METRICS, CONNECTION S AN D GLUIN G THEOREM S 17 is finite. Prov e tha t thi s operato r i s Fredhol m an d comput e it s kerne l and cokernel . Then , conside r th e sam e operator , bu t no w sendin g C 1 functions o n [0 , oo) to function s o n [0 , oo). No w prove that J ^ is no t Fredholm becaus e it s cokerne l i s infinite dimensional. ) This Fredhol m propert y ma y soun d lik e a technicality , bu t i t is , i n fact, extremel y important . I t i s one of the linc h pins of the whol e busi- ness of anti-sel f duality . Th e poin t i s that ther e i s a sor t o f "Fredhol m Principle" whic h goe s somethin g lik e this : Section s o f infinit e dimen - sional vector bundle s over infinite dimensiona l spaces which linearize t o Fredholm operator s "behave " lik e sections of finite dimensional bundle s over finite dimensional manifolds. Thi s is to say that wha t is true for th e latter i s probably tru e for the former i f the section in question linearize s to a Fredhol m operator . (I hav e deliberatel y avoide d statin g a forma l theore m her e becaus e the precise statement woul d require a certain number of quantifiers con - cerning the highe r orde r derivative s o f the section . I don't kno w of an y optimal se t o f quantifier s whic h invok e th e Fredhol m principle , an d I shy away from th e constructio n o f some mega-list o f all possible combi - nations which work. I f you have a vector bundl e section wit h Fredhol m linearization, I suggest tha t yo u wor k i t ou t fo r you r self. ) Here i s wha t yo u shoul d expec t fro m you r sectio n wit h Fredhol m linearization: 1) Manifol d Structures : Th e zer o se t want s t o b e a manifold . I n general, th e zer o se t shoul d b e homeomorphi c t o th e invers e imag e o f zero for a smooth ma p betwee n finite dimensiona l Euclidea n spaces . I n the contex t o f (2.14) , the local structure nea r an y [V ] i s that o f a finite dimensional , real algebraic set the zer o set o f a real analytic ma p between th e kernel(Vs|[v] ) an d cokernel(Vs|[v]) - Thi s loca l pictur e i s established usin g th e invers e functio n theorem . (Th e argumen t her e i s a simpl e extensio n o f the argumen t whic h prove s tha t th e zer o se t i s a manifold wher e the differentia l i s surjective. ) 2) Dimension : Wher e i t i s a manifold , th e dimensio n o f the zer o se t is equa l t o th e inde x o f th e linearization thi s inde x i s define d t o b e the differenc e o f the dimension s o f th e kerne l an d th e cokernel . I n th e finite dimensiona l context , thi s inde x i s simply th e differenc e o f the di - mensions o f th e domai n manifol d an d fiber o f th e vecto r bundle , an d so is insensitive t o deformation s o f the origina l section . I n a n abstract , infinite dimensiona l context , th e inde x i s unchange d b y an y homotop y of th e origina l sectio n through t section s whic h al l hav e Fredhol m lin - earizations.
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