whose entr ies are " c o l o r s " , in which the colors along each row (respect ively column) are a l l d i s t i n c t , and each 2x2 submatrix contains e i ther exactly two or exact ly four c o l o r s . Let M = (c. .) be one such matrix. We say that M can be signed by a (1 , - 1 ) - m a t r i x A = (a. .) to 1 > J s a t i s f y the norm condit ion i f 3 • • 3 • • i 3 • i • i 3 • § • ~ "~ 1 whenever c. • = c . t •,, i ^ i ' , j ^ j ' . Thus, the study of sums of squares formulae with integer c o e f f i c i e n t s i s combinatoric in nature. Most of the e f fo r t s and debates before the appearance of the c l a s s i c paper of Hurwitz [1898] were on the nonexistence of a [16,16,16] formula (with integer c o e f f i c i e n t s ) . In Section 12, we s h a l l "ed i t " these proofs into a very simple one by studying the structure of in terca late matrices of type (n,n,n) (Theorems 12.2 and 12.5) . Intercalate matrices of type (n,n,n) ex is t i f and k . . . only i f n = 2 and the matrix i t s e l f i s equivalent to the addi t ion table of the dyadic group = Z 2 © . . . © Z 2(k summands). The determination of r* zs is in general very d i f f i c u l t . For r < 9, or s ^ 9, however, i t has been known, in the f o l k l o r e , that r* zs = ros . This l a t t e r number ros can be determined very e a s i l y as 6 the least integer n for which the s t r i n g of binomial c o e f f i c i e n t s n-r < j < s , are a l l even (Shapiro [1984] or B4 below). We s h a l l give a simple proof of t h i s fact by rest r i c t ing the  m u l t i p l i c a t i o n table of a Cayley-Dickson algebra to the f i r s t r rows and the f i r s t s columns. Apart from s igns , t h i s i s an in te rca la te rxs matr ix . A simple enumeration of the number of "co lors" in t h i s matrix then f in i shes the proof (Theorems 12.4 and 13.2) . For the study of sums of squares formulae with integer c o e f f i c i e n t s , purely combinatoric methods, up to the present, have been desperately he lp less , f i r s t because of the d i f f i c u l t y in c l a s s i f y i n g in terca late matr ices, and then the no less d i f f i c u l t procedure of signing such matrices to y i e l d sums of -squares formulae. In Chapter 4, we s h a l l make use of the geometric and topological resu l ts establ ished in the preceding chapters to bypass much of combinatoric complexit ies and show that there i s no in tegra l sums of squares formulae of type [16,16,28] (Theorem 14.3) . We also prove that essent i a l l y there  are no [10,10,16] formulae with integer c o e f f i c i e n t s other than  the one recorded in Lam [1967] (Theorem 15.3) . U n t i l recent ly , the problem of determining r*s for given r and s has received much less at tent ion than the dual problem of determining, for given n and s , the greatest integer r, denoted by p (n ,s ) , for which there ex is ts an [ r , s ,n ] formula. This is equivalent to f inding sxn rea l matrices A ^ , . . . , A r s a t i s f y i n g A^A^ = Ig, i = 1 , . . . , r ; ' A.A1: + A .A^ = 0, 1 < i , j < r, -i / j . 7 The case n = s i s of course the c l a s s i c a l Hurwitz problem and was solved by Radon [1922] and Hurwitz [1923] h imsel f . However, these equations are very d i f f i c u l t to handle even i f n i s just a few uni ts greater than s . Berger and Fr iedland [1984] determined p(s+S,s) for 5 < 3 and also 6 = 4, s odd. We s h a l l adopt, in Chapter 5, the viewpoint of normed b i l i n e a r maps, and determine p(s + 6,s) for 6 < 5, with one possible except ion (Theorem 1 7 . 7 ) . More p r e c i s e l y , i f we l e t Pj- g s + g ] = max{p(m): s ^ m £ s + 6} and h(s+6,s) = minfj > 5: ^ S* 6J=1 (mod.2)} (cf . Lemma B5.2) , then for 5 < 5, r p [s ,s+5] ' i f p [s ,s+6] > 8 ' p(s+6,s) = \ U<s+6fs)f i f P [ S f S + 6 ] * 8 . except poss ib ly for the case 6 = 5 , s = 27 (mod.32). The proof of th i s theore-m in the case Pj- g s + § ] - 8 fol lows by a simple enumeration of r e s t r i c t i o n s of normed b i l i n e a r maps of the Hurwitz-Radon type (Proposit ion 17.2) . For Pj- g s + g j > 8, we s h a l l make use of Adams' resu l ts on stable vector bundles over rea l pro ject ive spaces (Adams [1974]) to show that none of the r e q u i s i t e nonsingular b i l i n e a r maps hidden behind a sums of squares formula of type ^ + p [ s s + 6 ] ' s , s + ^'' c a n e x ^ s t > Consequently, there i s no such sums of squares formula. Solut ions of the Hurwitz-Radon equations have been written down induct ive ly by many authors (Hurwitz [1923], Radon [1922], Wong [1961], Geramita-Seberry [1979] and others ) . In Chapter 5, we record an e x p l i c i t so lut ion of the Hurwitz-Radon equations (Theorem 16.1) , and show how t h i s can be obtained by signing an appropriate in te rca la te matrix to s a t i s f y the norm cond i t ion . 8 CHAPTER ZERO BACKGROUND INFORMATION A. History up to Hurwitz-Radon. We summarize the studies on sums of squares formulae up to the appearance of the milestone papers of Hurwitz [1923] and Radon [1922]. For subsequent developments, see Shapiro [1984]. A1. Ear ly studies on sums of squares concentrated on integer c o e f f i c i e n t s . As early as the second century A.D.,Diophantus already knew that the product of two sums of two squares is again a sum of two squares. In e f f e c t , he had 2 * ^ 2 = 2 and the [2,2,2] formula (x 2 + x 2 ) ( y 2 + y 2 ) = (x 1 y 1 - x 2 y 2 ) 2 + ( x ^ + x ^ ) 2 . Euler ( 1 7 7 0 ) rediscovered th i s formula as a consequence of the law of moduli of complex numbers: the product of the moduli of two complex numbers i s equal to the modulus of the i r product. E a r l i e r , in h is attempt (1748) to prove Fermat's c la im of the four-square theorem, l a t e r a t t r ibuted to Lagrange, he found the [4,4,4] formula where 2 2 1 + X 2 + X 3 + X 4 ) ( * 2 + 1 2 A 2^ y2+y3+ = f 2 + f 2 + f 2 3 r 4 ' f 1 " X 1*1 - x 2 y 2 - X 3?3 - x 4 y 4 ' f 2 = X1*2 + X2?1 + * 3 y 4 - x 4 y 3 , f 3 X 1*3 - * 2 y 4 + x3*1 + X 4 * 2 ' f 4 " x 1*4 + x 2 y 3 - X 3 Y 2 + * 4 y r 9 A2. In the ear ly nineteenth century, Degen [1822] announced an in tegra l [8,8,8] formula, and claimed (erroneously) that there k k k were general izat ions to in tegra l [2 ,2 ,2 ] formulae for k > 4. This work was not noticed u n t i l the years af ter Hamilton's discovery of the quaternions (1843). The discovery can be regarded as an outcome of Hamilton's attempt to f ind a vector product in three dimensions s a t i s f y i n g the law of moduli, or , in our terminology, to f ind a [3 ,3 ,3] formula, or , equ iva lent ly , 3 3 3 a normed b i l i n e a r map f : R x R >- R . Of course, Hamilton f i n a l l y rea l i zed that t h i s was impossible, and was forced to introduce the quaternions. (For an in terest ing account of Hamilton's discovery, see van der Waerden [1976]). Eu le r ' s [4 ,4 ,4] formula was then interpreted as the law of moduli of quaternions. Shortly thereaf ter , Cayley [1845] announced an algebra of octonions with 7. imaginary units e ^ . . . , e 7 2 2 s a t i s f y i n g e 1 = . . . = e^ = - 1 , and with m u l t i p l i c a t i o n given by the 7 t r i p l e t s 123, 145, 624, 653, 725, 734, 176, where the t r i p l e t i j k stands for the m u l t i p l i c a t i o n rules e i e j = _ e j e i = e k ' e j e k = " e k e j = e i ' e k e i = ~ e i e k = e j ' This a lgebra, since then referred to as the Cayley algebra, s a t i s f i e s the law of moduli , and. so gives an in tegra l [8,8,8] formula, as Cayley observed. In t h i s paper [1845], Cayley hinted at the p o s s i b i l i t y of extending t h i s algebra to one with k 2 -1 imaginary units for k > 4. 10 For la te r reference, we record the m u l t i p l i c a t i o n table of the Cayley a lgebra . Table A2.1 M u l t i p l i c a t i o n table of the Cayley algebra eo e 1 e 2 e 3 e 4 e 5 e 6 e 7 eo eo e l e 2 e 3 e 4 e 5 e 6 e 7 e 1 e l _ e o e 3 " e 2 e 5 " e 4 _ e 7 e 6 e 2 e 2 _ e 3 ~ eo e l e 6 e 7 _ e 4 _ e 5 e 3 e 3 e 2 _ e 1 " e 0 e 7 _ e 6 e 5 _ e 4 e 4 e 4 _ e 5 _ e 6 _ e 7 _ e o e 1 e 2 e 3 e 5 e 5 e 4 " e 7 e 6 _ e l _ e o " e 3 e 2 e 6 e 6 e 7 e 4 " e 5 _ e 2 e 3 " e 0 _ e 1 e 7 e 7 _ e 6 e 5 e 4 _ e 3 " e 2 e 1 _ e o Independently, Graves [1844] a lso came up with an algebra of octaves in h is search for an [8 ,8 ,8] formula with integer c o e f f i c i e n t s genera l i z ing Eu ler ' s [4 ,4 ,4] formula. Graves communicated h is r e s u l t s to Hamilton, and asked him to "send a paper on the subject to the P h i l . M a g . " (Halberstam-Ingram [1967]). In the meantime, Hamilton, preoccupied with h is own d i scover ies , d id not do as Graves requested, and the l a t t e r ' s discovery f a i l e d to be recorded in t ime. A3. Two problems natura l l y suggested themselves: to f ind genera l i zat ions of the in tegra l [8 ,8 ,8] formula and the Cayley a lgebra. It was very soon rea l i zed that there was no in tegra l k k k [16,16,16] formula and hence no in teg ra l [2 ,2 ,2 ] formulae 11 for k > 4; to th i s Cayley, Graves, Kirkman and Young a l l acknowledged, but a l l demonstrations were incomplete.(Young [1848], Kirkman [1848]). The nonexistence resu l t was, however, general ly accepted. A4. Kirkman [1848] invest igated the p o s s i b i l i t y of construct ing "pluquaternions" with 2m-1 imaginary u n i t s , m > 4. This i s to construct a normed b i l i n e a r map f : R 2 m x R 2 m > R 2 m sa t i s f y ing f (e1 ,ej ) = ey 1 < j < 2m; f ( e i , e 1 ) = e i , 1 < i < 2m; f ( e i , e i ) = - e 1 , 2 < i < 2m. Kirkman's arguments were, unfortunately , not s a t i s f a c t o r y : he "demonstrated" that such a map i s necessar i ly i n teg ra l and "showed" that for m > 4, a [2m,2m,2m] formula can ex is t only i f x 1 ' ' ' * ' x2m a n c ^ y^ i ' - **' v2m s a t i s f y some extra cond i t ions . Kirkman, however, did show how to construct , for m > 4, 2 an in teg ra l [2m,2m,m -3m+b] formula where b = 8 ,4 ,6 according as m= 0 , 1 , 2 (mod. 3) . In p a r t i c u l a r , he constructed in tegra l [10,10,16] , [12,12,26] , [14,14,32] , and [16,16,46] formulae. The f i r s t three of these are best possible up to the present. A5. In 1860, Genocchi, apparently unaware of these resu l ts and discussions in Engl ish and I r i s h journals , published a paper [i860] to prove, of course erroneously, that E u l e r ' s [4,4,4] formula can be extended to 2 squares, k > 4. Roberts [1879] attempted to c lear up the matter by d e t a i l i n g a proof of the nonexistence of an in tegra l [16,16,16] formula. His proof, unfortunately , l i k e that of Young [1848], was based on the 1 2 hi ther to u n j u s t i f i e d assumption that an in tegra l [16,16,16] formula i s necessar i ly an extension of an in tegra l [8,8,8] formula. Hence, i t was contended, one could s ta r t with a canonical [8 ,8,8] formula and show that the requ is i te extension to a [16,16,16] formula i s impossible. Cayley [1881b] objected that Roberts' proof was inadequate. He enumerated a l l possible in te rca la te matrices of type ( 1 6 , 1 6 , 1 6 ) , four in a l l , and showed that none of these can be signed to give an in tegra l [16,16,16] formula. This was, u l t imate l y , the f i r s t correct proof of the long conceded nonexistence of an integra l [16,16,16] formula. Roberts [18813 pointed out that Cayley's four in te rca la te matrices are a l l equivalent . Th is , however, d id not remedy h is own proof. A6. In the meantime, interest was sh i f ted gradually to the construct ions of algebras general i z ing the quaternions of Hamilton's and the octonions of Cay ley 's , not i n s i s t i n g on the law of moduli . Cayley observed as early as in the middle 1840s that the m u l t i p l i c a t i o n of octonions was not assoc ia t i ve . Later , he showed [ 1 8 8 1 a ] that there i s no way of def in ing an assoc iat ive 8 8 8 normed b i l i n e a r map f : R x R >- R to s a t i s f y (A6.1) f ( x , f ( y , z ) ) = f ( f ( x , y ) , z ) , even just for vectors from an orthonormal bas i s . In fac t , somewhat e a r l i e r , Frobenius had shown that the only assoc iat ive l inear algebra over R without d i v i so rs of zero are the f i e l d s of rea l numbers, complex numbers, and the skew f i e l d of quaternions. Now, a l inear algebra structure on R n i s a b i l i n e a r map f : R n x Rn- > R n . The algebra has no d i v i s o r s of zero i f and 1 3 only i f the map f i s nonsingular, namely, f (x ,y ) = 0 implies x = 0 or y = 0. Thus, Frobenius' theorem says that the only values of n for which there ex i s ts a nonsingular b i l i n e a r map f : R n x R n > Rn s a t i s f y i n g (A6.1) are n = 1 ,2 ,4 . A f i n i t e dimensional algebra without d i v i so rs of zero i s a d i v i s i o n algebra in the sense that the equations ax=b and ya=b are always solvable for any nonzero a. In the early twentieth century, attent ion was confined to l inear assoc iat i ve algebras over R and C and even f i n i t e f i e l d s , and we have c l a s s i f i c a t i o n theorems of Wedderburn, Cartan et a l . The existence of d i v i s i o n algebra structures on Rn was a much harder quest ion, and was not se t t led u n t i l the 1960's by topolog ica l methods (Bott -Mi lnbr [1958], Adams [ i960]) . A7. A d e f i n i t i v e paper on sums of squares formulae appeared in 1898. Hurwitz [ 1 8 9 8 ] demonstrated that the only values of n for which there ex is ts an [n,n,n] formula with real c o e f f i c i e n t s are n = 1 , 2 , 4 , 8 . In essence, he showed that i f such a formula e x i s t s , then there i s an imbedding of the C l i f f o r d algebra of Rn (with respect to the euclidean norm) in the algebra of nxn rea l matrices from which i t fol lows that n = 2 , 4 , 6 , 8 ; the case n = 6 was eas i l y e l iminated. Th is , therefore, se t t led elegantly the more general problem of searching for [n,n,n] formulae with real c o e f f i c i e n t s . Hurwitz posed, at the end of h is paper, the problem of determining r*s, and more s p e c i f i c a l l y ( i ) r*r; ( i i ) p(n,n) = p(n). The l a t t e r was, of course, the wel l known Hurwitz problem. In 1922, Radon solved the equations 1 4 A.A^= I , A.k' + A.A^ = 0, 1 < i , j < r, i ^ j l i n I j j 1 . ' J r / j for nxn matrices with complex e n t r i e s , and showed that , for n = 2 4 a + b ( 2 c + l ) , 0 < b < 3, there are a maximum of 8a+2b+2 of them, among which a maximum of p(n) = 8a + 2° can be r e a l . In the next year, there appeared the posthumous paper [1923] of Hurwitz, edited by Dickson, containing the same resu l ts obtained by e s s e n t i a l l y the same methods. The function p has since then been known as the Hurwitz-Radon funct ion . Dickson [1919] also summarized the h is tory of the eight-square theorem pr io r to the 1898 paper of Hurwitz. B. D e f i n i t i o n s and prel iminary r e s u l t s . B1. Tabulations of b i l i n e a r maps. Sums of squares formulae of type [ r , s ,n ] are equivalent to r s n normed b i l i n e a r maps of type R x R > R . There is an obvious notion of isometric equivalence of normed b i l i n e a r maps: we say that two normed b i l i n e a r maps f : R r x R s > R n and g: R r x R s > Rn are isometric i f there are isometries a : R r > R r , 0: R S > R S , and 7 : R n > R n , such that the diagram R r x RS -> R ax/3 R Y Rr 15 i s commutative. Two sums of squares formulae of type [ r , s ,n ] are said to be equivalent i f the corresponding normed b i l i n e a r maps are isometr ic . Let X, Y, and Z be euclidean spaces with orthonormal bases E = ( e ^ . . . , e r ) and £ = ( « , , . . . , £ ) respect i ve l y . We say that a b i l i n e a r map f : X x Y > Z i s tabulated by the rxs matrix M f = ( f ( e i ' c j ) ) ' r e l a t i v e to the bases E and £, of X and Y respect i ve ly . Lemma B1.1 Suppose a normed b i l i n e a r map f : X x Y > Z i s tabulated by the matrix . Then, ( i ) for each i = 1 , . . . , r, the vectors f (e^ ,e^ ) , 1 < j < s , are mutually orthogonal unit vectors ; ( i i ) for each j = 1,..., s , the vectors f ( e ^ , C j ) , . 1 < i < r, are mutually orthogonal unit vectors ; ( i i i ) for 1 < i , i ' < r, 1 < j , j ' < s, i / i ' , j =/ j ' , < f ( e i , e j ) , f ( e i , , e j , )> = - < f ( e i , e ^ , ) , f ( e i , , e^ )>. Proof. ( i ) and ( i i ) fol low from the fact that the induced l inear maps f x : Y > Z , x € X, and f y : X >• Z, y 6 Y are a l l orthogonal transformations, ( i i i ) i s a d i rec t consequence of the r e l a t i o n (B1.1) <f (x,y) , f (x ' ,y ' )> + <f (x ,y ' ) ,f (x' ,y)> = 2<x , x * x y , y' > , for any x 6 X and y 6 Y, which fol lows e a s i l y from the norm cond i t ion . • 16 Remark: In the l i t e r a t u r e , normed b i l i n e a r maps are also known as orthogonal m u l t i p l i c a t i o n s (Baird [1983], Eel ls -Lemaire [1983]) or orthogonal' pair ings (Yuzvinsky [1981 ], [1984]). B2. Hopf construct ions. Let f : R r x R s > R n be a nonsingular b i l i n e a r map. The mapping F: R r © R s > R n + 1 defined by F(x,y) = (|x| 2-|y| 2, 2f (x ,y ) ) r + s maps the punctured euclidean space R -{0} into the punctured euclidean space R n + 1 - { 0 } , and can, therefore, by normal izat ion, be regarded as a map S r + S 1 > S n between spheres. This is c a l l e d the Hopf construct ion of the nonsingular b i l i n e a r map f. If f i s a normed b i l i n e a r map, then the r e s t r i c t i o n of F to S r + S 1 maps genuinely into S n . The Hopf construct ion of a normed b i l i n e a r map i s , therefore, a map between spheres whose component functions are homogeneous quadratic funct ions. For n = 1 ,2 ,4 ,8 , there are normed b i l i n e a r maps of type [n,n,n] induced by the m u l t i p l i c a t i o n s of real numbers, complex numbers, quaternions and octonions (Cayley numbers) respect i ve ly . The Hopf constructions 1 1 3 2 7 4 of these are quadratic forms S > S , S > S , S > S 15 8 and S > S c a l l e d the c l a s s i c a l Hopf f i b r a t i o n s . (Hopf [1935]). These represent, as i s wel l known, homotopy elements of spheres of Hopf invar iant one (Adams [ i960] ) . The suspensions of these homotopy elements are generators of the image of the J-homomorphism in the stable 0 ,1 ,3 ,7 stems, l a b e l l e d 2t , TJ, V, and o respect ive ly . More genera l ly , for each integer k, there 1 7 k k k are normed b i l i n e a r maps of type [p(2 ),2 ,2 ]. The Hopf c o n s t r u c t i o n of any such normed b i l i n e a r map i s a q u a d r a t i c 2^"*"D(2^)*"1 2 ^  form S p > S , r e p r e s e n t i n g a g e n e r a t o r of the J-homomorphism i n the s t a b l e (p(2 )-1)-stem. B3. I n t e r c a l a t e m a t r i c e s . R e l a t i v e t o the c a n o n i c a l o r t h o n o r m a l bases of R r and R s the normed b i l i n e a r map c o r r e s p o n d i n g t o an [ r , s , n ] f o r m u l a i s t a b u l a t e d by M f = ( ^ a^ i -c k) , where •c-(c ^ ,..., i s k an o r t h o n o r m a l b a s i s of R , and f k a k , i , j X i y j i , j f o r k = 1, n. Ono [1977] and Y u z v i n s k y [ 1981 ] observed t h a t i f the c o e f f i c i e n t s a r e a l l i n t e g e r s , then the nonzero c o e f f i c i e n t s are .1 or - 1 . F u r t h e r m o r e , ( i ) f o r ev e r y i = 1,..., r , j = 1,..., s, t h e r e i s e x a c t l y one k = k ( i , j ) such t h a t a, . . i s nonzero; ( i i ) f o r e v e r y k = 1,..., n, i = 1,..., r , t h e r e i s a t most one j = 1,..., s such t h a t a, . . i s no n z e r o ; ( i i i ) f o r e v e r y k = 1,..., n, j = 1,..., s, t h e r e i s a t most one i = 1,..., r such t h a t a. . - i s nonzero. Thus, the t a b u l a t i n g m a t r i x of f i s of the form f i ,D M i , ] ) where a- • = ±1. I t i s c o n v e n i e n t t o t h i n k of Mf as an r x s m a t r i x whose e n t r i e s a r e " c o l o r s " c ^ , . . . , c n w i t h " s i g n s " . These c o l o r s and s i g n s s a t i s f y the f o l l o w i n g c o n d i t i o n s , which can be e a s i l y v e r i f i e d by u s i n g Lemma B1.1 above. 