APPENDIX B

A lette r fro m J-P . Serr e t o R.S . Garibald i

Paris, Septembe r 15, 2002

Dear Skip ,

Here ar e a fe w historica l comment s o n my involvemen t wit h cohomologica l

invariants an d trac e forms .

Let m e begin wit h th e trac e form .

I starte d bein g intereste d i n i t becaus e o f a lette r fro m J.D.Gray , se e [Se78] .

He wante d a modernize d (? ) accoun t o f Klein' s reductio n o f th e genera l quinti c

equation t o th e specia l for m x 5 + ax 2 + bx + c = 0 , togethe r wit h it s connection s

to modula r function s o f level 5 . I notice d tha t thi s i s possible thank s t o a formul a

relating th e tw o kind s o f Stiefel-Whitne y classe s W2 attached t o a n etal e algebr a

E, namel y th e Galoi s on e (whic h w e no w cal l wf

a

) , an d th e Delzan t on e fo r th e

quadratic trac e for m qE o f E. I wrot e thi s i n th e for m o f a lette r t o J . Martine t

(February 1982). I then went to Barcelona (Ma y 1982), where Nuria Vila, a student

of Pilar Bayer , was trying to solve the lifting proble m (fro m A

n

t o 2.A

n

) fo r specifi c

equations: trinomia l ones , mainly . Thi s amount s t o decidin g whethe r th e Stiefel -

Whitney clas s wf* i s 0 . I explaine d t o he r th e recip e o f my lette r t o Martinet ,

which reduce s th e computatio n t o tha t o f

W2(QE),

whic h i s easy. Thi s encourage d

me to publish th e formul a (i n [Se84]) , and peopl e immediately starte d usin g it (se e

the reference s i n 25.14). I n [Se84] , I ha d lef t ope n th e proble m o f generalizin g th e

formula t o an arbitrary wi\ thi s was soon solved by B. Kahn [Kahn84] . A few years

went by . The n I collaborate d wit h E . Baye r o n "self-dua l norma l bases" , a closel y

related topic . Thi s took plac e aroun d 1991-92 , but wa s published thre e year s late r

[BS94], becaus e o f th e heav y backlo g o f Amer . J . Math . A t abou t th e sam e tim e

(1991-92) I als o starte d bein g intereste d i n th e possibl e for m o f qE i n lo w rank .

Rank u p to 5 is easy, but i t mad e m e notice that

WS(QE)

i s always 0. T o go furthe r

I introduced i n a short not e (sen t i n July 1992 to E. Bayer an d a few other friends )

the notio n o f cohomologica l invarian t — wit h th e presen t definition , excep t tha t

Rost's compatibility theorem 11.1was then an axiom. Thi s allowed me to show tha t

the Wi are 0 beyond roughl y hal f th e rank . I also did th e sam e with wha t I dubbe d

the "Wit t invariants" , whic h ar e suppose d t o giv e mor e precis e informatio n tha n

the S t iefel-Whitney ones . I also started a t tha t tim e th e determinatio n o f all qE of

rank 7 , although the mos t precis e results were only obtained aroun d 1994. I gave

several courses on this: Harvar d (1990 and 1992), Ascona (1991), College de Franc e

(1992 an d 1994), Besangon (1993), Edinburgh (1995), the lates t bein g th e presen t

one (UCLA , 2001). Th e fac t tha t I did no t publis h unti l no w anythin g othe r tha n

short resume s wa s mostl y du e t o th e lac k o f prope r foundationa l materia l (found

here a s Chapter s I I through V) .

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http://dx.doi.org/10.1090/ulect/028/13