24 C. TAUBE S Now, play the role of a topologist an d as k yourself what , i f anything , can be done to exploit the preceding picture to garner information abou t X. Her e is an answer: An y cobordism invarian t o f M (a s a submanifol d of B) wil l be, apriori, a n invariant o f the smooth manifol d X. Thi s is to say that i f we have two smooth manifolds , X an d X\ an d i f we want t o determine that the y are not diffeomorphic , the n we need only show tha t some canonical cobordism invariant o f the M. fo r X i s different tha n th e corresponding invarian t fo r M fo r X' '. (I f X wer e diffeomorphic t o X\ then th e M. fo r X' woul d b e th e sam e a s a n M fo r X , bu t wit h som e different metric . Sinc e th e metric s o n X for m a conve x set , an y tw o are connecte d b y a smoot h path , an d s o an y th e pai r o f Ms woul d b e cobordant.) Now, the simples t suc h cobordis m invarian t woul d b e t o conside r M as a homolog y clas s i n B. Thi s i s precisely th e strateg y whic h lead s t o Donaldson's invariants. Mor e precisely, Donaldson showed that M nat - urally define s a dual to the cohomolog y o f B [14] . Thin k o f representin g a cohomology class by a closed form, sa y a , and then assig n the numbe r (2.22 ) / UJ JM Stokes theore m (integratio n b y parts ) provide s a forma l "proo f tha t this number i s independent o f the representing for m u an d o f the choic e of metric wit h whic h t o defin e M. The wor d proof , above , i s i n quote s because , a s remarke d above , the spac e M i s rarel y compact , whil e Stoke s theore m ca n b e applie d unerringly onl y o n compact spaces . Remarkabl y enough , th e geometri c compactification o f M (whic h wa s discusse d above ) serve s fo r Stoke s theorem whe n usin g a rather natura l choic e of closed forms to represen t the cohomolog y o f B . I n th e end , (2.22 ) make s goo d sens e inspit e o f the manifes t non-compactnes s o f A4. I won' t sa y muc h abou t 23' s close d form s sinc e th e reade r ca n fin d a ver y elegan t discussio n i n [14] , [20] . I wil l onl y sa y her e tha t th e cohomology o f B is determined b y the homotop y typ e of the underlyin g space X an d wit h thi s understood , on e shoul d thin k o f th e integra l i n (2.22) a s definin g a canonica l homomorphism , (2.23) V : Sym*(tfevenpO) ® A*(Jfodd(X)) —* Q , the Donaldso n invarian t o f X. Here , Sym*(VF ) i s define d a s th e di - rect su m o f th e symmetrize d tenso r products , 1 , W, Sym2(W),... etc while A*(W) i s define d a s th e direc t su m o f th e anti-symmetri c tenso r products.
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