18 (B3.1) The colors along each row ( respect ive ly column) are d i s t i n c t . (B3.2) If c. • = c . , . , for i / i ' , j / j ' , then c ., = c . , . 1 1 J 1 1 J 1 i J (B3.3) Under the hypothesis of (B1.3) above, 1 , 3 1 , ] ' i ' 1 ' , ] We s h a l l c a l l an rxs matrix an in te rca la te matrix of type ( r , s ,n ) i f i t s ent r ies are colors c ^ - . ^ c s a t i s f y i n g (B3.1) and (B3.2) above. Such a matrix i s sa id to be sighed by the ( 1 , - 1) - m a t r i x A = (a- •) to s a t i s f y the norm condit ion i f (B3.3) a lso holds. Thus, we have Lemma B3.1 An in te rca la te matrix (c. .) of type ( r , s ,n ) can be signed ^ 1 ] by a (1 , -1 ) -matr i x (a. .) to s a t i s f y the norm condit ion i f and only i f the "Hadamard product" (a. -c• .) tabulates a normed b i l i n e a r map corresponding to an f r , s , n ] formula with integer c o e f f i c i e n t s . • We s h a l l c a l l a sums of squares formula in tegra l i f i t i s equivalent to one with integer c o e f f i c i e n t s . Remark: The term " in te rca la te matrix" or ig inates from Norton [1939] in connection with the enumeration of Lat in squares of order 7. B4. Nonsingular b i l i n e a r maps. r • s n A normed b i l i n e a r map f : R x R > R i s necessar i ly nonsingular in that f (x ,y ) =0 impl ies x = 0 or y = 0. Hopf [1941] and S t i e f e l [ 1 941 ] proved the fo l lowing necessary condi t ion for the existence of a nonsingular b i l i n e a r map. 19 Theorem B4.1 (Hopf [1941], S t i e f e l [1941]) If there ex i s t s a nonsingular b i l i n e a r map R r x R s ——>- R n , then the binomial c o e f f i c i e n t s ( , n-r < j < s , are even. We s h a l l refer to th i s as the H o p f - S t i e f e l condit ion , and say that a t r i p l e ( r , s ,n ) s a t i s f i e s the H o p f - S t i e f e l condit ion i f the binomial c o e f f i c i e n t s i^j ' n_r < ^ K S' are a^ even* For given integers r and s , denote by r#s the least integer n for which there ex i s t s a nonsingular b i l i n e a r map f : R r x R s > R n . S i m i l a r l y , denote by ros the least integer n for which the t r i p l e ( r , s ,n ) s a t i s f i e s the Hopf-S t i e f e l cond i t ion . Theorem B4.2 For any integers r and s , r* zs > r*s ^ r#s > ros, If one or both of r and s i s not greater than 9, then, i t i s wel l known in the f o l k l o r e that these numbers are a l l equal (Shapiro [1984]). We s h a l l furn ish a simple combinatoric proof of t h i s fact in Chapter 4 (Theorem 13.2) . The determination of r#s i s in general very d i f f i c u l t . Better lower bounds than ros can sometimes be obtained by "sect ion ing vector bundles over rea l pro ject ive spaces". More s p e c i f i c a l l y , a nonsingular b i l i n e a r map R r x R s > R n gives 20 r sect ions of the n-plane bundle n£ . = £ , © . . . © £ , r s - 1 s-1 S ~ 1 (n summands), where ^ s - 1 i s the canonical l ine bundle over the rea l pro ject ive space RP . This "generalized vector f i e l d problem" has received much a t t e n t i o n . i n the l i t e r a t u r e (see, for example, Davis [1974]). Upper bounds of r#s can be obtained by exh ib i t ing examples of nonsingular b i l i n e a r maps R r x R s > R n . See, for examples, Milgram [1967], Lam [1967 ] , [1968], Adem [1968],[1970 ] ,[1971 ] , and Berr ick [1979]. B5. Binomial c o e f f i c i e n t s . Let N denote the set of nonnegative integers . For every subset I C N, we denote by 2 1 the integer 2* and write i e i 0 w 2 = 0 for the empty set 0. Conversely, for every nonnegative integer n, there i s unique subset I ( n ) C N such that 2 I ^ n ^ = n. We c a l l I(n) the dyadic set of n. An integer m i s said to be a p a r t i a l sum of n i f I (m)C I (n) . The par i ty of binomial c o e f f i c i e n t s can be e a s i l y determined in terms of dyadic sets . Lemma B5.1 (Lucas [1878]) The binomial c o e f f i c i e n t i s odd i f and only i f k i s a p a r t i a l sum of n. • There i s an algor i thm, according to P f i s t e r [1965], for the determination of ros (see also Shapiro [1984]). We f ind below an expression of ros in terms of p a r t i a l sums. For given integers n and s , denote by h(n,s) the greatest integer r for which the t r i p l e ( r , s ,n ) s a t i s f i e s the H o p f - S t i e f e l cond i t ion . Lemma B5.2 For given n and s, h(n,s) i s equal to the least p a r t i a l sum of exceeding n -s . In other words, h(n,s) = min{j > n -s : == 1 (mod. 2)}. • Lemma B5.3 Let 2t > r+s -1 . Then ros = 2t - h ( 2 t - r , s ) . Proof. This fol lows from Lemma B5.2 and the observation that the t r i p l e ( r , s ,n ) s a t i s f i e s the H o p f - S t i e f e l condi t ion i f and only i f the t r i p l e (2 f c -n , s , 2 t - r ) does(Yuzvinsky [1981 ]) . • CHAPTER ONE QUADRATIC FORMS BETWEEN SPHERES 1 . Geometric propert ies of quadratic forms between spheres. Let V = V n be an n-dimensional euclidean space with (pos i t i ve de f in i te ) inner product <,> and norm | | given 2 by |v| = <v,v> for every vector v £ V. The unit sphere S(V) = (v £ V: |v| = 1} has an induced Riemannian metr ic . We can ident i f y the tangent space T (S(V)) with (R.v)-L, the orthogonal complement in V of the 1 -dimensional subspace R.v spanned by v. A mapping F: V"1 > V"2 between euclidean spaces i s c a l l e d a quadratic form i f i t s component functions are homogeneous quadratic funct ions. This means 2 ( i ) F(av) = a F(v) for any a £ R and v £ V], ( i i ) the mapping B: x V1 > V 2 defined by B(v ,v ' ) = F(v+v') - F(v) - F (v ' ) i s b i l i n e a r . Remark: This associated b i l i n e a r map B: V^x V1 >• V 2 l s necessar i l y symmetric. We s h a l l say that a quadratic form F : V*1 > V 2 i s spher ica l i f i t maps the sphere S(V 1) into the sphere S (V 2 ) . I t fol lows from homogeneity that | F ( v ) | 2 = | v | 4 , v £ V 1 f for a spher ica l quadratic form. 23 In t h i s chapter, we study spher ica l quadratic forms as mappings between unit spheres. To emphasize th i s perspect ive, we say that a mapping F: S(V 1) > S(V 2) i s a quadratic form i f i t s r a d i a l extension F: > V 2 given by F(v) = !v|2 F ( - ^ T ) , i f v / 0 , 0, i f v = 0; i s a quadratic form between the underlying euclidean spaces. Throughout t h i s chapter, l e t F: SCV^) > S ^ V 2 ^ B E A quadratic form, with r a d i a l extension also denoted by F, i f there i s no danger of confusion, and associated b i l i n e a r map B: V1 x V1 > V 2 < Lemma 1 . 1 For any v ,v ' £ SiV^, a ' b € R ' F(av+bv*) = a 2 F ( v ) + b 2 F(v') + a b B(v,v'). • For any euclidean space V, the intersect ion of S(V) with a (k+1) -dimensional subspace of V i s c a l l e d a great k-sphere. Great 1 -spheres are simply great c i r c l e s . If v ,v ' £ S(V) are d i s t i n c t , nonantipodal po ints , then there is a unique great c i r c l e containing them. Proposit ion 1 . 2 Let v ,v ' G S(V 1) be orthogonal unit vectors . (a) If F(v) = F(v ' ) = q, then F maps the great c i r c l e through v and v' to the point q £ S (V 2 ) . 24 (b) If F(v) ji F ( v ' ) , then F "wraps" the great c i r c l e through v and v' uniformly twice around a c i r c l e on S^V^) which has F(v) and F(v ' ) as the endpoints of a diameter. Proof. Every point on the great c i r c l e through v ,v ' i s of the form cos 6,v + sin 0 . v ' , 6 £ R. By Lemma 1 . 1 , we have ( 1 . 1 ) F(cos 6. v + s in 0 . v ' ) - •2L(F(v)+F(v' ) ) = -jcos 26. (F(v) -F(v ' ) ) + ^s in 2 0 . . B ( v , v ' ) . (a) If F(v) = F(v ' ) = q, then, for any 6 € R, F(cos G.v + s in 6 . v ' ) = q + ^-sin 2 0 . B ( v , v ' ) . Since t h i s i s a unit vector for a l l 0 6 Rr B(v,vV) must be the zero vector , and we have F(cos 0 . v + s in 0 . v ' ) = q. (b) Suppose F(v) i F ( v ' ) . Then we claim that B(v ,v ' ) and F(v ) -F (v ' ) are l i n e a r l y independent; in p a r t i c u l a r , B(v ,v ' ) i s nonzero. Suppose, for a cont rad ic t ion , that these vectors are l i n e a r l y dependent. Then, i t fol lows from ( 1 . 1 ) above that the image of the great c i r c l e through v and v' i s contained in the segment through the d i s t i n c t points F(v) and F ( v ' ) . The quadratic form being s p h e r i c a l , t h i s image consists prec ise ly of the two iso lated points F(v) and F ( v ' ) , contradict ing the cont inui ty of F. This j u s t i f i e s the c l a i m . Thus, i t fol lows from ( 1 . 1 ) that the image of the great c i r c l e through v and v' i s contained in the a f f i n e plane obtained 25 by t r a n s l a t i n g the 2-dimensional span of (F(v ) -F (v ' ) ) and B(v ,v ' ) by ^(F(v)+F(v' ) ) . But, the in te rsect ion of S(V 2) with an a f f ine plane i s a c i r c l e , and i t i s elementary to check that i f q ,q ' £ V are such that cos 26.q + s in 26.q' l i e s on a c i r c l e for a l l 6 € R, then t h i s c i r c l e has center 0 and q, q' are orthogonal vectors of the same length. We conclude from t h i s ( i ) F (v ) -F (v ' ) and B(v ,v ' ) are orthogonal vectors of the same length; ( i i ) the quadratic form F "wraps" the great c i r c l e through v ,v ' uniformly twice around the c i r c l e cut by the t rans la t ion of the span of (F(v) -F(v ' ) ) and B(v,v*) by ^(F(v)+F(v' ) ) ; ( i i i ) t h i s c i r c l e has center ^(F(v)+F(v' ) ) ; consequently, F(v) and F(v ' ) are the endpoints of a diameter; and (iv) B(v ,v ' ) i s a lso orthogonal to F(v)+F(v' ) . • This resul t leads to the fo l lowing remarkable geometric propert ies of a quadratic form between spheres. Coro l lary 1.3 If v ,v ' £ S(V'1) are orthogonal unit vectors , then B(v ,v ' ) i s orthogonal to both F(v) and F ( v ' ) , and has the same length as F(v)-F(v* ) . • Coro l lary 1.4 If v ,v ' £ S t V ^ are d i s t i n c t , nonantipodal points such that F(v) = F (v ' ) = q, then the great c i r c l e through them i s mapped to the same point q . • 26 Theorem 1 .5 For every point q G S(V^) in the image of F, the inverse image F 1(q) i s a great sphere. Proof. For any nonantipodal v ,v ' 6 F 1 ( q ) , the great c i r c l e through them, by Coro l lary 1.4, l i e s in F 1 ( q ) . Thus, F 1(q) i s the in te rsect ion of S(V 1) with a l inear subspace of , and i s therefore a great sphere. • The main interest in studying quadratic forms between spheres a r i ses from the existence problem of sums of squares formulae, as indicated in the Introduct ion. Associated with a normed b i l i n e a r map f of type [ r , s , n ] i s the quadratic form F: S r + S _ 1 > S n given by F(x,y) = (|x| 2-|y| 2, 2f (x ,y ) ) c a l l e d the Hopf construct ion of f . Hefter [1982] proved by methods of d i f f e r e n t i a l geometry that the inverse image of a regular value of F is a great sphere. Lam [1984] removed the regular value r e s t r i c t i o n and proved Theorem 1.5 for the Hopf construct ion of any normed b i l i n e a r map. Lam's argument was based on considering r e s t r i c t i o n s of f to low dimensional subspaces, and making use.of known facts about the c l a s s i c a l Hopf f i b r a t i o n s . What we have shown above i s a very elementary proof that the same resu l t ac tua l l y holds for a rb i t ra ry quadratic forms between spheres. By Coro l lary 1 . 3 , we know that , for orthogonal unit vectors v ,v ' G V 1 ( the vector B(v ,v ' ) i s orthogonal to both F(v) and F ( v ' ) . This can, therefore, be regarded as a vector tangent to 27 F(v) . The fo l lowing proposit ion c l a r i f i e s the meaning of the associated b i l i n e a r map of a quadratic form between spheres: i t gives the d i f f e r e n t i a l s of F at various po in ts . For every unit vector v € S ( V 1 ) , denote by B v the l inear map given by B ( V ) = B(v ,v ' ) for every v' £ V . Proposit ion 1.6 For every v £ S(V ), t h e d i f f e r e n t i a l DF v i s the r e s t r i c t i o n of B v to (R.v)-L. Proof. This fol lows eas i l y from / » \ _ I im F(v + tv ' )-F(v)  D F v ( v } ~ t->0 t and Lemma 1.1. • We summarize t h i s section with a descr ipt ion of F 1 ( q ) . Corol lary 1.7 For every q £ S( V2^ ^ n fc^e i - m a 9 e °f F r F - 1 (q) = W OS ( V ) = S(W ) , where W = R.v © Ker B = R.v © Ker DF (orthogonal d i rec t sum) q v v 3 for any v £ F 1 ( q ) . Proof. The existence of W^  has been proved in Theorem 1.5. The f i r s t decr ipt ion of W follows from Corol lary 1.3. F i n a l l y , Ker B v = Ker DF^ since B v (v) = 2q. • 2. P a r t i t i o n of a quadratic form into sphere bundles. Let F: S ( V m + 1 ) > S ( V 2 + 1 ) be a quadratic form with associated b i l i n e a r map B: x V1 >- V"2 . We-have shown that for every point q £ S(V 2) in the image of F, the inverse image F 1 (q) i s a great sphere S(W^), where the subspace W^  can be determined, v ia any v £ F 1 ( q ) , as W^  = R.v © Ker B v > This subspace should not, of course, depend on the choice of v 6 F ^ q ) . We s h a l l analyze the associated b i l i n e a r map B in some d e t a i l to remove th i s apparent dependence of W^  on v £ F 1 ( q ) , and show that i f we r e s t r i c t to those q £ S(V"2) whose inverse images are great spheres of a f ixed dimension, then, in f a c t , W depends continuously on q. With t h i s , we s h a l l prove that every quadratic form between spheres i s a d i s j o i n t union of sphere bundles (Theorem 2.4) . Proposit ion 2. 1 For any v ,v ' £ V1 , |B(v ,v ' )| 2 = 4<v,v'> 2 + 2|v| 2|v' | 2 - 2<F(v),F(v' )>. Proof. This i s c lear i f one of the vectors i s zero. Suppose v and v' are nonzero and <v,v'> = 0. Then, applying v' (2.1) |B(v , v ' )| 2 = 2|v( 2|v' | 2 - 2<F(v),F(v')>. More genera l l y , write v' = av + u, where u i s orthogonal to ^ and a = < V f V 0 > . From t h i s , |v|2 (2.2) <v,v'> 2 = a 2 |v| 4 = a 2 |F(v )| 2 . V V Corol lary 1.3 to the unit vectors - i—r and i , i , we have 29 Furthermore, (2.3) B(v ,v ' ) = aB(v,v) + B(v,u) = 2aF(v) + B(v ,u ) . Since B(v,u) and F(v) are orthogonal by Corol lary 1.3, we have by (2.1) and (2.2) above, (2.4) |B(v,v ' )| 2 = 4<v,v'> 2 + 2( |v|2 j u| 2 -<F(v),F(u)>). But, | ut | 2 = | v ' | 2 - a 2 |v| 2 , and <F(v),F(u)> = <F(v), F(v' -av)> = <F(v), F(v ' )+a 2 F(v) -aB(v ' ,v )> = <F(v), F (v ' ) -a 2 F(v ) -aB(u ,v )> by (2.3) = <F(v),F(v')> - a 2 |F(v)| 2 = <F(v),F(v') - a 2 |v| 4 = <F(v),F(v' )> - | v | 2 ( |v' | 2 - | u | 2 ) . Thus, from (2 .4) , |B(v ,v ' )| 2 = 4<v,v'> 2 + 2( |v|2 j v' | 2 -<F(v) ,F(v ' )>). • For any l inear map f : >• V 2 between inner product spaces, we denote by f*: V 2 > the adjoint l inear map given by <f*(v), v'> = <v, f ( v ' )> , v Q V 2 , v' £ V 1 . Proposi t ion 2.2 For any q £ S ^ V 2 ^ * n t h e i m a 9 e o f F » a n d v £ F 1 ( q ) , W = Ker (B*oB -47r ) , q v v v ' where it^ denotes the orthogonal pro ject ion of onto R.v. Proof: Let T v : V1 > V"2 be the l inear map defined by T (v) = 0 and T (v') = B (v') for any v' orthogonal to v. Equiva lent l y , T y ( v ' ) = B v ( v ' ) - 2<v,v'>q, v'£ V}. Then, Ker T v = (R.v) ffi Ker B y , and, by Corol lary 1.7, W = Ker T = Ker T*oT . q v v v Since B^(v) - 2q, i t i s routine to check that T*oT = B*oB - 4 7 T , V V V V V ' and the resul t fol lows, Proposit ion 2.3 For any u,u' £ V , and v £ F 1 ( q ) , < (B*oB v -4^ v )u , u\> = 2<u,u'> - <q,B(u,u')>. Consequently, the endomorphism B * o B v " 4 7 r v £ Hom(V ,V ) depends on q = F(v) only. Proof. < (B*oB v -47r v )u , u'> = <B*oB v(u), u'> - 4 < 7 r v ( u ) , u'> = <B v(u), B v (u ' )> - 4 < v , u x v , u ' > = 1(|B y (u+u*)| 2 - |B v (u )| 2 - |B v (u ' ) | 2 ) - 4<v ,uxv ,u '> , Since v i s a unit vector , and F(v) = q, by Proposit ion 2 . 1 , <(B*oB - 4 7 r )u, u'> v v v ' 2 - 2 = 2<v,u+u'> +|u+u'| - <F(v),F(u+u')> -2<v,u> 2 l u i 2 + <F(v),F(u)> -2<v,u'> 2 - | u ' | 2 + <F(v),F(u'): -4<v, u x v , u' > 31 = 2<u,u'> - <q,F(u+u')-F(u)-F(u')> = 2<u,u'> - <q,B(u,u')>. • For every integer k, l e t Y k = {q £ S (V 2 ) : dim W^  = k}. This set i s possibly empty. Denote by G k ( V ™ + 1 ) the Grassmann manifold of k-dimensional subspaces of V . Theorem 2 . 4 For every k < m+1, i f Y^ i s nonempty, then the r e s t r i c t i o n of the quadratic form F to F 1 ( v k ) > Y k i s a great (k - l ) - sphere bundle p r o j e c t i o n . Proof. We have determined, in Proposit ion 2 . 2 , the subspace Wg as Ker (B*oB v~47r v) for any v £ F 1 ( q ) . The endomorphism B*oB v-47r v £ Hom(V1 ,V1 ) , however, i s independent of the choice of v £ F 1 ( q ) . Let W: Y^ > G k ^ V 1 ^ b e t h e m a ^ 9 * v e n ^ v W(q) = Wg. There i s a f a c t o r i z a t i o n W: Y k > H o m m + 1 " k ( V 1 , V 1 ) > G k ( V m + 1 ) , where Homm + 1 ^(V ,V ) denotes the ( topological ) subspace of Hom(V 1,V 1) cons is t ing of endomorphisms of constant rank m+1-k. The f i r s t map i s continuous by Proposit ion 2 . 