16 C. TAUBE S The relevant section , s , o f sends [V ] to [V,P + i? v ]. Thi s section is equivariant wit h respec t t o th e 577(2 ) action . And , (2.14) s' l (0)=M°. f) Fredhol m propertie s Here is a model to keep in mind when cogitating o n (2.14) : Le t B b e a smooth , finit e dimensiona l manifol d an d le t IT : E 5 b ea vecto r bundle. Le t s b e a sectio n o f E an d le t M = s -1 (0). Yo u shoul d b e thinking o f B her e a s a mode l fo r i?° , above, an d 5 here a s a model fo r 5, above . Then , M play s the rol e of the modul i spac e M°. Now M , bein g a subse t o f B, inherit s a topology . Question : Whe n is M a submanifol d o f B? Th e implici t functio n theore m tell s u s th e answer. Th e subse t M i s a submanifol d whe n V s : TB\M E\ M i s an everywhere surjective map . Here , V i s any covariant derivativ e on E near M . (Not e tha t th e ma p Vs\ M i s insensitiv e t o th e precis e choic e of V. For , i f V i s another choice , then V = V + a (wher e a is a sectio n of End(E) g T*B) an d s o V'S| M = Vs| M + a s\M = Vs\ M becaus e s vanishes o n M. ) I n addition , whe n s i s surjective, the n th e dimensio n of M nea r som e poin t i s equa l t o th e dimensio n o f th e kerne l o f th e linear ma p Vs . Thi s i s because th e kerne l o f V s a t a poin t o f M i s th e tangent spac e t o M (a s a subvector spac e o f TB.) There is a similar implicit function theore m in the infinite dimensiona l context, although one must take care with one's definition o f surjectivity . Rather tha n spea k abstractly , retur n t o (2.14) . Here , Vs|[v ] i s a linea r map fro m (TB°\[v])/TU(2) t o the space of su(V)-valued section s of A + . This ma p send s the su(\/)-value d 1 - form a (whic h obey s (2.11) ) t o (2.15) ^ 5 l [ v ] Q = P+dvCL. Generally, there is no guarantee that thi s map i s surjective. However , it doe s hav e th e followin g property : Us e th e L| + 1 Sobole v nor m t o complete th e spac e o f section s a o f su(V) ® T*X whic h obe y (2.11) . Use th e L\ Sobole v nor m t o complet e th e spac e o f sections o f su(V) ® A+. I f k i s non-negative , the n th e linea r operato r i n (2.15 ) extend s t o map the former completio n int o the latter completion , an d the resultin g extension i s a Fredhol m map . Tha t is , it s kerne l i s finite dimensiona l as i s its cokerne l (range/image) . Furthermore , bot h th e kerne l an d th e cokernel ar e represente d b y smoot h forms . (For thos e reader s wh o ar e unfamilia r wit h th e Fredhol m context , consider studyin g th e followin g example : Thin k o f th e operato r ~ a s sending C 1 function s o n a n interva l [0 , R) t o functions . Here , R 0
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