18 C. TAUBE S In th e gaug e theor y contex t o f (2.15) , th e inde x ca n b e compute d in terms o f certain characteristi c number s o f th e 4-manifol d X an d th e bundle V usin g th e celebrate d Atiyah-Singe r inde x theore m [5] . On e has: (2.16) inde x = 4 Pl (V) - 3 ( 1 - b^X) + 6+(X)), Here, P\(V) = Ci(V) 2 2• c2(V)^ interprete d a s an integer by evaluatio n on th e fundamenta l clas s o f X. Meanwhile , bi(X) i s th e ran k o f th e first cohomolog y of X, whil e b^iX) i s the hal f of the sum of the rank of H2(X) an d th e signatur e o f X. Th e numbe r i n (2.16 ) i s the dimensio n of M = Ai°/SU(2) wher e ever i s a manifold an d wher e ever SU(2) acts wit h stabilie r ±1 . 3) Sard' s theorem : Sard' s theore m fo r map s betwee n finit e dimen - sional manifold s ha s a generalizatio n t o th e Fredhol m contex t whic h i s due t o Smal e [43] . I n th e finit e dimensiona l case , Sard' s theore m as - serts tha t th e se t o f critica l value s o f a ma p hav e measur e zero . (Th e critical value s o f a ma p ar e th e point s i n th e rang e whic h ar e image s of points wher e the differentia l i s not surjective. ) I n th e infinit e dimen - sional case , the Smale-Sar d theore m assert s that th e non-critica l value s of a Predhol m ma p for m a Bair e set . ( A Bair e se t i s a countabl e unio n of open, dens e sets it i s dense, apriori. ) (Not e tha t th e manifold s her e must b e paracompac t wit h th e loca l mode l a separabl e Banac h space . Sometimes a tric k wil l allo w on e t o substitut e a n appropriat e Preche t space.) In th e gaug e theor y context , thi s Smale-Sar d theore m serve s a s th e crux o f Uhlenbeck' s proo f o f Theorem 2. 1 in [24]. 4) Cobordism : Suppos e tha t on e ha s a pai r o f sections , eac h wit h Predholm linearization , an d suc h tha t eac h zer o se t i s a n embedde d submanifold o f th e ambien t manifold . Suppos e tha t th e section s i n question ar e homotopi c throug h a 1-paramete r famil y o f sections , eac h of whic h ha s a Predhol m linearization . Doe s th e existenc e o f suc h a homotopy imply any relation between the starting and ending zero sets? In th e finit e dimensiona l context , th e answe r i s ye s - th e startin g an d ending zer o set s ar e cobordant . T o be mor e precise , le t {s t : t G [0,1]} denote the 1-paramete r family o f sections. Fo r a suitably generi c choic e of {s t }, th e se t (2.17) ^ = U t 6 | o,i,s t - 1 (0)c[0,l]xB defines a smoot h submanifol d wit h boundary , it s boundary bein g (2.18) ( { O J x r t J u H l J x , - 1 ! ! ) ) .
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