2 INTRODUCTION FIGURE 1 THEOREM 1.1 . Two compact connected surfaces are diffeomorphic if and only if their intersection forms are abstractly equivalent. Surfaces can be separated into two classes: type I and type U: Those of type I are nonorientable and can be decomposed into a connected sum of real projective planes. Those of type I I are orientable and can be decomposed into a connected sum of tori. The relevant dat a can be organized a s shown in Table 1. Type I Type II nonorientable (H-,*0) orientable M M B M \ I 1 / 0 1 1 0 i '• 0 1 1 0 / 2 s P 2(R)# •• • #P 2(R) I s ( 5 ' x S 1 )* •• • #(S l x S l) TABLE 1 2. Simply-connecte d 4-manifolds . On e o f th e glorious achievement s o f moder n topology i s th e recen t classificatio n o f compac t simply-connecte d topologica l 4-manifolds. A s wit h surfaces , th e classification i s stated i n term s of a n intersec- tion for m o n th e middle-dimensiona l homolog y group . No w a simply-connecte d manifold M 4 ca n b e oriented . Therefore , usin g th e orientations , th e intersectio n of tw o transversal , oriente d surface s ca n b e counte d a s a n integer . Thi s give s a symmetric bilinear for m /i on H 7 (M\ Z) . Foincare duality state s that thi s form is unimodular. That is , i f \i i s expresse d a s a n ( r X r)-matrix ((m iy )) wit h intege r entries (wit h respec t t o som e basi s o f th e fre e abelia n grou p H 2 (M Z)) , the n det((m,7)) = ±1. I t i s a classica l resul t tha t th e for m n determine s M u p t o homotopy type. THEOREM 2.1(J. H. C. WHITEHEA D (1949) [Wh]). Two compact simply-connected 4-manifolds are homotopy equivalent if and only if their intersection forms are equivalent.

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