2 Ronald Fintushel And

A basic question of 4-manifold topology is whether the complex projective plane, CP admits exotic smooth structures. Thus one is interested in knowing the smallest m for which CP#mCP 2 admits an exotic smooth structure. In the late 1980’s, Dieter Kotschick [K] proved that the Barlow surface, which was known to be homeomorphic to CP#8CP 2 , is not diffeomorphic to it. In following years the subject of simply connected smooth 4-manifolds with b = 1 languished because of a lack of suitable examples. However, largely due to a beautiful paper of Jongil Park [P], who found the first examples of exotic simply connected 4-manifolds with b = 1 and b = 7, the past year has found renewed interest in this subject. Peter Ozsvath and Zoltan Szabo proved that Park’s manifold is minimal [OS], and Andras Stipsicz and Szabo used a technique similar to Park’s to construct an exotic manifold with b = 1 and b = 6 [SS].