Preface These note s resul t fro m a meeting held i n Sant a Barbar a in August 1983. The purpose of th e meetin g was to bring together geometric topologists an d differen - tial geometers t o study in depth the recent work of Simo n Donaldson. Of course, due t o th e beautifu l an d profoun d result s o f Mik e Freedman , th e subjec t o f 4-manifolds ha d alread y become the focus of lively interest. Moreover, in light of Donaldson's result , th e Freedman-Casso n machiner y wa s abl e t o produc e th e startling fac t tha t ther e exis t exoti c differentiabl e structure s o n R 4. Fo r thes e reasons topologists have wanted to understand in depth the arguments of Donald - son, which are based on the theory of Yang-Mills fields . Consequently, th e principa l purpos e o f thes e lecture s (an d thes e notes ) i s t o present thes e argument s togethe r wit h al l th e backgroun d materia l require d b y someone wh o i s no t a n exper t i n th e field . Th e lecture s ar e aimed , however , a t mature mathematician s wit h som e trainin g i n geometr y an d topology . Th e tas k set ou t her e wa s alread y sufficientl y exactin g tha t n o tim e wa s availabl e fo r wide-ranging discussio n o r excursion s int o physics . O n th e othe r hand , thi s presentation attempts to be nearly complete. It shoul d b e mentione d tha t a seminar o n thi s subjec t wa s ru n las t sprin g a t M.S.R.l. b y M . Freedma n an d K . Uhlenbeck . Th e note s o f thi s semina r hav e been prepared fo r publication with the assistance of D . Freed. They also provide a detailed reference for this material. The success of th e conference was due to the enormous efforts of the organizing committee: Ken Milieu, Doug Moore, and Marty Scharlemaim, to whom all of us who participated hav e expresse d ou r gratitude. I would als o like t o than k Susa n Crofoot fo r he r beautifu l jo b o f preparin g th e manuscrip t an d th e organizin g committee for all the help they gave me with proofreading. It is a pleasure to report that two of th e conference participants, Ron Fintushe l and Ro n Stern, have subsequently succeeded in greatly generalizing the results of Donaldson while , a t th e sam e time , simplifyin g th e arguments . The y hav e als o applied thei r method s t o prove a spectacular resul t concerning homolog y cobor - disms of homolog y 3-spheres . In particular, they show that th e Poincard 3-spher e has infinit e orde r i n thi s group . Interestingly , th e ke y t o thei r argument s i s t o consider gaug e field s wit h S0 3 i n place of SU 2 . Fo r low instanto n numbers , th e moduli spac e o f self-dua l connection s i n thi s cas e i s actuall y compact. W e shal l say a bit more about this at the end of Chapter I. vn
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