Preface These notes paralle l a series o f te n lectures tha t I gave at th e Conferenc e Boar d o f th e Mathematical Science s Regional Conferenc e i n Rochester , Michigan , June 1979 , sponsored b y the National Scienc e Foundatio n an d th e Departmen t o f Mathematica l Sciences , Oakland University. I have taken th e liberty o f including some ne w results a s well a s some improve d arguments suggeste d b y colleague s a t th e conference . I a m mos t gratefu l t o Jac k Tsui , Stev e Wright , an d th e CBM S fo r organizin g th e conference an d t o th e Nationa l Science Foundation fo r its support. Th e facilities o f Meadow Brook Hall were elegant and conducive to the fruitful interactio n o f the participants. I am also indebted to Alain Connes, who provided me with an opportunity t o prepare these lectures in a visi t a t th e Institu t de s Haute s Etude s Scientifiques . Durin g th e precedin g yea r I ha d profited fro m a number o f conversation s with hi m regardin g his approach t o geometry , some aspects of which ar e reflected i n Chapte r 1 . I also wish t o than k th e participants , who pro- vided an audience that wa s both enthusiasti c an d provocativel y skeptical , as well as my col- leagues who attended a course a t UCL A in which I had th e opportunit y t o expan d th e lec- ture notes. Finally I wish to than k m y collaborato r an d forme r studen t Chao-Dan g Shen . Hi s un- flagging enthusias m an d ingenuit y hav e provide d considerabl e impetu s t o th e furthe r develop - ment o f dimensio n group s as a new mathematica l discipline . August 198 0 UCLA vn
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