4 INTRODUCTION The Whitehead Theorem 2.1naturally suggests the question: . Whic h form s ca n appea r a s intersectio n form s o n compac t * ' ' simply-connecte d 4-manifolds ? This is an existence question. There is also a uniqueness question. (2.7) Ho w man y inequivalen t manifold s ca n carry th e same form ? These question s ca n b e aske d fo r topologica l manifolds , wher e "equivalence " means homeomorphism , o r fo r differentiabl e manifolds , wher e "equivalence " means diffeomorphism . Fo r th e trivia l for m o f ran k zero , Questio n (2.7 ) i s th e 4-dimensional Poincar e Conjecture. It i s on e o f th e profoun d result s o f moder n topolog y tha t give s a complet e answer t o thi s questio n i n th e topologica l case . A s wit h surfaces , w e divid e th e simply-connected 4-manifold s int o two classes: those of type I which are nonspin and hav e intersectio n form s o f typ e I , an d thos e o f type I I whic h ar e spi n an d have intersection form s o f typ e II . Let^J o p denot e th e homeomorphism classe s of compac t oriente d simply-connecte d topologica l 4-manifold s o f typ e R. LetJ R denote the equivalence classes of unimodular symmetric bilinear forms of type R. Taking the intersection form of a manifold gives a map iR: ^JOP - J R . THEOREM 2. 8 (M . FREEDMA N (1982) [F]) . The map /„ : J?u° p-+Ju is a bijection. The map i}: JtJ°? - * Jl is exactly two-to-one and onto. Thus, ever y unimodula r for m i s th e intersectio n for m o f a simply-connecte d topological 4-manifold . Thi s manifold i s unique in the type I I case, and ther e are exactly tw o distinc t manifold s i n th e typ e 1 case. Th e tw o possibilitie s diffe r i n that on e of the m has a nonzero Kirby-Siebenmann obstructio n t o triangulabilit y (see IKS]). 3. Differentiabl e 4-manifolds . W e now examine the situation fo r the differentia - ble case . I t ha s bee n know n fo r som e tim e tha t no t ever y for m ca n appea r o n a differentiable 4-manifold . THEOREM 3. 1 (ROCHLI N [RO]) . Let M be a compact simply-connected differentia- ble 4-manifold of type II. Then signature(AZ) = 0 (mod 16). Recall tha t th e signature of a type II for m is always a multiple of 8 . However, forms of typ e II with signature 8 do exist—for example , the form £8 above . Thus, for a unimodular for m ix o f typ e II , we are led t o consider th e following Rochlin invariant p(/m ) s £ signature^ ) (mod2) . Form s wit h nonzer o Rochli n invarian t do no t occu r a s intersectio n form s o n compac t oriente d smoot h 4-manifolds . Until recently , little else was known about this question. We now come to the theorem whose proof i s the main focus of these lectures. THEOREM 3. 2 (S . DONALDSO N (1982) [D 12 ]). Let M be a compact simply- connected smooth 4-manifold whose intersection form fi is positive definite. Then ii is equivalent to the diagonal form i.e., /x s ( ] ) © © { ! ) .
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