2 , and the second map, which assigns the kernel to each l inear map of f ixed rank m+1-k, i s wel l known to be continuous. Thus, W i s continuous. The pul lback of the canonical k-plane bundle by W is prec ise ly the r e s t r i c t i o n of F to 3 £ Y k therefore a vector bundle p ro jec t ion , which further r e s t r i c t s to the great (k - l ) - sphere bundle pro ject ion F 1 (Y, ) > Y, . Wg > Y k . This l a t t e r map is 3 . When i s a quadratic form a Hopf form? Examples of polynomial maps between spheres are rare . The f i r s t n o n t r i v i a l ones are the c l a s s i c a l Hopf f i b ra t ions 3 0 7 A i c p S > S , S > S and S > S . These are quadratic forms. In general , we can obtain quadratic forms between spheres as the Hopf constructions of normed b i l i n e a r maps. For example, with normed b i l i n e a r maps of the Hurwitz-Radon types [p (n) ,n ,n ] , we have nonconstant quadratic forms sp(n)+n-1 ^ g n^ W o o d [ 1968] proved that every quadratic form between spheres i s , up to homotopy, the Hopf construct ion of a normed b i l i n e a r map. In th i s sect ion , we invest igate the question of whether these are necessar i ly Hopf construct ions up to isometr ies . The answer i s in general no (cf . Remark fol lowing Corol lary 3 . 3 ) , and the key factor i s the existence of "a pair of po les" . For any euclidean spaces X and Y, denote by X © Y the orthogonal d i rec t sum of X and Y. Let F: S(V 1) > S ( V 2 ) be a quadratic form with r a d i a l extension also denoted by F. Suppose there are orthogonal decompositions V1 = X © Y and V 2 = R.p © Z such that ( 3 . 1 ) F(x,y) = (|x| 2-|y| 2)p + 2 f ( x , y ) , where f (x ,y ) 6 Z i s orthogonal to p. Then, i t i s routine to v e r i f y that f i s a normed b i l i n e a r map. In th i s case, we say that F i s a Hopf form with poles ±p and underlying normed b i l i n e a r map f : X x Y > Z. Conversely, given a normed b i l i n e a r map f : X x Y ——>- Z, the quadratic form F: S(X © Y)—:—>• S(R.p © Z) given by (3.1 i s c a l l e d the Hopf construct ion of f with poles ±p. Lemma 3.1 Let v = (x,y) be a unit vector in S(X © Y) and v' = ( x ' , y ' ) a tangent vector at v. Then, B(v ,v ' ) = 2(<x,x'>-<y,y*>)p + 2 f ( x , y ' ) + 2 f ( x ' , y ) . • For each v= (x,y) £ S(X © Y) , x 0, y ^ 0, denote by v* y X x x t y -y) and c a l l i t the conjugate of the unit vector (— (see Roitberg [1975]). Lemma 3.2 Let v = (x,y) be a unit vector for which the conjugate v* i s wel l def ined. Then (a) v* i s orthogonal to v; (b) F(v*)= - F ( v ) ; (c) the great c i r c l e through v and v* i s "wrapped" uniformly twice around a great c i r c l e through q = F(v) and the poles ±p; (d) B(v,v*) = | x ] [ y | (-p + (|x| 2 -|y| 2 )q) . Proof. We prove (c) only ; the others are easy to v e r i f y . In p a r t i c u l a r , (d) fol lows d i r e c t l y from Lemma 3 . 1 . For (c ) , the great c i r c l e through v and v* i s "wrapped", according to Proposi t ion 1.2(b), twice around a c i r c l e on SiV^) which has q and -q as the endpoints of a diameter. This i s , . t h e r e f o r e , a great c i r c l e , and i t passes through p since F(|x|v-|y|v*) = F ( ( *_,())) = p. 34 Coro l lary 3.3 If q l i e s in the image of a Hopf form, then so does - q . • Remark: This i s not true for a rb i t ra ry quadratic forms between 1 2 spheres. Consider, for example, the quadratic form F: S > S 2 — 2 given by F ( x 1 f x 2 ) = (x 1 ,»/2x 1 x 2 , x 2 ) . The point ( 1 , 0 , 0 ) l i e s in in the image of F, but not the antipodal point ( - 1 , 0 , 0 ) . This , therefore, i s not a Hopf form by the lemma above. For a general quadratic form between spheres, we have Proposit ion 3.4 If q and - q are both in the image of F, then W and W are mutually orthogonal; i . e . , w _ g C W g ' Proof. Take v £ F 1(q) and v' 6 F 1 ( - q ) . These, vectors are l i n e a r l y independent. By Proposit ion 1 .2 (b ) , the great c i r c l e through them is "wrapped" uniformly- twice around a (great) c i r c l e through ±q. Let a> be the angle between v and v ' . Their images are at an angle 2u> to each other. Thus, cos 2u = -1, from which cos u> = 0 , and v ,v ' are orthogonal. • Proposi t ion 3.5 The fo l lowing statements are equivalent : (a) F i s a Hopf form with poles ±p; (b) p and - p are both in the image of F, and dim W + dim W = m+1; P "P (c) p and - p are both in the image of F and W = W-j- . Proof. C l e a r l y , (a) implies (b), and (b) i s equivalent to (c) . We need only prove that (b) implies (a) . By Corol lary 1 . 7 , 35 F 1(p) = S(Wp) and F 1 ( -p ) = S(W_p). We can choose an orthonormal basis e ^ . . . , e r , e^,...t e g of V1 , r + s = m+1, such that F (e i ) = p, 1 < i < r; F ( 6 j ) = - p , 1 < j < s . Let X = R.e, © . . . © R.e , and Y = R.e, © . . . © R.e„ so that i r 1 s V = X.® Y. For any x e X, y 6 Y, F(x,y) = F((x,0)+(0,y)) = F(x,0) + F(0,y) + B ( (x ,0 ) , (0 ,y ) ) = ( | x | 2 -|y| 2 )p + B ( ( x , 0 ) , ( 0 , y ) ) . Since B( (x ,0) , (0 ,y ) ) i s orthogonal to F(x,0) = p by Coro l lary 1.3, we conclude that F i s a Hopf form with poles ±p. • Let F: S(X © Y) > S(R.p © Z) be a Hopf form with poles ±p and underlying normed b i l i n e a r map f : X x Y > Z. For any point q £ S(R.p © Z) other than ±p, we c a l l the great c i r c l e through p and q the meridian through q (with poles ±p). It i s easy to see that i f q l i e s in the image of F, then so does every point on the meridian through i t . We show below that the inverse images of points on a meridian, save possib ly the poles , are great spheres of the same dimension. Furthermore, they are cut by a one-parameter family of mutually i s o c l i n i c subspaces (Wong [1961]). We remark that two subspaces A and B of a euclidean. space V are said to be i s o c l i n i c i f , for every nonzero vector a £ A, the angle between a and i t s orthogonal pro ject ion in B i s independent of the choice of a . 36 Theorem 3.6 Let q ^ ±p be a point in the image of F. The inverse images of points on the meridian through q, except possibly for the poles ±p, are mutually i s o c l i n i c . Proof. Without loss of genera l i t y , we assume that q i s a point on the "equator", namely, q = f (x ,y ) for some unit vectors x £ X, y G Y. Step 1. Every vector in F 1(q) i s of the form ( x ' , y ' ) , for some unit vectors x' £ X and y' £ Y. Step 2. If v = ^ = f ( x j ' Y i ) f 1 - i - k, i s an orthonormal basis of W , then, x^  ( respect ively y^) » 1 < i < k, are orthogonal unit vectors in X ( respect ively Y). For suppose i and j are d i s t i n c t indices among 1 , . . . , k; then, by Corol lary 1.3, B ( v ^ , V j ) = 0. From t h i s , we obta in , by Lemma 3 . 1 , <x^, X j > - <y^, y^> = 0. On the other hand, since v^ and v^ are orthogonal, we have <x., x•> + <y-, y-> = 0. It fol lows that <x., x.> = <y., y-> = 0, and th i s j u s t i f i e s the c l a i m . Step 3. The conjugates v* ,1 < i < k, of the vectors v^ are well def ined. For each 6 £ (0, —|—), and i = 1 , . . . , k, l e t V j ( 0 ) ' = cos 0.v. + s in 0.v*. We cla im that v^(0), 1 < i < k, i s an orthonormal basis of W^ ^ , where q{6) = cos 29.q - s in 26.p. In fac t , i t i s not hard to see that v^(0), 1 < i < k, are mutually orthogonal unit vectors . Furthermore, since F(v^) = q for 1 < i < k, F ( v i (0 ) ) = •cos 2 0.F(v. ) + s in 2 0.F(v*) + cos 0 . s in 0.B(v.,v*) = cos 26.q - s in 26.p = q(0) 37 by Lemma 1.1 and Lemma 3.2(d) . Thus, dim < dim W q(#)' Since 8 i s neither 0 nor —|- , the conjugates of v^(8), 1 < i < k, are wel l def ined, and v = cos 0.v (6) - s in 6.v (6)*. i i i By s i m i l a r reasoning, we have dim Wg(g) - dim W^. This j u s t i f i e s the c l a i m ; and i t fol lows that W , n S and W have the same q ( 8 ) q dimension. Step 4. Now, we ve r i f y that for any 8, 0 € (0, — ~^) , the subspaces Wq(g) an& Wq(^) a r e i s o c l i n i c . According to Step 3, we have orthonormal bases v^(0), 1 < i < k, and v^(cp), 1 < i < k, of these subspaces respect i ve ly . For each i = 1 , . . . , k, the orthogonal pro ject ion of v^(8) onto Wg(^) 1 5 cos (0 -0 ) . v^ (0 ) . Thus, these subspaces are i s o c l i n i c at an angle |0-0|. This completes the proof of the theorem. • 38 4. Quadratic forms with more than one pair of poles . We have shown (Theorem 2.4) that a quadratic form between spheres i s a d i s j o i n t union of sphere bundles, one over each ( topological ) subspace Y^ cons is t ing of points whose inverse images are great (k -1 ) -spheres. Not much i s known about the structure of Y^. However, i f the quadratic form admits more than one pair of poles , we s h a l l show in the present section that i t i s in fact a Hopf form, and the c o l l e c t i o n of a l l poles is a great sphere of low dimension. Proposi t ion 4.1 Suppose a quadratic form F: S (V m + 1 ) > S ( V 2 + 1 ) has more than one pair of poles. Then, (a) m+1 = 2r for some integer r; (b) F i s the Hopf construct ion of a normed b i l i n e a r map of type [ r , r , n ] . Proof. Suppose F has two pai rs of poles ±p and ±q. The pair ±p determines a normed b i l i n e a r map of type [ r , s , n ] , r+s = m+1 (Proposit ion 3 .5 ) . By Theorem 3 .6 , W and W have the same q - q dimension. Since ±q i s also a pair of poles , we have, m+1 = dim W + dim W , q - q ' by Proposi t ion 3 . 5 . From t h i s i t fol lows that r = dim W = dim W = s . • P ~P 39 Let p be the Hurwitz-Radon function defined by p(n) = 8a + 2 b i f n = 24a+b(2c+D, 0 < b < 3. Theorem 4.2 Let F: S ( V 2 r ) S C V ^ 1 ) be a quadratic form with more than one pai r of poles . The poles of F form a great sphere Y f of dimension / , ! < / < p ( r ) . Furthermore, the r e s t r i c t i o n of F to F 1 ( Y r ) Y^ i s a great ( r - l ) - sphere bundle over the sphere Y r . Proof. Let ±p and ±q be two pai rs of poles of F. Their inverse images are great ( r -1 ) - s p h e r e s . By Theorem 3 .6 , every point q' on the great c i r c l e through p and q i s also in the image of F, and i t s inverse image i s a lso a great ( r -1 ) - s p h e r e . It follows, from Proposit ion 3.5 that q' i s also a pair of poles of F. The poles of F, therefore , form a great sphere. By Theorem 3.6 again , (W^: q G Y f} i s a family of mutually i s o c l i n i c r -dimensional subspaces of V 1 . Thus, F 1 ( Y r ) > Y r i s a great ( r - l ) - s p h e r e bundle over the great sphere Y , with mutually i s o c l i n i c f i b e r s . Such f i b r a t i o n s have been studied in Wong [1961 ], and by Theorem 3.4 there in , the dimension of Y does not exceed p ( r ) . • Proposi t ion 4.3 Let F: > b e a nonconstant quadratic form which maps great c i r c l e s into great c i r c l e s or s ingle po in ts . Then, (a) the image of F i s a great sphere; (b) the r e s t r i c t i o n F' to the image i s a great-sphere bundle; (c) up to isometr ies , F' i s one of the c l a s s i c a l Hopf f i b r a t i o n s . 40 Proof. Let q be a point in the image of F, and F(v) = q . Since F i s nonconstant, W-L i s nonempty. By assumption and Proposi t ion 1.2(b), for any unit vector v' £ W-j-, the great c i r c l e through .v and v' i s "wrapped" uniformly twice around a great c i r c l e so that F(v ' ) = - q and W^  C By Proposit ion 3 .4 , W_g = W .^ It fol lows from Proposit ion 3.5 that F is a Hopf form, and every pair of ant ipodal points in the image i s a pair of poles . Hence, by Proposi t ion 4 . 1 , F i s the Hopf construct ion of a normed b i l i n e a r map of type [ r , r , n ] . By Theorem 4 . 2 , the image of F i s a great sphere of dimension, say, k; and the r e s t r i c t i o n of F to i t s image i s , by Theorem 2 .4 , a great sphere bundle r - 1 2 r - 1 k S > S > S . It i s wel l known that in t h i s case, r = n = 1,2 ,4 , or 8. • We close t h i s chapter with an app l i cat ion of the structure theorems on quadratic forms to a problem of Ee l ls -Lemaire [1983] on harmonic maps between spheres. A map 4>: M >• N between two Riemannian manifolds M and N i s said to be harmonic i f i t s Laplacian A<p vanishes (Eells-Sampson [1964]). A harmonic morphism i s a harmonic map c/>: M >• N universal in the sense that for every rea l valued harmonic function g: N > R, the composite qo<p: M > R i s harmonic. Eel ls -Lemaire [1983] posed the fo l lowing problem: For what integers r and n does there ex is t a normed b i l i n e a r map f : R r x R r >• R n whose Hopf construct ion i s a harmonic morphism? As a c r i t e r i o n for a map </>:M > N to be a harmonic morphism, we quote 41 Theorem 4.4 (Fuglede [1978]) Let M and N be manifolds with Riemannian structures g and h respect i ve ly . A harmonic map <t>: M > N i s a harmonic morphism i f and only i f i t i s hor i zonta l l y conformal. This means that for any x £ M, D0 i s su r jec t i ve , and the r e s t r i c t i o n of D</> to X X (Ker D0 )-L i s a s i m i l a r i t y , namely, there ex i s t s X(x) such that X h(D0 (a) , D 0 ( b ) ) = X(x) .g(a,b) X X for any a , b £ Ker(D0 )-k • Suppose the quadratic form F: S f V ^ > S(V 2) i s a harmonic morphism. By the remark fol lowing Theorem 7.1.1 of Baird [ 1 9 8 3 ] , F i s an open mapping and i s , therefore, s u r j e c t i v e . Furthermore, each i s m-n dimensional, since every DF v i s su r jec t i ve . By Theorem 2 .4 , F i s a great (m-n)-sphere bundle. Thus, n = 1 ,2 ,4 ,8 , and m = 2n -1 . In f a c t , F i s a Hopf form in which every pair of antipodal points i s a pair of poles . From t h i s , we have Theorem 4.5 A quadratic form i s a harmonic morphism i f and only i f i t i s , up to isometr ies , one of the c l a s s i c a l Hopf f i b r a t i o n s . Remark: Baird [1983] has shown that a normed map of type [ r , s , n ] , as a map between euclidean spaces, i s a harmonic morphism i f and only i f r = s = n = 1 , 2 , 4 , 8 . • 42 CHAPTER TWO HOMOTOPY CLASSES OF SPHERES REPRESENTED BY QUADRATIC FORMS 5. Nonsingular and normed bi1inear maps hidden in a quadratic form. In t h i s chapter, we further explore the geometry hidden in a quadratic form between spheres. Let F: SCV^ > S ^ V 2 ^ b e a quadratic form with associated b i l i n e a r map B: x V1 > V 2 . We show that corresponding to every point q £ ^ ( v ^ ) in the image of F, there i s a r e s t r i c t i o n of B to a nonsingular b i l i n e a r map, whose Hopf construction represents the same homotopy c lass of spheres as F does. Furthermore, t h i s nonsingular b i l i n e a r map "hidden" at q can be deformed in a canonical way, v ia nonsingular b i l i n e a r maps, into a normed b i l i n e a r map, so that we can actual ly speak of the "normed b i l i n e a r map hidden at q " . For a Hopf form, we g ive , in Section 6, a simple descr ipt ion of th i s hidden normed b i l i n e a r map. This leads to an easy determination of the homotopy c lass of spheres represented by a Hopf form. Proposi t ion 5.1 For any q £ S(V 2) in the image of F, the r e s t r i c t i o n of B to B : W\ x W-J- > T (S (V0 ) ) q q q q 2 i s a nonsingular b i l i n e a r map. Proof. The b i l i n e a r i t y of B^ i s c l e a r . Let v be a unit vector in W and v' a unit vector in W-J- . By Corol lary 1.4, B (v ,v ' ) = B(v ,v ' ) i s orthogonal to q . This vector cannot be zero; otherwise, F(v ' ) = F(v) and v' £ W ,^ a cont rad ic t ion . Thus, the b i l i n e a r map B^ i s nonsingular. • 43 We s h a l l c a l l the nonsingular b i l i n e a r map in Proposit ion 5.1 the nonsingular b i l i n e a r map hidden inside the quadratic form F at the point q £ S ( V 2 ) . The Hopf construct ion of th i s map (see Section B2) can be deformed, by normal izat ion, into a map between spheres, and hence represents a homotopy element of spheres. We show that t h i s must represent the same homotopy c lass of the quadratic form F. Bypassing the d e t a i l s of normal izat ion, i t i s enough to prove Proposit ion 5.2 Let q be a point in the image of F. There i s a homotopy H t : SiV^) > V2~{0} such that H Q = F (regarded as a map into V2~{0}) and H1 = the Hopf construct ion of ^B^ with poles ±q. Proof. The hidden nonsingular b i l i n e a r map B i s given by B g ( v , v ' ) = F(v+v') - F(V') - |v| 2 q. Thus, the Hopf construct ion of -^B with poles ±q i s the map F : S(V.) = S(W„ © W-L) > V_ given by q 1 q q 2 3 J F (v ,v ' ) = (|v|2- |v'| 2)q + B (v ,v ' ) = F(v+v') - F (v ' ) - | v ' | 2 q . Define a homotopy H f c : S(V 1) > V 2 , 0 < t < 1, by H t ( v , v ' ) = F(v+v') - tF (v ' ) - t | v ' | 2 q . Note that H t(v,v*) / 0 for any t £" [0, 1 ]. To see t h i s , observe that H ( v , v ! ) = 0 implies B(v ,v ' ) + ( l - t ) F ( v ' ) + ( |v| 2 -t i V ( 2 )q = 0. 44 Since, by Corol lary 1.3, B(v ,v ' ) i s orthogonal to both q and F ( v ' ) , such a l inear dependence i s impossible unless v = 0 and v' = 0. Thus, each Hfc maps into V^-{0], and the proposit ion fol lows by noting that H Q = F and H1 = F . . • Remark: In fac t , the proof above can be s l i g h t l y modified to show that the homotopy Hfc takes place in the orthogonal complement of q. Following Lam [1977a], we say that a stable homotopy c lass a £ ^ m^^^ * s b i l i n e a r l y representable i f i t l i e s in the image of a general ized J homomorphism J c s T . , (V„ J > 7T (S") , r s r+s = m+1; i . e . , there i s a nonsingular b i l i n e a r map f : R x R > R n whose Hopf construct ion represents a. Lam proved, for example, that every halvable element in the stable homotopy groups of spheres i s b i l i n e a r l y representable. On the other hand, A l - S a b t i and Bier [1978] exhib i ted homotopy elements of spheres that cannot be b i l i n e a r l y representable. As a consequence of Proposit ion 5 .2 , we have Coro l lary 5.3 (a) A l l hidden nonsingular b i l i n e a r maps of a quadratic form F between spheres represent the same homotopy c lass of F. (b) The homotopy c lass of a quadratic form between spheres i s b i l i n e a r l y representable. • Remark: The only unstable b i l i n e a r l y representable c lasses are the Hopf invar iant one elements represented by the c l a s s i c a l Hopf f i b r a t i o n s . 