Table Of ContentDedicated to Amanda, Becky and Jacob
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xi
Chapter0
HistoricalBackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Part I. Classical Compositions and Quadratic Forms
Chapter1
SpacesofSimilarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Appendix. Compositionalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Chapter2
AmicableSimilarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Chapter3
CliffordAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Chapter4
C-ModulesandtheDecompositionTheorem . . . . . . . . . . . . . . . . . . 73
Appendix. λ-HermitianformsoverC . . . . . . . . . . . . . . . . . . . . . . . . 81
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Chapter5
Small(s,t)-Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106
viii
Chapter6
Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
Chapter7
Unsplittable(σ,τ)-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . .119
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134
Chapter8
TheSpaceofAllCompositions . . . . . . . . . . . . . . . . . . . . . . . . . .135
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157
Chapter9
ThePfisterFactorConjecture . . . . . . . . . . . . . . . . . . . . . . . . . . .160
Appendix. Pfisterformsandfunctionfields . . . . . . . . . . . . . . . . . . . .167
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .170
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .175
Chapter10
CentralSimpleAlgebrasandanExpansionTheorem . . . . . . . . . . . . .176
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .196
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .202
Chapter11
HassePrinciples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .204
Appendix. Hasseprinciplefordivisibilityofforms . . . . . . . . . . . . . . . .218
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .220
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .223
Part II. Compositions of Size [r,s,n]
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227
Chapter12
[r,s,n]-FormulasandTopology . . . . . . . . . . . . . . . . . . . . . . . . . .231
Appendix. Moreapplicationsoftopologytoalgebra . . . . . . . . . . . . . . .252
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .256
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .264
ix
Chapter13
IntegerCompositionFormulas . . . . . . . . . . . . . . . . . . . . . . . . . .268
AppendixA.AnewproofofYuzvinsky’stheorem . . . . . . . . . . . . . . . .286
AppendixB.Monomialcompositions . . . . . . . . . . . . . . . . . . . . . . . .288
AppendixC.Knownupperboundsforr ∗s . . . . . . . . . . . . . . . . . . . .291
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .294
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .297
Chapter14
CompositionsoverGeneralFields . . . . . . . . . . . . . . . . . . . . . . . . .299
Appendix. Compositionsofquadraticformsα,β,γ . . . . . . . . . . . . . . .317
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .321
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .327
Chapter15
HopfConstructionsandHiddenFormulas . . . . . . . . . . . . . . . . . . . .329
Appendix. Polynomialmapsbetweenspheres . . . . . . . . . . . . . . . . . . .348
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .361
Chapter16
RelatedTopics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .363
SectionA.Higherdegreeformspermittingcomposition . . . . . . . . . . . . . .363
SectionB.Vectorproductsandcompositionalgebras . . . . . . . . . . . . . . .368
SectionC.Compositionsoverringsandoverfieldsofcharacteristic2 . . . . . .370
D.Linearspacesofmatricesofconstantrank . . . . . . . . . . . . . . . . . . . .372
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .375
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .380
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .381
ListofSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .407
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .413
Introduction
This book addresses basic questions about compositions of quadratic forms in the
sense of Hurwitz and Radon. The initial question is: For what dimensions can they
exist? Subsequentquestionsinvolveclassificationandanalysisofthequadraticforms
whichcanoccurinacomposition.
This topic originated with the “1, 2, 4, 8 Theorem” concerning formulas for a
product of two sums of squares. That theorem, proved by Adolf Hurwitz in 1898,
wasgeneralizedinvariouswaysduringthefolowingcentury,leadingtothetheories
discussed here. This area is worth studying because it is so centrally located in
mathematics: thesecompositionshavecloseconnectionswithmathematicalhistory,
algebra,combinatorics,geometry,andtopology.
Compositions have deep historical roots: the 1, 2, 4, 8 Theorem settled a long
standing question about the existence of “n-square identities” and exhibited some
of the power of linear algebra. Compositions are also entwined with the nineteenth
centurydevelopmentofquaternions,octonionsandCliffordalgebras.
Another attraction of this subject is its fascinating relationship with Clifford al-
gebras and the algebraic theory of quadratic forms. A general composition formula
involvesarbitraryquadraticformsoverafield,notjusttheclassicalsumsofsquares.
SuchcompositionscanbereformulatedintermsofCliffordalgebrasandtheirinvo-
lutions. Thereisalsoacloseconnectionbetweentheformsinvolvedincompositions
andthemultiplicativequadraticformsintroducedbyPfisterinthe1960s.
Alltheknownconstructionsofcompositionformulasforsumsofsquarescanbe
achievedusingintegercoefficients. Acompositionformulawithintegercoefficients
canberecastasacombinatorialobject: aspecialsortofmatrixofsymbolsandsigns.
These“intercalate”matriceshavebeenstudiedintensively,leadingtoaclassification
oftheintegercompositionswhichinvolveatmost16squares.
Finally this topic is connected with certain deep questions in geometry. For in-
stance,compositionformulasprovideexamplesofvectorbundlesonprojectivespaces,
of independent vector fields on spheres, of immersions of projective spaces into eu-
clideanspaces, andofHopfmapsbetweeneuclideanspheres. Thetopologicaltools
developedtoanalyzethesetopicsalsoyieldresultsaboutrealcompositions.
Let us now describe the original question with more precision: A composition
formulaofsize[r,s,n]isasumofsquaresformulaofthetype
(x2+x2+···+x2)·(y2+y2+···+y2)=z2+z2+···+z2
1 2 r 1 2 s 1 2 n
