I. Introduction It is always a wonderful even t in mathematics when the results of one discipline of though t hav e startlin g an d unexpecte d consequence s i n another . Thi s wa s recently the case when Simon Donaldson, arguing from deep results in gauge fiel d theory, prove d th e nonexistenc e o f differentiabl e structure s o n certai n compac t 4-manifolds. The resul t wa s timely , for , i n wha t mus t b e considere d on e o f th e ultimate achievements of topology, Mike Freedman had recently given a complete classification o f compac t topologica l 4-manifold s (i n th e simply-connected case. ) In fact , a s Freedma n an d Kirb y firs t observed , thi s theory , togethe r wit h Donaldson's result , implies the existence of exoti c differentiable structure s on R 4. The purpos e o f thes e lecture s is t o present Donaldson' s theore m togethe r with the foundationa l wor k i n gaug e fiel d theory , du e t o Uhlenbeck, Taubes , Atiyah , Hitchin, Singer, an d others, on whic h th e arguments are based. This first chapte r is a n introduction . W e begi n b y summarizin g th e curren t stat e o f affair s i n th e theory o f 4-manifolds . W e the n stat e Donaldson' s theore m an d giv e a brie f outline of its proof. 1. Connected surfaces . One of th e classical result s of topolog y is the classifica - tion o f compac t connecte d surface s (withou t boundary ) u p t o diffeomorphism . The resul t ca n b e presente d i n th e followin g way . Le t 2 b e suc h a surface an d consider close d curve s 7 , an d y 2 o n 2 . B y a smal l deformatio n w e ca n mak e y 2 transversal t o y 7 . Th e curve s the n intersec t i n a finit e numbe r o f points . Thi s number, modulo 2, turns out to depend only on the homology class of y} and y2 in //,(2 Z 2). W e thereb y ge t a symmetric bilinear for m jx oni/,(2 Z 2) calle d th e intersection form o f 2 . Poincar e dualit y say s tha t thi s form i s nondegenerate. W e define th e form t o be of type I I if /I(JC , x) = 0 for allJC otherwise, we say it is of type I. Note tha t i f y c 2 i s an embedde d curv e alon g which orientation i s reversed , then a tubula r neighborhoo d o f y i s a Mobiu s ban d an d M([Y][Y] ) 1 ( see Figure 1). Hence, \i is of type I. 1 http://dx.doi.org/10.1090/cbms/058/01
Previous Page Next Page