45 We now take a c l o s e r look i n t o t h e s e hidden n o n s i n g u l a r b i l i n e a r maps, and show t h a t they can be homotoped i n t o normed b i l i n e a r maps. T h i s w i l l then l e a d t o an improvement of C o r o l l a r y 5 . 3 ( b ) . r s n Let f : X x Y > Z be a b i l i n e a r map. For each x € X, l e t f : Y >• Z denote the l i n e a r map induced by f . Suppose f i s n o n s i n g u l a r . Then, f o r each nonzero x £ X, ( i ) the l i n e a r map vf i s i n j e c t i v e ; ( i i ) the endomorphism g = f * o f i s s e l f - a d j o i n t , and has X X X p o s i t i v e e i g e n v a l u e s ; ( i i i ) y £ Y i s an e i g e n v e c t o r of g = f*o f c o r r e s p o n d i n g t o X X X an e i g e n v a l u e X i f and o n l y i f < f ( x , y ) , f ( x , y ' ) > = X<y,y'> f o r every y' £ Y; ( i v ) i f y 1 f . . . , ' y are o r t h o g o n a l u n i t e i g e n v e c t o r s of g x , then f ( x , y 1 ) , . . . , f ( x , y ) are m u t u a l l y o r t h o g o n a l . Lemma 5.4 L e t f : X x Y > Z be a n o n s i n g u l a r b i l i n e a r map. Suppose the endomorphism g x i s independent of x £ S ( X ) . Then, f can be homotoped v i a n o n s i n g u l a r b i l i n e a r maps t o a normed b i l i n e a r map. P r o o f . L e t <P,= ( e . , .. ., e ) be an ort h o n o r m a l b a s i s of (common) u n i t e i g e n v e c t o r s of g w i t h p o s i t i v e e i g e n v a l u e s X.,..., X . X I o Take an or t h o n o r m a l b a s i s E = ( e ^ . . . , e r ) of X and t a b u l a t e the b i l i n e a r map f by the m a t r i x M^= ( f ( e ^ , e _ j ) ) . Note t h a t by the c h o i c e of the b a s i s of Y, f o r each i = 1 , . . . , r , the v e c t o r s f ( e . , e . ) , j = 1 , . . . , s, a r e m u t u a l l y o r t h o g o n a l . 46 For each t 6 [0,1], l e t M be the matrix M. = ( 1 f(e . ,e . ) ) . t /1-t+tXj 1 3 This tabulates a bi l i n e a r map f^.: X x Y > Z which is nonsingular, since every induced linear map f. : Y > Z, L , X x £ X, i s in j e c t i v e , being given by f t J O = 1 f(x,e.), 1 < j < s, t , x D i/1-t + tXj 11 and extension by l i n e a r i t y . The family f , 0 < t < 1, i s , therefore, a homotopy of nonsingular b i l i n e a r maps. Furthermore, If, J O | = l"^=f(x,e.)| = 1, 1 < j < s. : Thus, M 1 tabulates a normed b i l i n e a r map, and i s obtained simply by normalizing each vector in the matrix MQ = . • Remark: Not every nonsingular b i l i n e a r map can be homotoped into a normed bi l i n e a r map. For example, there are nonsingular b i l i n e a r maps of the types R 1 2x R 1 2 > R 1 7 and R 1 6 X R 1 6 ^ R 2 3, but normed b i l i n e a r maps of these types do not ex i s t (Lam [1984], [1985]). Hefter's theorem (Theorem 7.3) provides another proof in the former case: Lam [1977a] exhibited a nonsingular b i l i n e a r map R x R > R , and showed that i t represents v 6 K ^ , which i s not in the image of the J-homomorphism. Theorem 5.5 Every hidden nonsingular b i l i n e a r map of a quadratic form between spheres can be homotoped via nonsingular b i l i n e a r maps into a normed bi l i n e a r map. 47 Proof. For any q in the image of F, choose an orthonormal basis v ^ , . . . , w m + 1 _ k of . By Proposi t ion 2 . 3 , <B (v ,w.) ,B (v,w.)> = <B*oB (w.),w.> = 2<w. ,w . >-<q,B(w. ,w.)> q ' 1 ' q ] v v I ' 3 I R ^ 1 1 i s independent of the choice of v £ F 1 (q) = S ( W g) ' T n e resul t fol lows from Lemma 5 .4 . • Hereafter , we s h a l l c a l l the normed b i l i n e a r map obtained by e f f e c t i n g the homotopy in Lemma 5.4 to the nonsingular b i l i n e a r map B the normed b i l i n e a r map B hidden at q. We C[ C[ s h a l l f ind a simple descr ip t ion of th i s map in the next sect ion . We summarize the resu l ts in th i s sect ion by Theorem 5.6 Let F :S m > S n be a nonconstant quadratic form. For each point q in the image of F, there i s a hidden normed b i l i n e a r map of type [k,m-k+1,n], m-n+1 < k = dim W < n, representing the same homotopy c lass of F. • The existence of hidden normed b i l i n e a r maps inside a quadratic form between spheres a lso implies the fo l lowing in te res t ing and surpr is ing f a c t : hidden behind a sums of squares formula of type [ r , s , n ] , there are other sums of squares formulae of types [k , r+s -k ,n ] , r+s-n < k < n. Because of the l a t t e r cond i t ion , these hidden sums of squares formulae cannot be obtained simply by " r e s t r i c t i n g " the given [ r , s , n ] formula, namely, by set t ing some of the var iab les to. zero. I t i s not always possible to have a hidden sums of squares formula of type [k , r+s -k ,n] for each k in t h i s range. 48 This observation i s of c r u c i a l importance to the nonexistence resu l t s in Chapters 3 and 4. For example, the nonexistence of cer ta in sums of squares formulae hidden behind a [12,12,20] formula i s the key to the proof of the nonexistence of the [12,12,20] formula, as we s h a l l see in the proof of Theorem 11.1. 6. Retr iev ing hidden normed bi1inear maps. Let f : R r x R s > R n be a normed b i l i n e a r map, and F: S r + S 1 > S n i t s Hopf construct ion with poles +p . We know (Theorem 1.5) that for each point q in the image of F, the inverse image F 1(q) is a great sphere. Furthermore, there i s a normed b i l i n e a r map hidden at the point q (Theorem 5.6) . In the present sect ion , we compute the dimension of F 1(q) in terms of the tabulat ion matrices of the given normed b i l i n e a r map f, and show how to write down the hidden normed b i l i n e a r maps. Suppose the b i l i n e a r map f i s tabulated by a matrix r e l a t i v e to given orthonormal bases E = ( e ^ . . . , e r ) and = V s ( e ^ . . . , e g) of R and R respect i ve ly . Let q / i p be a point in the image of F. There ex ist a unit vector c £ R n and a. 6 (0,—^—) such that q = cos 2a.p + s in 2a .c . We s h a l l determine dim W in terms of the matrix q M f(c) = (<c , f (e i , e j )> ) . Note that for any v = (cos a . x , s i n a.y) 6 F 1 ( q ) , we have f (x ,y ) = c . We s h a l l f ind an a l t e r n a t i v e tabulat ion of the normed b i l i n e a r map f r e l a t i v e to other orthonormal bases whereby dim W and the hidden normed b i l i n e a r map B can be determined very e a s i l y . We do th i s by studying in some d e t a i l the endomorphism g^ = B*oB v-4ir v, F(v)= q, whose kernel W^  y ie lds the inverse image F 1(q) by r e s t r i c t i n g to unit vectors . By Proposit ion 2.3 and Lemma 3 . 1 , B ( ( e . , 0 ) , ( e . , , 0 ) ) = 26. ,,.p, 1 ^ ^ r; B ( ( e i , 0 ) , ( 0 , e ) = 2 f ( e i f 6 j ) , 1 < i < r, 1 < j < s ; B ( ( 0 , e • ) , ( 0 , e . , ) ) = - 2 6 - - , . p , 1 ^ * S . J J J , J We write down the matrix representing g = B*oB -47r r e l a t i v e r J q v v v to the orthonormal basis ( e ^ O ) , . . . , ( e r , 0 ) , ( O ^ ) , . . . , ( 0 , e ) of X © Y, as . 2 • 4sin o.I - 4 s i n acos a.M t 2 - 4 s i n acos a.M 4cos a.I s where M = M^(c). Proposit ion 6.1 There are orthonormal bases E = ( e . , . . . , e ) and f = ( e , , . . . , c ) 1 • r ^ 1 s of R and R respect ively such that the matrix (<c,f(e^,ej)>) i s a diagonal matr ix . Proof. Step 1. A vector (x ,y ) , x £ R , y £ R , i s an eigenvector of g = B*oB -4TT with eigenvalue X i f and only i f ' q v v v 3 2 2 t (4sin a - X)x = 4sin acos a.yM , (6.1) (4cos 2 a - X)y = 4sin acos a.xM. 2 Step 2. Let h = rank M. Suppose h < r. Then X = 4sin a i s an eigenvalue of g of m u l t i p l i c i t y r - h , each eigenvector being 50 of the form (x ,0 ) , where xM = 0. S i m i l a r l y , i f h < s , then 2 X = 4cos a i s an eigenvalue of g^ of m u l t i p l i c i t y s - h , each eigenvector being of the form (0 ,y ) , where yMfc = 0. 2 2 If X = —|—, then 4sin a = 4cos a = 2. In t h i s case, unless r = s = h, X = 2 i s an eigenvalue of g with m u l t i p l i c i t y r+s-2h, each eigenvector being of the form (x ,y ) , where xM = 0 and yM t=0. . 2 Step 3. Thus, we look for eigenvalues of g^ other than 4sin a 2 and 4cos a. Let X be one such eigenvalue. Every corresponding eigenvector i s of the form (cos 0.x , s in 6.y), where, by (6 .1) , ( i ) x i s a unit eigenvector of MM*" with eigenvalue 2 2 _ v (4sin a - X)(4cos a - X) (6.2) M = . 2 2 16sin acos a ( i i ) y i s a unit eigenvector of MfcM with the same eigenvalue M; ( i i i ) 6 = 0(X) i s the unique acute angle determined by 2 , „ 4 s m a - X (6.3) tan 6 = . 4cos a - X Step 4. Conversely, for any unit eigenvector x of MMt with eigenvalue M , y = —•—xM is a unit eigenvector of M^ M with the same eigenvalue. For th i s value of u , (6.2) i s a quadratic equation with two d i s t i n c t roots . We denote by X the smaller root , namely, (6.4) X = 2(1 - i/cos22a + jusin22a) , and determine a unique acute angle 6 according to (6 .3 ) . Then, (cos 6.x, s i n d.y) i s an eigenvector of g with eigenvalue X. Step 5. I t fo l lows eas i l y from (6.1) that i f (cos 6.x, s in d.y) 2 2 i s an eigenvector of g^ with eigenvalue X ^ 4sin a, 4cos a, 5 1 then i t s conjugate ( - s in 6.x, cos 0 .y ) i s an eigenvector with eigenvalue 4 - X . Thus, from the unit eigenvector x of MM*" with eigenvalue u, as in Step 4, we a lso have ( - s in 6.x, cos 0 .y ) for an eigenvector of g^ with eigenvalue ( 6 . 5 ) X + = 2 ( 1 + v/cos 22a + MSin 2 a ) . Step 6 . Thus, we consider the ( real symmetric) matrix MMfc and denote by ( 6 . 6 ) M1 > M2 > > MH the nonzero eigenvalues with corresponding orthogonal unit eigenvectors e ^ . . . , e^. Extend th i s to an orthonormal basis E = ( e . , . . . , e, , e, e ) of R r . The vectors e . , h+1 <i < r, 1 n n+1 r l are in the n u l l space of MM t , and so y i e l d eigenvectors (e^,0) . 2 of g^ with eigenvalue 4 s i n a. For each i = 1 , . . . , h, we determine e. = ——e.M as in ' ' ' I /— l Step 4. These, then, form a set of orthogonal unit eigenvectors of MfcM corresponding to the sequence ( 6 . 6 ) of nonzero eigenvalues, Extend t h i s set to an orthonormal basis £ = (e^,..., e g) of R s , where, e^, h+1 < j < s , are in the n u l l space of MfcM, and y i e l d eigenvectors (0 ,e_j) of g^. Step 7 . Now, we determine the matrix (<c , f (e^ , e.. )> ) . For each i = 1 , . . , r, and j = 1 , . . . , s , <c f f (e. ,ej)> = e^elj, where M = M f (c) = (<c,f (e i ,e^)>) . If i > h, then e i MM t = 0 impl ies e^M = 0 so that 52 (6.7) < c , f ( e i f 6 j ) > = 0 , for every j = 1 , . . . , s . S i m i l a r l y , (6.7) i s true for any j > h and i = 1 , . . , r . Step 8. It remains to consider i < h and j < h. For each i < h, we have according to Step 4, an eigenvector (cos 6^.e^, s in 0 ^ ' ^ ) with eigenvalue X^  determined by (6.4) and 6^ by (6 .3) . It fol lows <c , f ( e i , e j ) > = ejMe^ 4cos 2 a - X^  - - t = -.—: -r s in 6 ..e.e. 4sin acos acos 6^ l l j = JJTh. • according to (6 .3) . It fol lows that the matrix (<c,f(e^,e^)>) i s the diagonal matrix whose diagonal entr ies are the nonnegative square roots of the eigenvalues of MM*" arranged in descending order. • We now proceed to describe the normed b i l i n e a r map hidden at the point q = cos 2a.p + s in 2a.c in the image of F: S r + S _ 1 > S n . Let E = ( e , , . . . , e ) and £ = (e^,..., e ) be orthonormal bases set up as in Step 6 above. Suppose n = . . . = M k = 1 in the sequence (6.6) of eigenvalues of MMfc. The dimension of the subspace of corresponding eigenvectors is given by k = r - rank(I-MM f c), where M = (<c,f(e^,e^)> ) . These eigenvectors g ive, according to Step 4, eigenvectors of g with eigenvalue 0 and corresponding 6 = a from (6 .3 ) . Thus, we have an orthonormal basis E' of W cons is t ing of the vectors u^ = (cos a.e^, s in a .e^) , 1 < i < k Before proceeding fur ther , we record Corol lary 6.2 t The dimension of W i s equal to r - rankd -MM ). • q r Remark: This i s also given by s - rank(I S~ M fcM). C l a s s i f i c a t i o n of the eigenvalues of MMfc into 1 = „ = . . . = M k > M k + 1 > . . . > M h > 0 = M h + 1 = • induces a p a r t i t i o n of the tabulat ion matrix M^  = (f(e^,e_j)) r e l a t i v e to the bases E and into k h-k s -h M11 M 12 M 13 M21 M 22 M 23 M31 M 32 M 33 k h-k r-h where the numbers appearing in the border are the numbers of rows and columns of the b locks . According to Steps 4 and 5, we also have an orthonormal basis of cons is t ing of the fo l lowing r + s-k unit vectors : ( - s in a .e^, cos a .e^) , 1 < i < k; (cos 0^.e^, s in d^.e^), k+1 < i < h; ( - s in e . , e . ,cos (?^.e^), k+1 < i < h; ( 0 , ^ ) , h+1 < j < s ; (e i ,0) , h+1 < i < r. To tabulate the hidden nonsingular b i l i n e a r map B , we compute the fol lowing vectors by Lemma 3 . 1 , ( i ) Bg((cos a.e^, s in a . e ^ ) , ( - s i n a . e ^ , , cos c e . , ) ) 2 - 2 = 2(-25^ ^ , s in acos a.p + cos a . f (e^ ,e^ , ) - s in a . f (e^ , 2q*, i f i = i ' 2 f ( e i f e . , ) i f i / i ' ; for 1 < i ' < k; ( i i ) Bg((cos a.e^, s in a .e . ) , ( cos 0j.e_j, s in 6y e^) ) = 2cos asin dyt(e^.e^) + 2sin acos dyfieye^)) for k+1 < j < h; ( i i i ) Bg((cos a.e^, s in a . e ^ ) , ( - s i n 6yey cos 6 y e ^) ) = 2cos acos c 9 j . f ( e i,Cj) - 2sin asin dytieye^)); for k+1 < j < h; ( iv) B ((cos a . e - , s in a .e^ ) , (0 ,e . )) = 2cos a . f (e^ ,e j ) for h+1 < j < s ; (v) Bg((cos a.e^, s in a . « . ) , ( e . , , 0 ) ) = 2sin a . f ( e . , , e . ) for h+1 < i ' < r. Consequently, the hidden nonsingular b i l i n e a r map B i s tabulated by the matrix M ' = ( M ; , M ; 2 M 2 1 M ; 3 M - , ) where ( i ) M n = 2 M i 1 except that each diagonal entry i s 2 q * , the conjugate of q ; 55 ( i i ) M^2 = 2cos a . M 1 2 P + 2sin a .M 2 1 Q, P and Q being respect ively the diagonal matrices P = d iag(s in d^+^,..., s in 6^' Q = diag(cos 6^+^,..., cos d^); ( i i i ) M 2 1 = 2cos a.M 1 2 Q - 2sin a . M 2 1 P ; ( iv) = 2cos a . M ^ ; (v) = 2sin a . M ^ . The hidden nonsingular b i l i n e a r map maps into the orthogonal complement of q, and s a t i s f i e s the condit ion of Lemma 5 .4 , as noted in Theorem 5 .5 . It fol lows that each vector in the tabulat ion matrix M' , apart from the diagonal entr ies of M| 1 , i s orthogonal to both q and q*. Thus, by e f fec t ing an appropriate isometry in R n , we can assume that = 2 M 1 1 . The hidden normed b i l i n e a r map B^ can be obtained, according to the proof of Lemma 5 .4 , by normalizing each entry in the tabulat ion matrix M. In p a r t i c u l a r , i f the point q l i e s on "the equator" of S n , namely, a = —^—, then, each 8^ = —^—, k+1 < j < h, according to (6 .3 ) . In t h i s case, the tabulat ion matrix of B i s p a r t i c u l a r l y s imple: Proposi t ion 6.3 If a = —|— so that q = c 6 R n r the hidden normed b i l i n e a r map B i s tabulated by q ( M 1 1 M 12 M 21 M 13 M31> where M" 2 ( respect ive ly M^) i s obtained by normalizing each >ntry of M 1 2 + M 2 i ( respect ive ly M 1 2 " M 2 ^ * 56 7. A l l quadratic forms between spheres represent Im J elements. Baum [1967] showed that every stable homotopy element of spheres can be represented by a quadratic map, not necessar i ly homogeneous, between punctured euclidean spaces. Such maps are not spher ical in general . In fac t , Wood [1968] showed that spher ica l polynomial maps between euclidean spaces must be homogeneous. Wood [1968] a lso proved that every quadratic form between spheres is homotopic to a Hopf form. On the other hand, Hefter [1982] showed that the.Hopf construct ion of a normed b i l i n e a r map must represent an element in the image of the c l a s s i c a l J homomorphism. It fol lows that a l l quadratic forms between spheres represent elements in the image of J . Following Hefter , we s h a l l of fer a very simple proof of t h i s un i f i ed resul t (Theorem 7.3 below) by making use of the hidden normed b i l i n e a r maps discovered in the las t sect ion . We s h a l l , however, f i r s t r e c a l l the d e f i n i t i o n of the c l a s s i c a l J homomorphism. Let 0(n) be the group of nxn orthogonal matr ices. For each ~m-n / •> j r - -t / \ r - m -m-n* _n-1 _n , map g: S > 0(n) , define J (g ) : S = S * S > S by J(g)(cos a.x, s in a.y) = (cos 2a, s in 2a.g(x ) (y ) ) . The homotopy c lass of J(g) depends only on that of g, and th i s construct ion y ie lds a homomorphism J : 7r (O(n)) > ir ( S n ) . J r m-n m If m < 2n-2, then both homotopy groups are s tab le , and we regard J(g) as an element of the stable (m-n)-stem 7r „ = I i m ir (S ) r m-n , m+k The homotopy groups of the i n f i n i t e orthogonal group 0 = lim 0(n) n-»°° were ca lcu lated by Bott [1959]. 57 Theorem 7.1 (Bott [1959]) The stable homotopy groups of the i n f i n i t e orthogonal group 0 are as fo l lows. k (mod. 8) 0 1 2 3 4 5 6 7 ?rk(o) z 2 z 2 0 Z 0 0 0 Z Each of these homotopy groups i s , therefore , a c y c l i c group. For each integer k, l e t a k be the least integer n for which p(n) = k, namely, the least integer n for which there ex i s t s a normed b i l i n e a r map of type [ k , n , n ] . S p e c i f i c a l l y , a 1 = 1; a 2 = 2; a 3 = a 4 = 4; a 5 = a 6 = a 7 = a 8 = 8 ; ak+8 = 1 6 a k * Insofar as homotopy i s concerned, the image of J in stem k i s n o n t r i v i a l only i f k = 0 ,1 ,3 ,7 (mod.8). Generators of "^(0) can be chosen as the adjoint of any normed b i l i n e a r map of type [k+1,a k + , a k + 1 ] . For a canonical cho ice , we can adopt, for example, the normed b i l i n e a r maps of Hurwitz-Radon types tabulated in Proposit ion 16.3, and labe l the corresponding generators of the image of J as in the fol lowing t a b l e . Table 7.2 Generators of image of J . k ] 3 8r -1 8r 8r+1 8r + 3 generator 2 ,. of im J k 71 v p r Pr71 p r * *r Remark: The homotpy c lasses r\, V , and p1 = a are the suspensions of the Hopf invar iant one elements, which are the only b i l i n e a r l y representable unstable c lasses . 58 For k = 0,1(mod.8), the image of has order 2. For k = -1 (mod.4), the order of the Im J generator i s given by the Adams "conjecture" (see Theorem 8 . 