These are the lecture notes from the Conference 'Inde x Theory, Coarse Geometry ,
and Topolog y o f Manifolds ' whic h wa s hel d i n Boulde r i n Augus t 1995. I hav e
adhered fairl y closel y t o th e origina l schem e o f the lectures , althoug h th e note s
naturally contai n rathe r mor e detail an d i n a few places (especiall y i n lecture 7 )
describe construction s tha t wer e finalized onl y afte r th e lectures wer e delivered .
It i s o f th e natur e o f thi s subjec t t o requir e a rathe r divers e background . I
attempted to deal with this in the lectures by including a number of mini-surveys,
and i n this boo k mos t o f lectures 1 and 6 and larg e parts o f lectures 2 and 3 are
devoted t o suc h surveys . I hope tha t thes e will be useful .
The wor k reporte d o n her e ha s develope d ove r a doze n year s o r s o an d
during tha t tim e I hav e ha d th e pleasur e o f learnin g fro m a grea t numbe r o f
colleagues. Amon g the m I woul d mentio n Alai n Connes , Mish a Gromov , Stev e
Hurder, Jonatha n Rosenberg , Stepha n Stolz , an d Guolian g Yu. Mor e recently I
have benefited fro m th e advic e o f Stev e Ferry , Eri k Pedersen , Andre w Ranicki ,
and Shmue l Weinberger , al l o f who m hav e (amon g othe r things ) answere d th e
numerous question s o f a surger y neophyt e wit h exemplar y patience . Finally , a
major deb t o f gratitude i s owed to Nigel Higson, with whom I have collaborate d
on man y o f the project s tha t ar e summarized i n thi s book .
The Conference was made possible by the organizational effort s o f Jeffrey Fox ,
Guoliang Yu , an d Carl a Farsi , an d b y th e financial suppor t o f th e Conferenc e
Board fo r the Mathematical Science s and the National Scienc e Foundation. Th e
manuscript wa s completed while I was Ulam Visiting Professor a t th e Universit y
of Colorado , Boulder . I would lik e to expres s m y thanks t o al l these peopl e an d
organizations fo r thei r generou s support .
John Roe , Octobe r 1995