METRICS, CONNECTIONS AND GLUING THEOREMS 19 In the infinit e dimensional , Predhol m context , th e se t W shoul d als o be a smoot h manifol d wit h boundar y (give n b y (2.18) ) fo r a suitabl y generic choic e o f homotop y {s t }. Thin k o f thi s cobordis m theore m a s a non-linea r versio n o f th e assertio n tha t th e inde x o f a Predhol m op - erator i s invariant unde r homotopie s o f the operato r throug h Predhol m operators. Indeed , th e cobordis m theore m reduce s t o th e assertio n o f constant inde x in the case where the base space is a linear Banach spac e and wher e th e sectio n i s a linea r map . In th e gaug e theor y context , tw o smoot h .M 0 's (correspondin g t o generic metric s o n X a s discusse d above ) ar e smoothl y cobordan t b y taking a n appropriat e 1-paramete r famil y o f metric s an d the n th e cor - responding 1-paramete r famil y o f «M°'s. 5) Orientation s an d othe r characteristi c classes : Conside r th e finite dimensional contex t an d suppos e tha t M = s -1 (0) i s a smoot h sub - manifold o f B. I s thi s manifol d orien t able? I s i t spin ? Wha t ar e it s Pontrjagin classe s (o r Chern classe s when complex) ? Man y of the char - acteristic classe s of the tangen t bundl e t o M ca n b e compute d i n term s of the characteristic classes of the tangent bundl e to B and of the vecto r bundle E. Thi s i s because TB\ M « TM ®E\ M . One coul d as k simila r question s i n th e infinit e dimensiona l context . Here, man y o f th e characteristi c classe s o f M ca n b e compute d usin g the Atiyah-Singe r inde x theore m fo r familie s o f operators [7] . Thi s ap - plication o f [7 ] is based o n th e followin g observations : If , a s on e varie s through th e bas e 5 , th e operato r V s i s always Predholm , the n it s ker - nel an d cokerne l are , everywhere , finite dimensional . And , th e forma l difference o f thes e vecto r space s (Index(Vs) ) fits togethe r ove r B a s an elemen t i n the K- theor y o f B (see , e.g. , [3]) . Th e Stieffel-Whitne y classes and Pont r agin classes exist in the real K- theory, while the Cher n classes sit i n the comple x if-theory , an d s o Index(Vs) ha s such charac - teristic cohomolog y classe s o n B. (Thes e cohomolog y classe s ar e rigi d under perturbation s o f Vs throug h Predhol m operators. ) What' s more , if V s i s homotopi c t o a n ellipti c differentia l operator , the n thes e co - homology classe s hav e a topologica l interpretatio n vi a th e inde x the - orem fo r familie s o f operator s i n [7] in practic e thi s make s the m em - inently computable . Then , wit h th e preceedin g understood , remar k that Index(Vs)| M ~ TM an d s o the if-theor y characteristi c classe s of Index(Vs) determin e th e characteristi c classe s of M's tangen t bundle . In th e gaug e theor y context , Donaldso n [15 ] (se e [20] ) ha s exploite d the preceedin g observatio n (wit h th e Atiyah-Singe r inde x theore m fo r families o f Predholm operators ) t o prov e that th e space s A4° ar e alwa y orientable whe n the y ar e manifolds .
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