2 ) . With these reco l lec t ions about the J homomorphism, we prove Theorem 7.3 (Hefter [1982], Wood [1968]) Every quadratic form between spheres represents an element in the image of the J homomorphism. Proof. Let a £ 7r m (S n ) be a homotopy c lass represented by a quadratic form F: S m > S n . If F i s not su r jec t i ve , then i t i s necessar i ly t r i v i a l ; consequently, a = 0. Suppose then F i s s u r j e c t i v e . By Sard's theorem (Sard [1942]), there i s a regular value q . Hidden at th i s regular value q, there i s , according to Theorem 5 .6 , a normed b i l i n e a r map of type [m-n,n ,n] , which represents the same homotopy c lass of F. Obviously, t h i s hidden normed b i l i n e a r map represents an element of the image of J : 7r (0) > n ( S n ) . • ^ m-n m Remark: It i s in te res t ing to note that , apart from the c l a s s i c a l Hopf f i b r a t i o n s , the homotopy c l a s s of a quadratic form S m > S n i s always s tab le . In f a c t , we s h a l l show that i f m > 2n-1 , and (m,n) (1 , 1 ), (3,2) , (7,4) , (15,8) , then the quadratic form must be constant (Proposit ions 9.3 and 9 .5 ) . Lam [1977a] gave an example of a normed b i l i n e a r map of type [10,10,16] , and determined, by the method of framed cobordism, the homotopy element represented by the [10,10,16] formula to be ±2v. We s h a l l make use of the simple tabulat ion of hidden normed b i l i n e a r maps (Proposit ion 6.3) to show that in fact 59 every [10,10,16] formula with integer c o e f f i c i e n t s represents ±2v (Theorem 15.5) . Adem [1975] has discovered a [17,18,32] formula with integer c o e f f i c i e n t s as a r e s t r i c t i o n of the m u l t i p l i c a t i o n of the Cayley Dickson algebra (X^ (see Section 13). This [17,18,32] formula c l e a r l y represents the zero homotopy c lass in the stable 2-stem. We record below a useful lemma, which fol lows e a s i l y from the representation theory of C l i f f o r d algebras (Eckmann [1942]). Lemma 7.4 Suppose n = (2m+1)a^ + l. Then every [k+1,n,n] formula represents an odd mul t ip le of the image of J generator in stable stem k. 60 8. Some homotopy theorems relevant to the study of sums of  squares formulae. We conclude t h i s chapter with a c o l l e c t i o n of theorems in homotopy theory that are relevant in the subsequent chapters to the nonexistence theorems on cer ta in sums of squares formulae. Let F: S m > S n be a nonconstant quadratic form with a hidden normed b i l i n e a r map of type [k,m-k+1,n]. By Theorem 5.6, th i s hidden normed b i l i n e a r map represents the same homotopy c lass of F, which l i e s in the images of the general ized J -homomorphisms ir, ,(Vn m , , ,) > ^ (S n) and 7r ,(V" .) > r k-1 n,m-k+1 m m-k n,k 7 r m ( S n ) . In the notations of Paechter [1956], we write these as m-n s j m-n s _ . • -, 7r, m . , m . . . >• 77 ^ and irn , , > ^ respect i ve ly . k-m+n-1,m-k+1 m-n n-k,k m-n J We s h a l l study the hidden normed b i l i n e a r maps in re la t ion to the 2-orders of the homotopy elements they represent. In general , for an element g in an abel ian (addit ive) group G, we mean by the 2-order of g the least integer e for which 2 g = g + . . . + g (2 summands) has f i n i t e odd order. The 2-exponent of G i s the least integer e for which every 2 e g , g € G, has f i n i t e odd order. If no such e e x i s t s , we say that g ( respect ively G) has i n f i n i t e 2-order (2-exponent). The 2-exponent of a c y c l i c group i s the 2-order of a generator. Theorem 8.1 Suppose an [ r , s , n ] formula represents an odd mul t ip le of the generator of the image of J r + s _ n - i ' a n c ^ t n e 2-exponent of one or both of the groups T ( V n ^ r + s _ k ) and * r + s _ k _ , ( V n ^ k ) i s smaller than the 2-exponent b( r+s -n - l ) of the generator of 61 Im J r + n - 1 . Then, the [ r , s , n ] formula contains no hidden normed b i l i n e a r map of type [k , r+s -k ,n ] . • Homotopy groups of S t i e f e l manifolds were ca lcu lated by Paechter [1956] and Hoo-Mahowald [1965]. On the other hand, the 2-exponents of the image of J generators are given by the Adams "conjecture" : Theorem 8.2 (Adams [1966], Qui l len [1971], Becker-Gottl ieb[1975]) The 2-exponent b(k) of the Im generators are as fo l low. k(mod.8) 0 1 2 3 4 5 6 7 b(k) 1 1 0 3 0 0 0 1+MU+1) where 2 * ^ k + 1 ^ is the highest power of 2 d i v id ing k+1. • We s h a l l f ind an app l i ca t ion of t h i s resul t in Theorem 11.1. Sometimes, i t i s possib le to obtain further r e s t r i c t i o n s on the types of normed b i l i n e a r maps hidden in an [ r , s , n ] formula by invoking a commutative diagram at t r ibuted to James [1976] in the l i t e r a t u r e . It gives a f a c t o r i z a t i o n of the generalized J-homomorphism. Theorem 8.3 (James [1976], Raussen-Smith [1979]) If s < n, there i s a commutative diagram 7T (V ) 1 > TT (S") r-1 n,s r+s-1 S + 1 in which ( i ) E i s the (s+1)-fold suspension, and i s an isomorphism i f 2n > r+2s+2; ( i i ) 3 i s the boundary homomorphism in the homotopy exact sequence of the f i b r a t i o n . n - s - 1 -> v n, s+1 -> V n, s We s h a l l f ind an app l i cat ion of t h i s proposi t ion in Theorem 11.2. A l - S a b t i and Bier [1978] formulated an extension of t h i s commutative diagram, and used i t to deduce that cer ta in stable homotopy elements of spheres are not b i l i n e a r l y representable. Theorem 8.4 ( A l - S a b t i - B i e r [1978]) If s < n, then the fo l lowing diagram commutes. * r 1 ( V n c } r - l n , s Y /_n-s- 1 x 7T r _ 2 ( S ) V » r - 2 ( v n , s V • W l < s " > E S+1 r~2 n - s - 1 * * r - 2 ( P n - s - 1 ) In the diagram, ( i ) j : S n " s - 1 • P , i s inc lus ion into the bottom c e l l of n - s - 1 the stunted pro ject ive space p n _ s _ - | ? ( i i ) the bottom maps are obvious isomorphisms. 63 C o r o l l a r y 8.5 I f 2n > r+2s+2, then t h e r e i s an ex a c t sequence \ - , ( V n , s > — W 5 " " 5 " ' ) — 'r -2«C-1>- " The k e r n e l s of j * : 71 0 ( S n _ S 1) > ?T -(P°° _ ,) were * r _ z 1 ~ z n — s - i c a l c u l a t e d by M i l g r a m - S t r u t t - Z v e n g r o w s k i [1977]. Suppose an [ r , s , n ] f o r m u l a c o n t a i n s a hidden normed b i l i n e a r map of type [ k , r + s - k , n ] . A p p l y i n g C o r o l l a r y 8.5 w i t h r and s r e p l a c e d by r+s-k and k r e s p e c t i v e l y , we have Theorem 8.6 Suppose an [ r , s , n ] f o r m u l a c o n t a i n s a hidden normed map of type [k, r + s - k , n ] . I f the 2-exponent of K e r ( j * : * r + s _ k _ 2 < S n " k _ 1 ) ^ * r + s-k-2 ( Pn-k-1>> i s s m a l l e r than t h a t of Im J r + S _ n _ 1 , then the [ r , s , n ] f o r m u l a cannot r e p r e s e n t an odd m u l t i p l e of the image of J g e n e r a t o r . • We s h a l l f i n d an a p p l i c a t i o n of t h i s theorem i n Theorem 11.1. 64 CHAPTER THREE NONEXISTENCE OF SUMS OF SQUARES FORMULAE 9. Nonexistence of cer ta in nonconstant quadrat ic forms. In Chapter 2, we have shown that a nonconstant quadratic form between spheres contains many hidden normed b i l i n e a r maps of types [k,m-k+1,n], each corresponding to a point q £ S n in the image of F, with m-n+1 < k = dim W^  < n (Theorem 5.6) . It i s not necessar i ly true that there be a hidden normed b i l i n e a r map corresponding to each value of k in t h i s range. In the language of Section 2, the set Y^ i s poss ib ly empty. The c l a s s i c theorem of Hopf [1941] and S t i e f e l [1941] (Theorem B4.1) places a r e s t r i c t i o n on th i s dimension. Proposi t ion 9.1 (a) Let F: S m > S n be a nonconstant quadratic form, and q G S n a point in the image of F, with dim W^  = k. Then, the t r i p l e (k,m-k+1,n) s a t i s f i e s the Hopf -S t ie fe l cond i t ion . (b) Consequently, i f an [ r , s , n ] formula contains a hidden normed b i l i n e a r map of type [k , r+s -k ,n ] , then the t r i p l e (k , r+s-k,n) s a t i s f i e s the Hopf -S t ie fe l cond i t ion . • In the present sect ion , we explore various consequences of t h i s r e s u l t . We f i r s t record, however, a simple lemma that helps e l iminate cer ta in values of k from being the dimension of w g . 65 Lemma 9 . 2 For given integers m and n, the t r i p l e (k,m-k+1,n) s a t i s f i e s the H o p f - S t i e f e l condit ion i f and only i f there i s no p a r t i a l sum of n in the range k-(m-n) < j < k - 1 . • Proposit ion 9.3 Suppose there ex i s ts an integer / such that m > 2l > n. Then, every quadratic form F: S m > S n i s constant. Proof. Suppose, for a c o n t r a d i c t i o n , that there ex i s ts a nonconstant quadratic form S m > S n . By composing with appropriate inc lus ion maps, we obtain a nonconstant quadratic 2l 2l-\ form F: S > S . The hidden normed b i l i n e a r maps are of types [ k f 2 ' - k + 1 , 2 l - \ ] , 2 < k < 2 7 - 1 . But, by Lemma 9 . 2 , for values of k in th i s range, none of the t r i p l e s (k , 21-k+1,2'- 1 ) s a t i s f i e s the H o p f - S t i e f e l cond i t ion . Thus, the requ is i te hidden normed b i l i n e a r maps do not e x i s t , and the quadratic form i s constant. • Remark: Wood [1968] showed that under the same hypothesis of Proposit ion 9 . 3 , every polynomial map F: S m > S n i s constant. His proof made use of in te res t ing theorems of Cassels [1964] and P f i s t e r [1965]. Proposi t ion 9.3 gives a simpler proof for quadratic forms. Wood also asked whether for an even integer n, any element of 7r n(S n) other than 0, ±1, can be represented by polynomial maps from S n > S n . The fo l lowing proposit ion gives a p a r t i a l s o l u t i o n . 66 Proposi t ion 9 . 4 If n i s even, every quadratic form F: S n >• S n i s t r i v i a l in homotopy. Proof. Suppose F i s n o n t r i v i a l . It must be su r jec t i ve , and thus must admit a regular value q by Sard's theorem. The hidden nonsingular b i l i n e a r map i s of the type R1 x R n > R n . By Proposi t ion 2 . 4 of Lam [1976], t h i s represents 0 £ TT (S n ) . The resu l t now fol lows from Coro l lary 5.3(b) . • Wood [1968] a lso asked i f there are integers m and n, other than those in Proposi t ion 9 . 3 , for which every quadratic form F :S m S n i s constant. The fo l lowing proposit ion provides an example. Proposi t ion 9.5 2 / + 1 - 1 2l (a) If / > 4 , every quadratic form S > S i s constant 2 / + 1 - 1 2l (b) If / ^ 3 , every nonconstant quadratic form S > S i s isometric to the Hopf f i b r a t i o n . Proof. Suppose there e x i s t s a nonconstant quadratic form 2 / + 1 - 1 2l F: S S . The hidden normed b i l i n e a r maps are of types [ k , 2 1 + 1 - k , 2 l ], and, by Lemma 9 . 2 , the only admissible value of k i s 2l. Thus, every hidden normed b i l i n e a r map i s of type [ 2 / , 2 / , 2 / ] , and 0 < / < 3 by the c l a s s i c a l theorem of Hurwitz-Radon. This proves (a) . For 0 ^ / ^ 3 , the above argument shows that the image of F i s Y, , which is a great sphere of dimension p ( 2 ^ ) = 2 ' . 2 ' - 1 2 / + 1 - 1 Furthermore, by Theorem 2 . 4 , S > S > Y^  i s a (2l-1)-sphere bundle over the great sphere Y, . From t h i s , 2l Yj - S and F is isometric to the Hopf f i b r a t i o n . • 67 Proposit ion 9 . 6 2 / + 1 - 1 2 Z +1 (a) If / > 4 , every quadratic form S > S is constant. 2 / + 1 - 1 2 Z +1 (b) If / ^ 3 , every nonconstant quadratic form S »• S is e i ther in fact mapping into S , or , in case / = 1 , the Hopf construct ion of a [ 1 , 3 , 3 ] formula. 2 / + 1 - 1 2 * + 1 Proof. Suppose F: S > S i s a nonconstant quadratic form, with a hidden normed b i l i n e a r map of type [ k , 2 / + 1 - k , 2 ^ + 1 ] . There are only three possible values for k , namely, 2 ' - 1 , 2^ , 2 ' + 1 . If k = 2 ^ - 1 , then we have a normed b i l i n e a r map of type [21 -1 ,2l + 1 ,2l + 1 ] , from which 2 / - 1 < p ( 2 ' + l ) = 1 , a contradict ion unless / = 0 , 1 . Thus, for / ^ 2 , (a) and the f i r s t part of (b) follow by the same argument of the previous p ropos i t ion . 3 . 3 Consider the case / = 1 . The quadratic form maps S into S . As above, the hidden normed b i l i n e a r maps are of type e i ther [ 1 , 3 , 3 ] or [ 2 , 2 , 3 ] , By Proposit ion 2 . 4 of Lam [ 1 9 7 6 ] , nonsingular 1 3 3 3 b i l i n e a r maps of type R x R > R represent ±2L £ )• On the other hand, every normed b i l i n e a r map of type [ 2 , 2 , 3 ] can 2 2 2 3 be factored as R x R > R c >' R , and so represents zero homotopy c lass (Smith [ 1 9 7 8 ] ) . Thus, these two types of hidden normed b i l i n e a r maps are mutually exc lus ive , by Theorem 5 . 6 . In the f i r s t case, F is the Hopf construct ion of a normed b i l i n e a r map of type [ 1 , 3 , 3 ] , and, up to isometr ies , i s given by 2 2 2 2 F ( x ^ , X 2 , x ^ , x ^ ) = (x.|—x2~x^~ ^  4 f ^ ^ 1 ^ 2 ' 2 ^ 1 ^ 3 ' 2 ^ ] ^ 4 ^ * If the hidden normed b i l i n e a r maps are of type [ 2 , 2 , 3 ] , then i t fol lows from the previous proposit ion that , up to 3 2 3 isometr ies , F i s S S c > S , where the f i r s t map i s the c l a s s i c a l Hopf f i b r a t i o n and the second one the inc lus ion map. 68 Lam [1984],[1985] studied the d i s t r i b u t i o n of the possible dimensions of for the Hopf construct ion F: S > S of a normed b i l i n e a r map of type [ r , s , n ] . Denote by r#s the least integer n for which there ex i s ts a nonsingular b i l i n e a r map f : R r x R s > R n . Lam [1984], [1985] proved that ( i ) there i s a point q in the image of F such that dim W < r + s - r # s ; ( i i ) there i s at least one hidden nonsingular b i l i n e a r map of type R k x R r + s - k > R n , where r + s-n < k < r + s - r#s. In view of the resu l ts of Section 5, we have Theorem 9.7 (a) Let F: S r + S n > S n be the Hopf construct ion of an [ r , s ,n ] formula. There i s at least one hidden normed b i l i n e a r map of type [k , r+s -k ,n ] , where r+s-n < k < r+s-r#s. (b) Suppose there ex is ts a sur jec t i ve normed b i l i n e a r map of type [ r , s , n ] . Then, r+s < n+p(n), p being the Hurwitz-Radon funct ion . • Theorem 9.8 (Lam [1984]) There i s no [12,12,19] formula. Consequently, 12*12 > 20. Proof. If such a sums of squares formula does e x i s t , then , since 12#12 = 17, by Theorem 9 .7(a ) , there must be a hidden normed b i l i n e a r map of type [ k , 2 4 - k , l 9 ] , 5 £ k < 7. But by Lemma 9 .2 , none of the t r i p l e s ( k , 2 4 - k , l 9 ) , k=5,6,7, s a t i s f i e s the H o p f - S t i e f e l cond i t ion , a cont rad ic t ion . • Proposi t ion 9.9 (Lam [1985]) For r and s between 10 and 17, the values of r*s i s no less than the amount shown in the fo l lowing tab le : Table 9.10  Lower bounds of r*s for 10 < r ,s < 17. 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 10 16 20 20 20 20 24 24 26 11 20 20 20 24 24 24 27 1 2 20 24 24 24 24 28 1 3 24 24 24 24 29 1 4 24 24 24 30 1 5 24 24 31 1 6 24 32 17 32 We conclude th i s sect ion with a nonexistence resu l t on nonconstant quadratic forms between spheres, providing an example to the question of Wood [1968] mentioned in the remark fo l lowing Proposi t ion 9 .4 . Theorem 9.11 25 23 There i s no nonconstant quadratic form S > S Proof. By using the H o p f - S t i e f e l cond i t ion , we e a s i l y see that the only possib le normed b i l i n e a r map in a nonconstant 25 23 quadratic form F: S >- S are of types [ k , 2 6 - k , 2 3 ] , 10 < k < 13. But, k*(26-k) > 24 for 10 < k < 13, according to Table 9.10. Thus, none of the requ is i te normed b i l i n e a r maps e x i s t s , and the quadratic form must be constant. • 10. A lower bound of (2 a+l)*2 . k k The determination of 2 *2 i s p a r t i c u l a r l y i n t e r e s t i n g . We know from the nonimmersion resu l t of James [1963] that 2 k#2 k > 2 k + 1 - p ( 2 k ) . By Theorem 9.7(b) , we can deduce e a s i l y that 2 k * 2 k > 2 k + 1 - p ( 2 k ) + 1 . Lam [1985] undertook a c loser analys is and obtained Theorem 10.1 (Lam [1985]) Let c = max{p(/): 1 < / < p(2 )}. Then, we have 2 k * 2 k > 2 k + 1 - c . • Remark: Let 2 m be the least power of 2 exceeding k+1. Then, in f a c t , c = p (2 m ) . We l i s t , in Table 10.2," the lower bounds of 2 k * 2 k for 4 < k < 10, determined by Theorem 10.1 . Table 10.2  Lower bounds of 2 k * 2 k , 4 < k < 10. k 4 5 6 7 8 9 1 0 2 1 6 32 64 128 256 512 1 024 2 k*2 k> 24 56 1 20 247 503 1015 2039 Theorem 10.3 (2 a+1)*2 b > 2 a +2 b unless (2 a+l) < p ( 2 b ) , or a < min{b,4}. 71 Proof. Case 1: a > b. Suppose there ex i s ts a normed b i l i n e a r map of type [ 2 a + 1 , 2 b , 2 a + 2 b - 1 ] . Since ( 2 a + l ) # 2 b > 2 a+1, we have, by Theorem 9.7(a) , a hidden normed b i l i n e a r map of type [k ,2 a +2 b -k+1,2 a +2 b -1 ] for some integer k s a t i s f y i n g 2 < k < 2 b - 1 . But, by Lemma 9 .2 , none of the t r i p l e s (k ,2 a +2 b -k+1,2 a +2 b -1 ) , 2 < k < 2 b - 1 , s a t i s f i e s the Hopf cond i t ion . Thus, the said normed b i l i n e a r map does not e x i s t . Case 2: a = b. This i s the easiest case. We have ( 2 a + l ) # 2 a a +1 = 2 . This fol lows from the H o p f - S t i e f e l condit ion and the 2a+1 2 a 2 a + 1 existence of a nonsingular b i l i n e a r map R x R >- R induced by m u l t i p l i c a t i o n of polynomials with rea l c o e f f i c i e n t s . Thus, ( 2 a + l ) * 2 a > ( 2 a + D # 2 a = 2 a + 1 = 2 a + 2 a . Case 3: a < b. In th i s case, i f a < 4, then 2a+1 < 9, and by Theorem 13.2, ( 2 a + l ) * 2 b = (2 a+1)o2 b = 2 b (see Remark below). Thus, we assume a S 4. Milgram [1967] has constructed a family of nonsingular b i l i n e a r maps (data corrected in Berr ick [1979]) 2a+2 2 b 2 b +2 a -1 which includes one of the type R x R > R From t h i s , i t fol lows that (2 a+1)#2 b > 2 b + 2 a - 1 . We can assume 2a+1 > p(2 b ) so that ( 2 a + l ) # 2 b > 2 b . Suppose there e x i s t s a normed b i l i n e a r map of type [ 2 a + 1 , 2 b , 2 b + 2 a - 1 ] . By Theorem 9.7(a) , there ex is ts a hidden normed b i l i n e a r map of type [k ,2 b +2 a - k+1 ,2 b +2 a -1 ] , 2 < k < 2 a . However, none of the corresponding t r i p l e s s a t i s f i e s the H o p f - S t i e f e l condit ion by Lemma 9 . 2 . Thus, the said normed b i l i n e a r map does not e x i s t . 72 Remark: Theorem 13.2 has been known in the f o l k l o r e (see also Shapiro [1984]). The proof of th i s theorem does not depend on Theorem 10.3 . By Proposi t ion 9 .4 , even i f there ex i s ts a normed b i l i n e a r b i l i n e a r map of type [ 2 a + 1 ,2° , 2 a + 2 b ] , i t s Hopf construct ion is necessar i ly t r i v i a l in homotopy. 