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

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 1985 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.