11. Nonexistence of [12,12,20] and [16,16,24] formulae. In t h i s sect ion , we improve Lam's resu l ts 12*12 > 20 (Theorem 9.8) and 16*16 > 24 (Theorem 10.1) by showing that in fact [12,12,20] and [16,16,24] formulae do not e x i s t . We sha l l make use of the theorems in homotopy theory reco l lec ted in Section 8. Theorem 11.1 There i s no [12,12,20] formula. Consequently, 12*12 > 21. Proof. Let us suppose that there i s a normed b i l i n e a r map of 9 3 on type [12,12,20] , with Hopf construct ion F: S > S , and derive a contradict ion in the fo l lowing steps. Step 1. The only poss ib le hidden normed b i l i n e a r maps are of types [ k , 2 4 - k , 2 0 ] , where k = 4 , 8 , 9 , 1 0 , 1 1 , 1 2 . This fol lows eas i l y by checking the H o p f - S t i e f e l cond i t ion . Step 2. F represents an odd mul t ip le of the generator v £ 7r^ and i s consequently s u r j e c t i v e . Since 12#12 = 17, by Theorem 9.7(a) , there must be at least one value of k in the range 4 ^ k < 7. The only poss ib le value i s k = 4, according to Step 1 . . The corresponding hidden normed b i l i n e a r map, being of type 73 [4 ,20,20] , represents an odd mul t ip le of v by Lemma 7.4 , and is n o n t r i v i a l . Consequently, F i s s u r j e c t i v e . Step 3. The values k = 9, 10, 11 are inadmiss ib le . The n o n t r i v i a l homotopy c lass of F l i e s in the image of the 20 generalized J homomorphisms ^ - i ^ ^ O 24-k^ " 2 3 ^ ^ a n < ^ 20 7 r 23-k^ V 20 k^  ~ — ^ ff23^S ^' W e l * s t b e l ° w t n e homotopy groups of S t i e f e l manifolds, according to Paechter [1956]. k 8 9 10 11 12  * k - 1 ( V 2 0 , 2 4 - k ) Z 1 2 ® Z 8 ® Z Z 2 Z 12 Z 2 Z24®Z8 7 r 2 3 - k ( V 2 0 , k ) Z4®Z48 Z 2 Z 12 Z 2 Z24®Z8 Since the 2-exponent of the generator in the stable 3-stem is 3, by Theorem 8 . 2 , t h i s rules out k = 9,10,11 by Theorem 8 . 1 . Step 4. k = 8 i s also inadmiss ib le . With k = 8, we f ind from the tables in Milgram-Strutt -Zvengrowski [1977] that Ker( j* : T T 1 4 ( S 1 1 ) > ^ ^ P " , ) ) = Z^. This has 2-exponent 2. Thus, k=8 i s ruled out by Theorem 8.6 . Step 5. We are l e f t with only two possible values of k, namely, k = 4,12. The quadratic form F, by Theorem 2 .4 , is 3 - 1 the union of two great sphere bundles, S > F ^ Y 4 ' and S 1 1 > F _ 1 ( Y 1 2 ) > Y]2. Now, by Theorem 4 . 2 , Y 2 i s a great sphere of dimension / < p(12) = 4. Since the map F is 20 / 19-/ su r jec t i ve , Y 4 = S -S has the homotopy type of S . The fol lowing homology c a l c u l a t i o n shows that th i s i s inadmiss ib le . Denote M = F 1(Y^) for s i m p l i c i t y . We have two sphere bundles 11 / 3 23 19-/ over spheres, S > M > S , and S > S -M > S Since / < 4, we have H*(M) = H*(SZ x S 1 1 ) , and H*(S 2 3-M) = H * ( S 3 x S 1 9 1 ) . By Alexander d u a l i t y , HMM) = H 2 2 _ I ( S 2 3 - M ) Z i f i = /, /+3, 19; . 0 otherwise. But, from H*(M) = H*(SZ x S 1 1 ) , we have (M) = Z for i = /, 11,11+/. This then requires / = 8, a cont rad ic t ion . The proof of Theorem 11.1 is complete. • Theorem 11.2 There i s no [16,16,24] formulae. Consequently, 16*16 > 25. Proof. Suppose, for a cont rad ic t ion , that there i s a normed b i l i n e a r map f of type [16,16,24] with Hopf construct ion 3 1 ? 4 F: S J > S . Since 16#16 = 23, by Theorem 9.7(a) , there is a hidden normed b i l i n e a r map of type [k, 32-k, 24] , 8 < k < 9. But k = 9 i s impossible by the H o p f - S t i e f e l cond i t ion . Thus, there i s a hidden [8,24,24] "formula, which, by Lemma 7.4, represents an odd mult ip le of o £ , and i s n o n t r i v i a l . This holds also for the normed b i l i n e a r map f of type [16,16,24] , by Theorem 5.6 . The Hopf construct ion F, according to Theorem 8 . 3 , i s homotopic to the 17-fold suspension of 3 (ad . f ) , where 15 • 15 ad.f :S > V"24 1 6 i s the r e s t r i c t i o n to S of the adjoint 7 of f . No element of " '•j^S )/ however, can be suspended to a c lass of odd Hopf invar iant , since every such homotopy c lass is detected by S q 8 , which vanishes on any 2 - c e l l complex S ^ l j e 1 ^ . Remark: In the next chapter, we s h a l l show (Theorem 14.3) that i f only integer c o e f f i c i e n t s are permitted, then, not even a [16,16,28] formula can e x i s t . 75 CHAPTER FOUR SUMS OF SQUARES FORMULAE WITH INTEGER COEFFICIENTS 1 2 . Intercalate matrices and the H o p f - S t i e f e l c o n d i t i o n . In th i s chapter, we study sums of squares formulae with integer c o e f f i c i e n t s . As noted in Section B 3 , an [ r , s , n ] formula with integer c o e f f i c i e n t s i s equivalent to an i n t e r c a l a t e matrix M = (c. •) of type ( r , s , n ) , signed by a ( 1 , - 1 ) - m a t r i x A = (a. .) to s a t i s f y the norm condi t ion , i . e . , ( 1 2 . 1 ) a. - a - , - , a . •, a . , . = - 1 1 , 3 i ' ,]* \,y i ' ,D whenever c, . = c . , . , , i / i ' , j ^ j ' • In fac t , we can assume, by mul t ip ly ing cer ta in rows and columns by - 1 i f necessary, that the entr ies on the f i r s t row and the f i r s t column are a l l + 1 . We c a l l such a ( 1 , - 1 ) -matrix a standard signing matr ix . There i s an obvious notion of equivalence of i n t e r c a l a t e matrices of type ( r , s , n ) : two such matrices are equivalent i f one can be brought to the other by permutations of rows and columns, and r e l a b e l l i n g of c o l o r s . There i s also an obvious notion of tensor product of i n t e r -ca late matr ices. Let C = (c! .) and C" = (cV •) be in te rca la te matrices of types ( r ^ s ^ n ^ and ( r 2 , s 2 , n 2 ) r e s p e c t i v e l y . There i s an in te rca la te matrix CSC" = (c^ j) of type (r 1 r 2 , s 1 s 2 , n 1 n 2) , where 76 c • i ' ,1 i f i = i ' r~ + i 0 < i " < r 2' and j = j ' s 2 + j " , 0 < j " < s 2 . The c l a s s i f i c a t i o n of in te rca la te matrices per se i s very complicated even in low dimensions, say r = s = 10. We s h a l l , however, make use of the topolog ica l resu l ts in the previous chapters to bypass most of the combinatoric complexi t ies in analyzing those in te rca la te matrices tabulat ing in teg ra l sums of squares formulae. In the meantime, we formulate a useful lemma on the structure of in te rca la te square matrices admitting two "ubiquitous" c o l o r s , each of which appears in every row and every column of the matr ix . * Let M be an rxs matr ix . For given subsets I CZ { l , . . . r } and J CZ {1 , • . . f s} , we denote by M(I;J) the submatrix of M cons is t ing of rows and columns whose indices are in I and J respect i ve ly . Lemma 12.1 Let M be an in te rca la te matrix of type ( r , r , n ) with two colors each appearing in every row and every column. Then, (a) r and n are both even, say r = 2 r ' , n = 2n'; (b) M i s equivalent to a tensor product M1 fi M 2 , where M1 and M 2 are in te rca la te matrices of types ( r ' , r ' , n ' ) and (2,2,2) respect i v e l y . Proof. Let c 1 and c 2 be two such "ubiquitous" c o l o r s . Step 1. We begin by showing that the colors c 1 and c 2 induce pa i r ings of the rows and the columns of M. More p r e c i s e l y , for each i = 1,..., r, there ex is t unique i ' , a n d j , j ' such that the submatrix M ( i , i ' ; j , j ' ) i s an in te rca la te matrix of type (2,2,2) with co lors c . and c 0 . 2 77 For given i , there ex is t unique j , j ' such that c- •= c. and c. .,= c 0 . Let i ' be the unique integer for which c . , •= c~. Since c. ., = c - , . , we have, by d e f i n i t i o n , c•, •,= c- • = c . . Thus, M ( i , i ' ; j , j ' ) i s an in terca late matrix of type (2,2,2) with co lors c 1 and c 2 « It fol lows that r i s even, and we write r = 2 r ' . Step 2. Since the in te rca la te property of M i s not destroyed by permutations of rows and columns, we can assume, by e f fec t ing such permutations i f necessary, cifi= c]f 1 < i < 2r' ; ( 1 2 ' 2 ) c = c . . = c 1 < i < r' c 2 i - 1 , 2 i c 2 i , 2 i - 1 c 2 ' ~ * We remark that i f the diagonal entr ies of a square in terca late matrix are the same c o l o r , then the matrix i s symmetric. Step 3. We show that i f c 1 and c 2 are arranged as in (12.2) , then the matrix M i s par t i t i oned into 2x2 in te rca la te sub-matrices More p r e c i s e l y , for any i , j = 1 , . . . , r ' , the submatrix M ( 2 i - 1 , 2 i ; 2 j - 1 , 2 j ) i s in te rca la te of type ( 2 , 2 , 2 ) . C l e a r l y , we need only consider the case i / j . Since c 2 i - i 2 i = c 2 j 2 j - 1 = c 2 ' we have, by (12.1) , ^2i-] 2 j -1 = C 2 j 2i = c 2 i 2j* F r o m t h i s , t h e c laim fol lows e a s i l y . Step 4. Further , we note that the colors c 1 and induce a pa i r ing of the colors of M as w e l l . Consequently,the number n i s even, say, n= 2 n ' . We formulate th i s more p r e c i s e l y : i f the submatrices M ( 2 i - 1 , 2 i ; 2 j - 1 , 2 j ) and M ( 2 i ' - 1 , 2 i ' ; 2 j ' - 1 ,2 j ' ) contain a color in common, then they in fact contain the same two c o l o r s . To see t h i s , f i r s t suppose c 2 i - - | 2 j - 1 = C 2 i ' - 1 2 j ' - l 78 = c 2 i , 2 j • T h u s » c 2 i - l , 2 j ' = c 2 i ' 2j-1 b y t h e i n t e r c a l a t e property. This l a t t e r color i s the same as C 2 i<-1 2 j . Thus, f ° H ° w s from the in terca late property again that c 2 ^_ i 2 j = c 2 i » - i 2 j ' ' S i m i l a r l y , the other case c 2 ^_ 1 2 j - 1 = c 2 i ' - 1 2 j ' ^ e a < ^ s t 0  c 2 i - 1 , 2 j = 0 2 i ' - 1,2j'-1* Step 5. The above consideration leads to a "contract ion" of the 2x2 blocks of M into s ingle c o l o r s , thus g iv ing an r ' x r ' matrix of n' co lo rs , which inher i t s the in te rca la te property from M. From th i s the conclusion of the proposit ion fo l lows . • Let us consider an in te rca la te matrix of type (n ,n ,n) . Repeated appl icat ions of Lemma 12.1 shows that in. fact n i s a power of 2, say, n = 2 , and M i s a k - f o l d tensor product M.j ® . . . ® , where each , 1 < i < k, i s an in te rca la te of t y p e(2,2,2). A prototype in terca la te matrix of type (2,2,2) i s the "addit ion table" of the c y c l i c group Z 2 of two elements 0 and 1 , namely, 0 1 1 0_ In f a c t , up to equivalence, th i s i s the only in te rca la te matrix of type ( 2 , 2 , 2 ) . Thus, every in te rca la te matrix of type (n,n,n) is equivalent to the addit ion table M fc of Z 2 © . . . © Z 2(k copies) . We labe l th i s group and c a l l i t the k-th dyadic group. Thus, we have M 1 -79 Theorem 12.2 Every in te rca la te matrix of type (n,n,n) i s equivalent to the addit ion table of the dyadic group D^, where n = 2 . • Let N be the set of nonnegative integers and denote by D the i n f i n i t e dyadic group I lDk* T h e r e * s a n obvious b i j e c t i o n k=1 N > D which induces a group structure on N. The group operation ffl, which we refer to as the dyadic addit ion of integers , can a lso be character ized in terms of dyadic sets (see Section B5) as fo l lows . For any m, n £ N, Km ffl n) = Km) A I(n) = ((Km ) - I ( n ) ) U ( (I (n )-I (m)) . For any integers a ^ b, denote by [a,b] = {k £• N: a < k < b] . The fo l lowing lemma provides a succinct connection between dyadic addit ion and the H o p f - S t i e f e l cond i t ion . Lemma 12.3 The t r i p l e ( r , s ,n ) s a t i s f i e s the H o p f - S t i e f e l condit ion i f and only i f n ^ [ 0 , r - 1 ] ffl [ 0 , s - 1 ] . Proof. ( = 7 * ) . Suppose n = h ffl k, 0 < h < r, 0 < k < s . We can f ind k' such that n = hfflk' = h+k'. E x p l i c i t l y , l e t k' = 2 I ( k ) ~ I ( h ) Then k' i s a p a r t i a l sum of n and i s odd. Furthermore, n-k' = h < r impl ies that n-r < k' < s . ( < r = ) If the t r i p l e ( r , s ,n ) does not s a t i s f y the H o p f - S t i e f e l cond i t ion , then there ex i s t s k such that n - r < k < s , and £^j i s odd. This means that I(k) C I (n ) , and n = (n-k)ffl k € [ 0 , r - l ] + [ 0 , S - 1 ] . • 80 Theorem 12.4 The addi t ion table of [ 0 , r - l ] ffl [ 0 , s - l ] i s an in te rca la te matrix of type ( r , s , r o s ) . Proof. This fol lows from [ 0 , r - l ] ffl [ 0 , S - 1 ] = [ O , r o s - 1 ] , which i s an easy consequence of Lemma 12.3. • Remark: Yuzvinsky [1981] conjectured that an in te rca la te matrix of type ( r , s ,n ) ex is ts i f and only i f the t r i p l e ( r , s ,n ) s a t i s f i e s the H o p f - S t i e f e l cond i t ion . Theorem 12.4 s e t t l e s the suff icency par t . The necessity part i s s t i l l unresolved. In t h i s context, we note that not every in terca late matrix i s equivalent to the dyadic addi t ion table of two sets of integers . Here i s an example. 1 2 3 4 2 1 4 3 5 6 7 8 . 6 5 9 10 As noted in Section A, there were many p r i o r i t y disputes and debates on the proof of the nonexistence of a [16,16,16] formula with integer c o e f f i c i e n t s . Cayley [1881b] was the f i r s t to give a correct proof of t h i s r e s u l t ; a l l previous proofs depended on the u n j u s t i f i e d assumption that such a formula necessar i l y "contains" an [8,8,8] formula. This f a c t , though co r rec t , i s not a p r i o r i obvious. It fo l lows , as we now see, from Theorem 12.2 by analyzing the structure of an in te rca la te matrix of type (16,16,16) . Theorem 12.5 (Graves, Young, Kirkman, Roberts, Cayley [1881b]) There i s no sums of squares of type [16,16,16] with integer c o e f f i c i e n t s . 81 P r o o f . I t s u f f i c e s t o show t h a t the i n t e r c a l a t e m a t r i c cannot be s i g n e d by any ( 1 , - 1 ) - m a t r i x t o s a t i s f y the norm c o n d i t i o n . Step 1. The p r i n c i p a l 8x8 s u b m a t r i x can, of c o u r s e , be s i g n e d by a s t a n d a r d s i g n i n g m a t r i x A_ = (a. .) t o s a t i s f y the norm c o n d i t i o n . S i n c e M, i s symmetric, a. • = -1, i / 1. Suppose we put a 2 , 3 " a l ; a 2 , 5 " a 2 ; a 2 , 8 " a 3 ; a 3 , 5 ~ a 4 ' Then the r e m a i n i n g e n t r i e s of A a r e a l l d e t e r m i n e d by the norm c o n d i t i o n ( 1 2 . 1 ) , and A^ i s the m a t r i x 1 1 1 1 1 1 1 - 1 a 1 " a1 a 2 " a 2 _ a 3 a 3 -a 1 -1 a 1 a 4 a 5 _ a 4 _ a 5 a. _ a 1 -1 a 6 " a 7 a 7 " a 6 _ a 2 _ a 4 ' a 6 - 1 a 2 a 4 a 6 a 2 _ a 5 a 7 " a2 -1 ~ a7 a 5 a 3 a 4 _ a 7 _ a 4 a 7 -1 _ a 3 " a 3 a 5 a 6 _ a 6 " ' a5 a 3 -1 3 a4' a 6 = a l a 3 a 4 and a 7 = a,a 2 a4* Step 2. In f a c t , i t i s p o s s i b l e t o augment A 3 t o a 10x10 m a t r i x A = (a. •) s i g n i n g the p r i n c i p a l 10x10 su b m a t r i x of M g i v e a [10,10,16] form u l a w i t h i n t e g e r c o e f f i c i e n t s . Suppose a 1 9 = a 1 I0 = a 9 l = 3 1 0 1 = 1 s o t h a t A s t a n d a r d . L e t a 2 9 = a 8 " T n e n 82 a 2 , l 0 " a 9 , 2 " a 8 ; a 1 0 , 2 ~ a 8 ' The ent r ies g and a^ i n correspond to the "co lors" 10 and 11 respect i ve ly . These appear now for the f i r s t time; and we can endow them with +1 ; i . e . , a^ g = a 3 10 = 1• F r o m t h i s , a 9 , 3 = a l 0 , 3 = " 1 ; a 4 , 9 = a l 0 , 4 = a 1 a 8 ; a 4 , l 0 = a 9 , 4 = " a l a 8 * S i m i l a r l y , we can assume a^ g = a^ 10 = a 7 and determine the matrix A completely as 9 ~ a 7 , 1 0 = 1 , 1 1 1 1 1 1 1 1 1 1 1 -1 a 1 ~ a 1 a 2 _ a 2 " a 3 a 3 a 8 _ a 8 1 - a , -1 3 1 a 4 a 5 " a 4 _ a 5 1 1 1 a , ~ a l -1 a 6 " a 7 a 7 " a 6 a 9 _ a 9 1 - a 2 _ a 4 _ a 6 -1 a 2 a 4 a 6 1 1 ( 1 2 . 3 ) ' 1 a 2 " a 5 a r " a 2 -1 " a 7 a 5 a i o _ a i o 1 a 3 a 4 " a 7 _ a 4 a 7 -1 ~ a 3 1 1 1 - a 3 a 5 a 6 " a 6 _ a 5 a 3 -1 " a l 1 3 1 1 1 " a 8 -1 _ a 9 -1 _ a i o -1 a l 1 -1 a 8 1 a 8 - 1 a 9 -1 9 1 0 -1 " a 1 1 " a 8 -1 where ag = a ^ g , a 10 = a 2 a 8 and a 1 1 = a 3 a 8 * Step 3 . Now, we show that t h i s s igning matrix A cannot be augmented to a (standard) 10x11 matrix signing the p r i n c i p a l 10x11 submatrix of to s a t i s f y the norm cond i t ion . Let A' = (a- •) be such an augmented matrix with a. = 1. Then, 1 , 3 I , i i i t fol lows from the norm condit ion that 2 ,11 a 3 , 1 1 _ 1 ; a 4 , 1 1~ a 9 ; a 9 , 1 1 _ 1 ; a 1 0 , 1 1 _ a 9 ' 83 C o n s i d e r now the e n t r y 1 1 . The c o r r e s p o n d i n g c o l o r i n M^is 14, which a l s o appears i n p o s i t i o n s (9,7) and ( 1 0 , 8 ) . C o n s i d e r the 2x2 s u b m a t r i x M 4(5,9;7,11) and the c o r r e s p o n d i n g s u b m a t r i x of A 4, we have (12.4) a 5 f l l = a 4 . On the o t h e r hand, the s u b m a t r i x M 4(5,10;8,11) y i e l d s a 5 , 1 1 a 6 ( _ a 1 1 ) ( " a 9 ) = ~ 1" From t h i s , i t f o l l o w s t h a t a5,11 = " a 6 a 9 a 1 1 = _ ( a i a 3 a 4 ) ( a 1 a 8 ) ( a 3 a 8 ) = ~ a 4 ' T h i s i s i n c o m p a t i b l e w i t h ( 1 2 . 4 ) , and we o b t a i n the d e s i r e d i n c o n s i s t e n c y . • 13. D e t e r m i n a t i o n of r * z s f o r r < 9. As an a p p l i c a t i o n of Theorem 12.4, we g i v e a s i m p l e d e t e r m i n a t i o n of r * z s r o r r < 9. The a l g o r i t h m i s w e l l known i n the f o l k l o r e (see a l s o S h a p i r o [ 1 9 8 4 ] ) . However, we p r e s e n t a s i m p l e p r o o f u s i n g C a y l e y - D i c k s o n a l g e b r a s ( S c h a f e r [1954], Adem [ 1 9 7 5 ] ) . For each n o n n e g a t i v e i n t e g e r k, the C a y l e y - D i c k s o n a l g e b r a ( X ^ i s a 2 k d i m e n s i o n a l a l g e b r a w i t h an a d d i t i v e b a s i s k . . . . e^, 0 ^ l ^ 2 - 1 , and m u l t i p l i c a t i o n g i v e n by e . e - = a - - e - m -where a- • i s e i t h e r +1 or - 1 . 84 Thus, we can think of the m u l t i p l i c a t i o n table of the algebra ^ as the addi t ion table of the dyadic group signed by a (1 , -1 ) -mat r i x A, = (a. . ) , where K 1 i ] A Q = (1 ) ; 1 -1 A 2 " 1 1 1 1 1 - 1 1 - 1 1 - 1 - 1 1 1 1 - 1 - 1 and the others can be determined induct ive ly as fo l low . k k Suppose we have the 2 x2 matrix A^. Then A^ + 1 i s the matrix B 1 1 B 12 B 21 B 22 where B,,= A, and each of B , , B~,, and B0^ i s -A, with the 1 1 k 12' 21' 22 k fo l lowing modi f icat ions : ( i ) the ent r ies on the f i r s t row and the f i r s t columns of B 1 2 are a l l +1, and the other diagonal ent r ies are a l l - 1 ; ( i i ) the ent r ies on the p r i n c i p a l diagonal and the f i r s t column of B 2 1 are a l l +1, and the other ent r ies on the f i r s t row are - 1; ( i i i ) the ent r ies on the p r i n c i p a l diagonal and the f i r s t column of B 2 2 are a l l - 1 , and the other ent r ies on the f i r s t row are +1. For k > 4, the (1 , -1 ) -mat r i x A^, of course, does not sign the in te rca la te matrix to s a t i s f y the norm condit ion (Theorem 12.5) . However, we prove 85 Proposit ion 13.1 The submatrix of cons is t ing of the f i r s t 9 rows can always be signed by the corresponding submatrix of A^ to give a normed k k b i l i n e a r map of type [9,2 ,2 ]. Proof. We say that an element x = aQeo + a l e 1 + ••• + a n _ i e n _ i ' n = 2 - 1 , i s purely imaginary i f aQ = 0. A l so , for any three elements x, y, z £ d k , denote by {x,y,z} the associator (xy)z -x (yz) . Adem [1975] has shown that i f X 1 , X 2 , Y1 and Y 2 are subspaces of Q, k + 1 such that ( i ) the m u l t i p l i c a t i o n of Q. r e s t r i c t e d to each X. x Y . , K 1 J i = 1,2, i s a normed b i l i n e a r map; ( i i ) each x ^ y ^ x ^ y ^ , x i € X i , £' Y i . i = 1,2, is purely imaginary, then, the m u l t i p l i c a t i o n of (^) c + 1 r e s t r i c t e d to (X1 xX 2 ) x (Y1 xY2 ) i s also a normed b i l i n e a r map. For k = 4, le t X.) = Y1 = Y 2 be the Cayley-Diekson algebra Q^2' the Cayley algebra with m u l t i p l i c a t i o n table (A2.1), and X 1 ~ & Q ' t n e a l 9 e h r a of rea l numbers. Since x 2 i s a s c a l a r , we have { y 1 , x 2 , y 2 ) = 0, and the condit ions ( i ) and ( i i ) above are s a t i s f i e d . Thus, the r e s t r i c t i o n of the m u l t i p l i c a t i o n of (X^ to X ' xY ' , X' = the subspace spanned by e^, eg, and Y' = , i s a normed b i l i n e a r map. This proves the proposit ion for k = 4. For k > 4, we proceed by induct ion. Let X1 = the subspace of (Jik spanned by e Q , . . . , eg, X 2 = 0, Y1 = Y 2 = Q k « Since X 2 = 0, condit ion ( i i ) i s t r i v i a l l y s a t i s f i e d , whereas condit ion ( i ) fol lows by induction hypothesis. Thus, we conclude that the the r e s t r i c t i o n of the m u l t i p l i c a t i o n of to X^  = X^O = the subspace of spanned by e^, eg, and = Y 1 xY 2 = CL^+i 1 S a lso a normed b i l i n e a r map. • For k > 4, the Hurwitz-Radon number p(2 ) exceeds 9. However, we can only r e s t r i c t the m u l t i p l i c a t i o n table of the Cayley-Dickson algebra Q^y to a maximum of nine rows to give an in tegra l sums of squares formula. We omit the d e t a i l s . In the next chapter, we s h a l l show that for the in terca la te matrix , i t i s possible to choose p(2 ) rows for which the corresponding submatrix can be signed by an appropriate ( 1 , - 1 ) -matrix to y i e l d an in tegra l sums of squares formula of type [ p ( 2 k ) , 2 k , 2 k ] . Theorem 13.2 For r < 9, r* zs = r*s = r#s = ros. Proof. By Theorem B4.2, i t i s enough to exh ib i t an integra l sums of squares formula of type [ r , s , r o s ] . Let k be such that r ,s ^ 2 . By Theorem 12.4 and Proposit ion 13 .1 , the r e s t r i c t i o n of to i t s f i r s t r rows and f i r s t s columns, signed by the corresponding submatrix of A^, provides one such example." 87 s 14. Nonexistence of [16,16,28] formulae with integer coef f ic ients, In Section 6, we have shown, for a normed b i l i n e a r map with a given tabu la t ion , how to record the normed b i l i n e a r map hidden at each point in the image (Proposit ion 6 . 3 ) . If the normed b i l i n e a r map ar ises from a sums of squares formula with integer c o e f f i c i e n t s , the normed b i l i n e a r maps hidden at the " l a t t i c e points" can be read off immediately. We make t h i s precise in the fo l lowing propos i t ion . Let f : R r x R s > R n be a normed b i l i n e a r map tabulated by a signed in terca late matrix of type ( r , s , n ) , with respect to orthonormal bases E = ( e ^ . . . , e r ) of R r , ( e ^ , . . . , e ) of R and ^~C= (c 1 , . . . , c ) of R n respect i ve l y . Let c = c^ for some h < n. We can assume, by permuting the rows and columns of the tabulat ion matrix i f necessary, that the "color" c appears in f i r s t k rows and the f i r s t k columns and nowhere e l s e . We s h a l l say that the color c has frequency k. Proposi t ion 14.1 Suppose the tabulat ion matrix of f i s par t i t ioned into M11 M 12 M21 M 22 and a color c of frequency k appears as the diagonal entr ies of the p r i n c i p a l kxk submatrix M 1 1 . Then the normed b i l i n e a r map hidden at (0,c) i s of type [ k , r + s - k , n ] , and i s tabulated by <Mn M l 2 M ^ ) . 88 Proof. The numerical matrix M = M f (c) = (<c,f(e^,ej)>) i s a ( 0 , 1 1 ) - m a t r i x in which every row (respect ively column) has at most one nonzero entry . Up to permutation equivalence, MMt i s the matrix The resu l t now fol lows f rom .Corol lary, , 6.2 and Proposit ion 6 .3 . • We now prove the resul t promised at the end of Chapter 3: there i s no [16,16,28] formula i f only integer c o e f f i c i e n t s are permitted. We f i r s t prove, by a simple combinatoric argument, the easier case that no in tegra l [16,16,27] formula can e x i s t . For the harder case of [16,16,28] , we s h a l l have to invoke the the deep theorems in algebraic topology reco l lected in Section 8. Proposi t ion 14.2 There i s no in tegra l [16,16,27] formula. Consequently, 16* z16 > 28. Proof. This i s equivalent to proving there i s no in terca la te matrix of type (16,16,n) , n < 27, that can be signed to s a t i s f y the norm condit ion (12.1) . By Theorem 11.2, we know that n > 25. Suppose there i s one such in te rca la te matrix M of type (16,16, 25+6), where 6 = 0 , 1 , 2 . Let c be a color with frequency k. By Proposi t ion 14.1 above, the t r i p l e (k, 32-k,25+6) s a t i s f i e s the H o p f - S t i e f e l cond i t ion . The only possib le values of k are 8 and 16. Thus, each of the colors of the in te rca la te matrix appears e i ther 8 or 16 times in the matr ix . Suppose there are a colors with frequency 8 and b colors with frequency 16. Then, 89 a + b = 25 + 6 8a + 16b = 16x16 = 256. It fol lows that a = 18+26 and b = 7 -6 . Since 5 < 2, there are at least 5 ubiguitous colors (of frequency 16). By Lemma 12.1, M i s a tensor product M1 ® M 2 , where M1 and M 2 are respect ive ly in te rca la te matrices of types (4,4,m) and ( 4 , 4 , 4 ) . It i s easy to ve r i f y that each in terca la te 4x4 matrix with at least one ubiguitous color i s equivalent to one of the fol lowing 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 1 2 3 4 2 1 5 6 3 5 1 7 4 6 7 1 It fol lows that m = 4 or 7. However, m = 7 would lead to 4m = 28 > 27 colors in M. On the other hand, i f m = 4, then M i s an in terca la te matrix of type (16,16,16) ; and i t i s wel l known that t h i s cannot be signed to y i e l d a [16,16,16] formula (Theorem 12.5) . Thus, the said in te rca la te matrix does not e x i s t . • Theorem 14.3 There i s no in tegra l [16,16,28] formula. Consequently 16*z16 ^ 29, Proof. We assume, that there i s one such formula given by a signed in terca la te matrix M of type (16,16,28) , and derive a contradic t ion in the fo l lowing steps. Step 1. Every hidden normed b i l i n e a r map i s of type [ k , 3 2 - k , 2 8 ] , and the only possible values of k are 4 ,8 ,12 ,16 , by Lemma 9 .2 . Step 2. We show that no [16,16,28] formula, i f i t ever e x i s t s , can represent an odd mul t ip le of ^ g 7 ^ . According to Corol lary 8 .5 , such a formula represents an element of Ker( j* : T T 1 4 ( S ) > ^ 1 4 ^ * 1 ^ * T ^ ^ s 9 r o u P i s Z 4 according to the tables of Milgram-Strutt -Zvengrowski [1977], and has 2-exponent equal to 2, which i s less than that of the image of J in the stable 3-stem (Theorem 8 .2 ) . By Theorem 8 . 6 , t h i s cannot represent an odd mul t ip le of v. Step 3. Consequently, the value k = 4 i s excluded; otherwise, the [16,16,28] formula represents an odd mult ip le of v by Lemma 7.4. The only possible values of k are 8 ,12,16. Step 4. It fol lows that the Hopf construct ion of a [16,16,28] formula, i f i t e x i s t s , cannot be s u r j e c t i v e . Hence, such a formula necessar i l y represents zero homotopy c l a s s . Step 5. We further exclude the p o s s i b i l i t y k = 12, by noting that every (hidden) normed b i l i n e a r map of type [12,20,28] i s necessar i l y su r jec t i ve . This fol lows from 12#20 = 28, to be j u s t i f i e d in the next step. Granted t h i s , every hidden normed b i l i n e a r map of type [12,20,28] would represent a n o n t r i v i a l homotopy c lass by Sard's theorem, and Lemma 7.4 again. Since the hidden normed b i l i n e a r map represents the same homotopy c lass as the normed map does, t h i s contradicts Step 4. Step 6. We j u s t i f y the c laim 12#20 = 28 made in Step 5. According to Milgram [1967], there i s a nonsingular b i l i n e a r map R 1 2 X R 2 0 > R 2 8 , showing that 12#20 < 28. On the other 12 20 27 hand, a nonsingular b i l i n e a r map of type R x R > R cannot e x i s t ; otherwise there would be at least 20 sect ions of the vector bundle 2 7 ^ over the rea l pro ject ive space R P 1 1 . However, by Theorem 1.1 of Lam [1972], 27?;^ admits a maximum of 16 s e c t i o n s . It fol lows that 12#20 > 27 and 12#20 = 28. 91 Step 7. Only two possible values of k remain, namely, 8 and 16. Note that we have not imposed the "integer c o e f f i c i e n t s " r e s t r i c t i o n up to th i s po int . Step 8. Now, we make use of the assumption that t h i s [16,16,28] formula has integer c o e f f i c i e n t s , and consider the signed i n t e r c a l a t e matrix M tabulat ing i t . There are 28 co lors each appearing e i ther 8 or 16 times in M. By a counting procedure s imi lar , to that in Proposit ion 14.2, we conclude that there are 4 colors each appearing in every row and every column. Repeated appl icat ions of Lemma 12.1 show that M i s a tensor product of the form M1 ® M 2 , where M1 and M 2 are in te rca la te matrices of types (4,4,7) and (4,4,4) respect i ve l y , and M1 has one ubiguitous c o l o r . Step 9. Every in te rca la te matrix of type (4,4,7) with one ubiguitous color contains an in te rca la te submatrix of type ( 3 , 3 , 4 ) . It fol lows that M contains an in te rca la te submatrix M' of type (12,12,16) . Since 12*12 > 16 (cf . Theorem 11.1) , t h i s in te rca la te matrix M' cannot be signed to y i e l d a [12,12,16] formula. Accordingly, the in te rca la te matrix M of type (16,16,28) cannot be signed to y i e l d a sums of squares formula with integer c o e f f i c i e n t s . • 92 15. Structure of [10,10,16] formulae with integer coef f ic ients . In t h i s sect ion , we show that there i s e s s e n t i a l l y only one type of [10,10,16] formula with integer c o e f f i c i e n t s . We begin by studying the homotopy c lass represented by such a sums of squares formula. Proposit ion 15.1 Every sums of squares formula of type [10,10,16] , not necessari ly with integer c o e f f i c i e n t s , represents a n o n t r i v i a l homotopy c l a s s . Proof. This fol lows e a s i l y from the two steps below. Step 1. The generalized J homomorphism J : ^ g ( V 1 g 1Q ) > ^ ^ ( S 1 6 ) i s i n j e c t i v e . This homomorphism maps Z 1 2 into Z 2 4 » Lam [1977a] has exhibi ted a [10,10,16] formula (with integer c o e f f i c i e n t s ) representing ±2v in the stable 3-stem. Thus, the image of J i s a group of order 12. Consequently, the kernel i s zero, and the homomorphism is i n j e c t i v e . Step 2. Let f be a normed b i l i n e a r map of type [10,10,16] with 9 adjoint h: S > 1 Q . We c laim that h must be n o n t r i v i a l . If h i s t r i v i a l , then i t admits an extension to the upper hemi-sphere which can be further extended to an equivariant map S 1 ^ > V^g with adjoint a skew l inear map S1^xR1 <^ > R 1^, g iv ing 10 sections of the bundle 1 6 £ i n o v e r the rea l project ive space R P 1 ( \ This i s impossible according to the tables of Lam [1972]. • Proposit ion 15.2 The hidden normed b i l i n e a r maps of a [10,10,16] formula are of types [ k , 2 0 - k , 1 6 ] , where k = 4 , 8 , 1 0 . 93 P r o o f . We e x c l u d e the o t h e r v a l u e s of k i n the range 4 ^ k < 10 i n two s t e p s . Step 1. The r e l e v a n t homotopy groups of S t i e f e l m a n i f o l d s , from the t a b l e s i n P a e c h t e r [ 1 9 5 6 ] , a re as f o l l o w . 8 10 , r k - l ( V 1 6 , 2 0 - k ) Z2 Z ^  2®Z 4©Z " l 9 - k ( V 1 6 - k , K } Z4® Z48 Z 2 ® Z 2 Z12 Z 2 Z24® Z8 1 2 12 By Theorem 8.1, the v a l u e s k = 5, 6 a r e e l i m i n a t e d , Step 2. We l i s t the groups K 3 k _ 5 = K e r ( j * : ffk_2(Sk 5 ) V 2 l P k - 5 , ) a n d K 3 , 1 5 - k = K e r ( ^ * : w l 8 - k ( s l 5 " k ) OO * " i B - k ^ l S - k * * from the t a b l e s of M i l g r a m - S t r u t t - Z v e n g r o w s k i [ 1 9 7 7 ] . K K 3,k-5 3,15-k Z 2 Z4 0 '8 0 z. 1 0 By C o r o l l a r y 8.5, the v a l u e s k = 7, 9 are i n a d m i s s i b l e . • Theorem 15.3 Every i n t e g r a l [10,10,16] f o r m u l a i s o b t a i n e d by s i g n i n g the i n t e r c a l a t e m a t r i x (15.1) 1 2 3 4 5 6 7 8 9 10 2 1 4 3 6 5 8 7 10 9 3 4 1 2 7 8 5 6 1 1 12 4 3 2 1 8 7 6 5 1 2 1 1 5 6 7 8 1 2 3 4 1 3 14 6 5 8 7 2 1 4 3 1 4 13 7 8 5 6 3 4 1 2 15 16 8 7 6 5 4 3 2 1 16 15 9 10 1 1 1 2 13 14 15 16 1 2 10 9 1 2 1 1 1 4 1 3 16 1 5 2 1 to s a t i s f y the norm c o n d i t i o n . 94 Proof. Let M be an in te rca la te matrix of type (10,10,16) that can be signed to s a t i s f y the norm cond i t ion . By Proposit ions 14.1 and 15.2, each color of M has frequency 4, 8, or 10. By a procedure s imi la r to that in Proposi t ion 14.2, we enumerate the various possible d i s t r i b u t i o n s of the frequencies of colors in M. Table 15.4 D i s t r i b u t i o n of co lors in an in te rca la te matrix of  type (10,10,16) with prescr ibed frequencies Frequency 4 8 1 0 T ~ T T H 7 9 0 Number ; . . v „ c ~ „ f ( i i ) 8 6 2 ? r ( i i i ) 9 3 4 c o l o r s ( iv) 10 0 6 The las t two cases can be eas i l y el iminated as fo l lows . We know, from Theorem 4 . 2 , that i s a great spheres of dimension no greater than p(l0) = 2. Thus, there cannot be more than two l a t t i c e points appearing 10 times in the tabulat ing matr ix , and c < 2. (A l te rna t i ve l y , we conclude that c ^ 2 by observing that i t i s impossible to arrange 3 or more colors in each row and each column of an in te rca la te 1 Ox 1 0 matr ix ) . We postpone the lengthy e l iminat ion of case ( i ) to Proposit ion 15.6. Granted t h i s , we know, by Lemma 12.1, that M i s a tensor product M 1 ® M 2, where M1 and M 2 are respect ive ly in terca late matrices of types (5,5,8) and (2,2,2) respect i ve ly . Furthermore, the frequency d i s t r i b u t i o n of the co lors in M1 i s as fo l lows. Frequency Number of co lors It i s easy to check that every such in te rca la te matrix is equivalent to 95 2 3 4 4 3 2 5 6 7 8 2 3 4 5 4 3 6 2 7 8 Thus, the in te rca la te matrix M i s equivalent to that in the statement of the propos i t ion . The p r i n c i p a l 8x8 submatrix i s an in te rca la te matrix of type ( 8 , 8 , 8 ) . By the ( f i r s t two steps of the) proof of Theorem 12.5, t h i s in te rca la te matrix M of type (10,10,16) can be signed by the matrix (12.3) to s a t i s f y the norm cond i t ion . • Theorem 15.5 Every in tegra l [10,10,16] formula represents ±2v. Proof. We consider the in te rca la te matrix (15.1) signed by (12.3) to s a t i s f y the norm c o n d i t i o n . By Proposi t ion 14.1, the color 9, of frequency 4, corresponds to a regular value. We determine the hidden normed b i l i n e a r map by Proposit ion 6 . 3 . Re labe l l ing the colors in (15.1) according to the scheme 1 2 3 4 5 6 7 8 9 10 11 12 13 1 4 1 5 16 2 3 4 5 8 9 12 13 0 1 6 7 10 11 14 15 we tabulate t h i s hidden normed b i l i n e a r map as the dyadic addi t ion table of [0,3] ffl [0,15] signed by the (1 , -1 ) -matr ix (A Afc A A)P where A i s the 4x4 matrix A = -1 -1 -1 -e e 1. e -e e 96 and P i s the 16x16 diagonal matrix with diagonal (1 1 1 1 1 - 1 - 1 -ae 1 1 1 e 1 ce 1 e ) . Now, i t i s straightforward to check that an in te rca la te matrix of type ( 4 , 4 , 4 ) , signed by the matrix A, tabulates a normed b i l i n e a r map representing ev in the stable 3-stem. On the other hand, s igning by Afc y ie lds -ev. Thus, the [10,10,16] formula in question represents the homotopy c lass ( -1)( -1)( -ae)(e)(ce)(e)(e-e+e+e)v = -2acev. • Now we carry out the e l iminat ion of in te rca la te matrices of type (10,10,16) with frequency d i s t r i b u t i o n given by ( i ) in Table 15.4, thereby completing the proof of Theorem 15.3. Proposit ion 15.6 There is no in te rca la te matrix of type (10,10,16) with nine colors of frequency 8 and seven colors of frequency 4. Proof. Assume that there i s one such matrix M with colors 1 , . . , 16. We s h a l l derive a contradict ion in the fol lowing steps, making in Step 4, a c r u c i a l observation on the " l inkage" of the frequency 8 c o l o r s , and showing, in the subsequent steps, that these colors cannot be " l i n k e d " as expected. Step 1. Consider two c o l o r s , say 1 and 2, of frequency 8. These two colors appear in at least 8+8-10 = 6 common rows. Within 6 common rows, they appear in at least 6+6-10 = 2 common' columns. Thus, M has an in te rca la te 2x2 submatrix with colors 1 and 2, which can be assumed to appear in the upper l e f t hand corner of M. 97 Step 2. Each of the colors 1 and 2 appears 6 times in the in terca late 8x8 submatrix in the lower r ight hand corner, which can be par t i t ioned into ~ N n N 12~ N N _ 21 2 2 J ' where is 6x6 with each diagonal pos i t ion occupied by color 1. Color 2 must appear in H at least 6-2-2 = 2 t imes. Thus, contains an in terca la te 2x2 matrix with colors 1 and 2. Step 3. It fo l lows, from above and Lemma 12.1, that for every pair of colors 1 and 2 with frequency 8, the 10x10 matrix M contains an in terca late 4x4 submatrix with colors 1 ,2 ,x , y , for some colors x, y of M. Step 4. Permute the rows and columns of M i f necessary so that the diagonal entr ies of the p r i n c i p a l 8x8 submatrix M' are a l l color 1. C l e a r l y , M' i s symmetric. It a lso fol lows from Step 3 that every color with frequency 8 appears at least 4 times in M'. Furthermore, for any two such co lors a and b, M' contains an in terca la te submatrix of type (4,4,4) with colors a , b , x , y , for some co lors x,y of M. Step 5. We claim that , apart from color 1, at least one color of frequency 8 appears more than four times in t h i s p r i n c i p a l 8x8 submatrix M'. If not, then every color of frequency 8 appears exact ly four times in M'. By permutations of rows and columns i f necessary, we can assume that color 2, which has frequency 8, appears in pos i t ions ( 1 , 2 ) , ( 2 , 1 ) , ( 3 , 4 ) and (4,3) of M'. By Lemma 12.1, the p r i n c i p a l 4x4 submatrix of M' i s in te rca la te with co lo rs , say 1 , 2 , 3 , 4 . 98 4 * * * * 3 2 1 1 1 1 1 None of the colors 2,3,4 appears again in M' . It fo l lows , by the- symmetry of M' , that the co lors in the upper r ight hand corner are a l l d i s t i n c t . (If the colors at ( i ,4+j) and ( i ' , 4 + j ' ) are the same, 1 ^ i , i ' , j , j ' < 4, i ^ i ' , j ^ j ' , then, by symmetry, the colors at (4+j , i ) and ( i ' , 4 + j ' ) are the same, and by the in terca late property, the color at ( i ' , i ) in the upper l e f t hand corner appears again at (4+j,4+j') in the lower r ight hand corner, a c o n t r a d i c t i o n ) . Consequently, the 8x8 matrix M' would have contained at least 4+16 = 20 c o l o r s , contrary to the assumption that M has 16 c o l o r s . This j u s t i f i e s the c laim that at least one color of frequency 8, apart from color 1, appears more than four times in M ' . Step 6. We assume that color 2, which has frequency 8, appears s ix (or more) times in the p r i n c i p a l 8x8 submatrix M' , and show that the other frequency 8 co lors cannot be " l i n k e d " as expected in Step 4. Now, by permutations of the rows and columns i f necessary, we can put color 2 in pos i t ions (1,2 ) , ( 2 , 1 ) , ( 3 , 4 ) , ( 4 , 3 ) , ( 5 , 6 ) and (6,5) of M' , and two other pos i t ions of M, possibly inside M' as w e l l . The p r i n c i p a l 6x6 submatrix of M' i s , by Lemma 12.1, of the form 1 2 3 4 5 6 2 1 4 3 6 5 3 4 1 2 7 8 4 3 2 1 8 7 5 6 7 8 1 2 6 5 8 7 2 1 . 99 Since M contains nine colors of frequency 8, there must be one such c o l o r , say color 9, appearing (exactly) four times in the 8x8 submatrix M' . This color does not appear in the lower r ight corner of M' ; otherwise we cannot f ind four pos i t ions in M' to accomodate i t . Up to equivalence, there are two possib le ways of arranging color 9 in M' , namely, (15.2) 1 2 3 4 5 6 9 * 2 1 4 3 6 5 3 4 1 2 7 8 4 3 2 1 8 7 5 6 7 8 1 2 6 5 8 7 2 1 * 9 or (15.3) 1 2 3 4 5 6 9 * 2 1 4 3 6 5 3 4 1 2 7 8 4 3 2 1 8 7 5 6 7 8 1 2 6 5 8 7 2 1 We s h a l l make use of the c r u c i a l observation in Step 4 to el iminate these two cases. Step 7. We consider (15.2) and determine the colors in the remaining pos i t ions by the in terca la te property. L 1 2 3 4 5 6 9 10 2 1 4 3 6 5 10 9 3 4 1 2 7 8 1 1 12 4 3 2 1 8 7 1 2 1 1 5 6 7 8 1 2 1 3 1 4 6 5 8 7 2 1 1 4 13 9 10 1 1 1 2 1 3 1 4 1 2 10 9 1 2 1 1 14 1 3 2 1 1 00 There i s only one in te rca la te submatrix of type (4,4,4) containing color 9, namely the submatrix formed by rows 1,2, 9,10 and columns 1 ,2 ,9 ,10 . Thus,according to Step 4, each of the colors 3,4 , 5 ,6 ,7 ,8 ,11 ,12 ,13 ,14 has frequency 4 in the matrix M. But then there are more than seven of these co lo rs ! Step 8. S i m i l a r l y , we consider (15.3) by f i r s t determining the colors in the remaining pos i t i ons . 1 2 3 4 5 6 9 1 1 2 1 4 3 6 5 10 1 2 3 4 1 2 7 8 1 1 9 4 3 2 1 8 7 1 2 10 5 6 7 8 1 2 1 3 1 5 6 5 8 7 2 1 1 4 16 9 1 0 1 1 1 2 1 3 1 4 1 3 1 1 12 9 10 1 5 1 6 3 1 In t h i s case, there i s no in te rca la te matrix of type (4,4,4) containing the colors 2 and 9, both of frequency 8, contradict ing the observation in Step 4. This completes the proof of the nonexistence of an in terca late matrix of type (10,10,16) with the nine colors of frequency 8 and seven colors of frequency 4. • 101 CHAPTER FIVE SUMS OF SQUARES FORMULAE NEAR THE HURWITZ-RADON RANGE 16. An e x p l i c i t so lut ion of the Hurwitz-Radon equations. As mentioned in A7, the Hurwitz problem of determining, for a given integer n, the greatest poss ib le integer r, denoted by p(n), for which there ex i s ts an [ r ,n ,n ] formula, was f i r s t solved by Radon [1922] and Hurwitz [1923] h imself , by studying the system of equations A^ = -A . , A? = - I , 1 < i < r-1 , ( 1 6 , 1 ) A.A.= - A . A . ; 1 < i , j < r - 1 , i ^ j , in nxn matrices A ^ . . . , A 1 # This system of equations, known as the Hurwitz-Radon equations, has been much studied in the l i t e r a t u r e (Eckmann [1942], Wong [1961], Geramita-Seberry [1979]). In t h i s sect ion , we write down an e x p l i c i t so lut ion of the equations (16.1) . Let G r _ 1 be the C l i f f o r d group with generators e, a^,..., a r _ 1 and re la t ions 2 2 e = 1; a^ = e, ea^=a^e, 1 < l < r - 1 ; a ja j = 1 < i , j < r - 1 , i / j . According to Eckmann [1942], every [ r ,n ,n] formula i s equivalent to an n-dimensional spin representation of G r _ 1 . This means that there are isometries h A = i . h. h . 0 n 1 r-1 on R n s a t i s f y i n g 1 02 h? = - t n , 1 < i < r-1 ; h j O h i = - h . o h j , 1 < i , j < r - 1 , i / j , where i denotes the ident i t y operator on R n . We s h a l l c a l l th i s a family of r C l i f f o r d operators on R n . k n Let n = 2 (2m+l). Every family of C l i f f o r d operators on R i s of the form ^ l 2m+1® h i : 0 ~ i ~ r _ 1 K where {h^: 0 < i < r-1} 2 k i s a family of r C l i f f o r d operators on R . Thus, i t i s enough n k to construct C l i f f o r d operators on R for n = 2 . Ident i fy R with the Cayley algebra with m u l t i p l i c a t i o n 2 b table given by A2 .1 . For 0 < b < 2, l e t R be the subalgebra generated by eg=1, e ^ . . . , e n - 1 , n = 2°. This can be i d e n t i f i e d with the algebra of rea l numbers, complex numbers, or quaternions for b = 0,1,2 respect i ve ly . h ? b For 0 < b < 3, l e t h. •, 0 < i < 2 - 1 be the operator on R o, 1 induced by l e f t m u l t i p l i c a t i o n by e^. Then, the family = {h, . : 0 ^ i ^ 2 -1} cons is ts of 2 C l i f f o r d operators. From b, l t h i s , p(2 b ) = 2 b , 0 < b < 3. Theorem 16.1 2 Let X , M , and e be operators on R defined by X(e Q) = e 1 , X ( e 1 ) = e Q ; i i (e Q ) = e n , M(e1 ) = - e 1 ; e(e Q) = e 1 , e(e 1 ) = - e Q ; and r = X®tg on R 1 6 = R2®R8 1 03 (a) There i s a f a m i l y H 4 of p(16) = 9 C l i f f o r d o p e r a t o r s on R 1 6: h4,0 = l2*lB = I 1 6 ; h 4 , i = «® h3,i' 1 * 1 5 7; h4,8 = e ® l 8 ; (b) For k = 4a+b, 0 < b < 3, t h e r e i s a h 7k 8a+2 C l i f f o r d o p e r a t o r s on R : f a m i l y of p ( 2 k ) = h k , 0 = l 2 k h k , 8 / + i " ' d * h 4 , i f t T ° ' ' ° " 1 ~ h k , 8 a + j " h b , j ^ a ' 1 s 3 * 1 < i < 8, d ,k-4/-4 where T = T ® T . . . ® T (/ f a c t o r s ) . P r o o f . The o p e r a t o r s X,M, and e s a t i s f y the f o l l o w i n g r e l a t i o n s , as a r e e a s i l y v e r i f i e d . 2 2 2 X = M = T 2 ' E = - t 2 ' e o y = - y o e = X ; Xoe = - e o X = M; Xovu = - M o X = e . Suppose { h Q = i , h 1 n r _ i J i s a f a m i l y of r C l i f f o r d o p e r a t o r s on R n. Then, l2® ln = i 2 n ; M®h i, 1 < i < r - 1 ; e®t n i s a f a m i l y of r+1 C l i f f o r d o p e r a t o r s on R = R ®R . Us i n g 8 the f a m i l y of 8 C l i f f o r d o p e r a t o r s on R , we o b t a i n the f a m i l y H 4 of 9 C l i f f o r d o p e r a t o r s on R 1 6 as i n ( a ) . C o n s i d e r the o p e r a t o r r = h^ ^o...oh^ 1 = X®t R on R 1 ^ . 2 T h i s commutes w i t h e v e r y h., . , 1 < i < 8, and s a t i s f i e s r' o , 1 = I16' 1 04 ' If (h.Q= i n, h 1 , n r_1} i s a family of r C l i f f o r d operators on R n , then l n ® l l 6 = l 16n' l n ® h 4 , i ' 1 * 1 * 8 ; h f T > ^ 3 ^ r-1 i s a family of r+8 C l i f f o r d operators on R l 6 n = R n ®R 1 6 . Let k = 4a+b, 0 < b < 3. Star t ing with the family H f c , we k b obta in , by induct ion, the family H, of p(2 ) = 8a+2 C l i f f o r d 2 k operators on R in (b). • Remark: The [9,16,16] formula given by the family H 4 in (a) i s tabulated by the addit ion table of [0,8] + D with sign matrix (16.2) A = The [p(2 ),2 ,2 ] formula given by the family in (b) can a lso be tabulated in a simple way. We f i r s t formulate a lemma. Lemma 16.2 r 2 a 2 a Suppose we have in tegra l b i l i n e a r maps f : R x R > R s 2 b 2 b g: R x R > R tabulated by the addi t ion tables of X ffl D a and Y ffl with sign matrices A and B respect i ve ly . Then f®g i s tabulated by (X.2b+Y) ffl D a + b with sign matrix ASB. • For each integer /, l e t X z = { 1 6 ' i + — Y 5 ( 1 6 Z - 1): 1 < i < 8}. A l so , for each b < 3, l e t X[ fa = {16 Z i+ -yg (16 Z -1 ) : 1 < i < 2 b - l } . Denote also by E, the 1x2^ matrix whose ent r ies are a l l +1. 105 Proposition 16.3 k k k The [p(2 ),2 ,2 ] formula given by the C l i f f o r d operators in Theorem 16.1 is tabulated by signing the addition table of Y k ffi D R where Y R = {0} [) ( J j ' x . ) \J X ^ . This matrix is signed as follows. (i) The submatrix corresponding to 0 i s signed by E k < ( i i ) The submatrix corresponding to X., 0 ^ i ^ a-1, is signed by E k_4/_4® T® E/' which consists of blocks of TSE^, T being the the 8x16 matrix consisting of the bottom 8 rows of A (cf.(16.2)). ( i i i ) The submatrix corresponding to X , is signed by T, ®E , a . f D D o where T b i s the (2 b-1)x2 b matrix A(2,...,2 b;1,...,2 b). • The C l i f f o r d operators in Theorem 16.1 in fact form an an ascending chain: ^ C H 2 C C H R C so that we can speak of the (direct) l i m i t of t h i s chain as an i n f i n i t e Hurwitz-Radon family H. This i n f i n i t e family can OD be tabulated as the addition table of X B D, X = {0} \J ( M X. ) , i= 1 signed as follows. (i) The f i r s t row corresponding to 0 i s signed by E, the i n f i n i t e sequence each of whose terms i s +1. ( i i ) For each i , the submatrix corresponding to X^ is signed by blocks of T®E i. From th i s table, examples of sums of squares formulae of types [r,s,n], for arb i t r a r y r and s, can be written down by r e s t r i c t i n g to any r rows and s columns. 1 06 17. Determination of p(s+6,s) for 6 ^ 5 . In t h i s sect ion , we consider the problem of determining, for given n and s , the greatest integer r, denoted by p (n ,s ) , for which there ex i s t s a sums of squares formula of type [ r , s , n ] . It i s easy to f ind lower and upper bounds for p (n , s ) . A normed b i l i n e a r map i s necessar i ly nonsingular. It fol lows from the H o p f - S t i e f e l condit ion that p(n,s) ^ h (n , s ) . On the other hand, for each integer m in the in te rva l [ s , n ] , we can interpret an [p(m),m,m] formula of the Hurwitz-Radon type as an [ r , s , n ] formula. These are not, of course, genuine [ r , s , n ] formulae i f s < n. Nevertheless, we derive from t h i s Proposi t ion 17.1 For any integers s < n, p [ s , n ] ~ P ( n ' s > - b (n , s ) , where p r = max{p(m): s ^ m < n}. • L s , n j Suppose p j s n j < 8. Then, there i s no mult ip le of 16 in the i n t e r v a l [ s , n ] . In t h i s case, h(n,s) < 8 as w e l l . By Theorem 12.4 and Proposi t ion 13.1, for r ^ 8, there i s a normed b i l i n e a r map of type [ r , s , n ] i f and only i f the t r i p l e ( r , s ,n ) s a t i s f i e s the H o p f - S t i e f e l cond i t ion . Thus, there i s an fh (n , s ) , s ,n ] formula, and we have Proposi t ion 17.2 If p j s n-j ^ 8, then p(n,s) = h (n , s ) . • In g e n e r a l , d e t e r m i n i n g the v a l u e of p(n,s) f o r p j s n j > 8 i s v e r y d i f f i c u l t . B e r g e r - F r i e d l a n d [1984] s t u d i e d the system of g e n e r a l i z e d Hurwitz-Radon e q u a t i o n s A.A^ = 1 , 1 < i < r ; A.A^ + A .A^ = 0, 1 < i , j < r , i / j i i s' ' 1 J J 1 ' ' J i n sxn m a t r i c e s A 1,..., A r, and de t e r m i n e d p(s+6,s) f o r 5 ^ 3 , 8 = 4, .n odd. The case 6 = 0 i s of c o u r s e the c l a s s i c a l H u r w i t z Radon case c o n s i d e r e d i n the l a s t s e c t i o n . We s h a l l adopt, i n the p r e s e n t s e c t i o n , the v i e w p o i n t of normed b i l i n e a r maps and deter m i n e p(s+6,s) f o r 8 < 4, and a l s o 5 = 5 , s ^  27(mod.32). The c o r r e s p o n d i n g q u e s t i o n f o r n o n s i n g u l a r b i l i n e a r maps has been s t u d i e d i n Lam [1966]. Denote by /3(n,s) the g r e a t e s t r s n i n t e g e r r f o r which a n o n s i n g u l a r b i l i n e a r map R x R > R e x i s t s . Lam [1966] proved Theorem 17.3 For 6 = 1,2, /3(s + S,s) = P [ s s + g]« P r o o f . A n o n s i n g u l a r b i l i n e a r map R rx R s R s +^ y i e l d s s independent s e c t i o n s of the v e c t o r bundle (s+6)£ .j (see S e c t i o n B 4 ) . The complement of the s e s e c t i o n s would be an 6- d i m e n s i o n a l v e c t o r bundle T? such t h a t (17.1) (s+6) ^ r _ 1 = sE © I J r - 1 where E i s the t r i v i a l l i n e bundle over RP I f 6 = 1,2, TJ i s n e c e s s a r i l y a sum of l i n e bundles ( L e v i n e [ 1 9 6 3 ] ) . From the o r d e r of the c a n o n i c a l l i n e bundle (Adams [ 1 9 6 2 ] ) , t h i s f o r c e s r = p j s +$] * n case 8 = 1 , and r = max{3,p r e , r i } i n case 6 = 2 . • 1 08 For 6 = 3,4, the s e t u p i n the pro o f above remains v a l i d . However, the bundle r? may not s p l i t as a sum of l i n e b u n d l e s . In t h e s e c a s e s , we can make use of Adams' r e s u l t s on s t a b l e v e c t o r b u n d l e s of low g e o m e t r i c dimensions (Adams [ 1 9 7 4 ] ) . For g i v e n s, denote by s' the l e a s t m u l t i p l e of 16 not l e s s than s. Assume p^s s + g - j > 8 so t h a t s + gj = p ( s ' ) . We quote Theorem 17.4 (Adams [1974]) L e t x 6 KO(RP p) be the s t a b l e c l a s s of the c a n o n i c a l l i n e bundle over RP P. The o n l y e l e m e n t s i n KO(RP p) t h a t (a) have geometric d i m e n s i o n s ^ 3 are mx, 0 ^ m < 3, i f p > 13; (b) can be r e p r e s e n t e d by S p i n ( 4 ) - b u n d l e s a r e 0,4x; (c) can be r e p r e s e n t e d by S p i n ( 5 ) - b u n d l e s w i t h nonzero S t i e f e l -Whitney c l a s s w 4 a r e ( i ) 4x i f p > 13; ( i i ) 4x, -12x i f 9 < p < 11; ( i i i ) 4x, -12x, -60x i f p = 12. • Remark: A v e c t o r bundle i s a s p i n bundle i f the f i r s t two S t i e f e l - W h i t n e y c l a s s e s w and w- are z e r o ( c f . Husemoller [ 1 9 6 6 ] ) . For c o n v e n i e n c e , we s h a l l s i m p l y denote p ^ g s + § ] by P> W i t h Adams' theorem, we can now have Theorem 17.5 For 6 = 3,4, i f p > 8, 1 0 9 except poss ib ly for ( i ) S E E 6 2 , 6 3 (mod.128) i f 6 = 3 ; ( i i ) S E E 1 4 ( m o d . 1 6 ) , 61 , 6 2 (mod. 1 2 8 ) i f 6 = 4 . Proof. If p < 8, t h i s fol lows from the considerat ion leading to Proposi t ion 1 7 . 2 above. It remains to consider s and 6 with p > 8. These are ( i ) 6 = 3 , s== 1 3 , 1 4 , 1 5 , 1 6 ( m o d . 1 6 ) ; ( i i ) 6 = 4 , S E E 1 2 , 1 3 , 1 4 , 1 5 , 1 6 (mod. 1 6 ) . We only need to prove that there i s no nonsingular b i l i n e a r R 1 + P x R s > R s + ^ . A l l cases being s i m i l a r , we s h a l l only treat the case 6 =4 and s = l 3 ( m o d . 1 6 ) . In t h i s case, s'=s+3 and ( 1 7 . 1 ) becomes ( 1 7 . 2 ) ( s '+1 ) £ p = ( s ' - 3 ) E © TJ, for a 4 -p lane bundle TJ. It fol lows that the stable c lass of 7j i s ( S ' + 1 ) X and that of { = ( T J © E ) ® £ ^ i s ( 4 - s ' ) x . Now, from St iefe l -Whitney c lass c a l c u l a t i o n s , $ i s a S p i n (5) - b u n d l e with nonzero w4 . This contradicts Theorem 1 7 . 4 above i f p ^ 13 or p = 1 0 . For p = 9, ( 1 7 . 2 ) would imply 13 sections of the bundle I7£g, which contradicts Theorem 3 . 1 of Lam [ 1 9 7 2 ] , The remaining case p = 1 2 corresponds to the possible exception s =. 6 1 ( m o d . 1 2 8 ) . • Remark: I t i s not known whether nonsingular b i l i n e a r maps can ac tua l l y ex is t in the undecided cases in Theorem 1 7 . 5 above, namely, of the types 1 1 0 ( 1 7 . 3 ) R 1 3 x R 1 2 8 / + 6 2 R 1 2 8 / + 6 5 Q R E V E N  R 1 3 X R 1 2 B / + 6 1 _ ^ R 1 2 8 / + 6 5 ( 1 7 . 4 ) R 1 3 x R 1 2 8 / + 6 3 _ ^ R 1 2 8 / + 6 6 Q R E V E N  R 1 3 X R 1 2 8 / + 6 3 ^ R 1 2 8 / + 6 7 ( 1 7 . 5 ) R 1 + P x R 1 6 / " 2 > R 1 6 / + 2 , p = P ( 1 6 / ) . However, we s h a l l exp lo i t the propert ies pecul iar to normed b i l i n e a r maps to show that normed b i l i n e a r maps of these types do not e x i s t . Proposi t ion 1 7 . 6 There are.no normed b i l i n e a r maps of types ( 1 7 . 3 , 4 , 5 ) above. Proof. A l l cases being s i m i l a r , we treat ( 1 7 . 5 ) only . Suppose, for a c o n t r a d i c t i o n , that there i s a normed b i l i n e a r map f of type [ 1 + p , 1 6 / - 2 , 1 6 / + 2 ] . Observe that ( 1 + p ) # ( 1 6 / - 2 ) > 1 6 / + 1 . th i s i s because the existence of a nonsingular b i l i n e a r map R 1+p x R 1 6 / - 2 ^ R 1 6 / W O U L D I M P L Y 1 6 / £ P = ( 1 6 / - 2 ) E © T ? for a 2 - p l a n e bundle 77, which, by St iefe l -Whitney c lasses considerat ion, i s necessar i l y 2 E , a cont rad ic t ion . Accordingly , at least one of the hidden normed (or nonsingular) b i l i n e a r maps of f i s of type [ p - 2 , 1 6 / + 1 , 1 6 1 + 2 ] or [ p - 3 , 1 6 1 + 2 , 1 6 / + 2 ] . Since p > 8 , these maps do not ex is t by Proposit ion 1 7 . 2 . • For 6 = 5 , we can proceed along the same l i n e , and show that for p > 8 , normed b i l i n e a r maps of type [ l + p , s , s + 5 ] do not e x i s t . The r e s t r i c t i o n s 2 7 (mod .32) i s not easy to remove. It would be des i rab le to show that nonsingular b i l i n e a r maps of type 11 39/+97 ^°/+1? R x R > R do not e x i s t . But, we have not yet managed to do so. In t h i s regard, the work of D a v i s - G i t l e r --Mahowald [ 1 9 8 1 ] may be relevant . 111 We summarize the nonexistence resu l ts on normed b i l i n e a r maps establ ished in th i s sect ion in the fol lowing theorem. Theorem 17.7 If 0 < 6 < 5, then \ h ( s + 6 , s ) i f P [ B j S + s ] « 8; except possibly for the case 6 = 5, s =27(mod.32). • Remark: Under the condit ions of Theorem 17.6, suppose p < 8 so that p(s+6,s) i s given by h(s+6,s) . 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