Exotic smooth structures on 3CP2♯8CP¯2

Abstract

It is an intriguing and very tricky problem in differential topology to construct exotic smooth 4-manifolds with small Euler characteristics. For example, given an ordered pair of positive integers (i, n), we can ask whether there exists a smooth closed simply-connected 4-manifold that is homeomorphic but not diffeomorphic to mCP2#nCP . Note that such an exotic 4-manifold has e = m + n + 2, b+ = m, and b= n. Probably the most famous example of this kind is the Barlow surface, which corresponds to the pair (1, 8) (cf. [B] and [Ko]). The Barlow surface has the smallest Euler characteristic among all known closed oriented simply-connected 4manifolds with more than one smooth structure. In this paper we construct three exotic 4-manifolds corresponding to the pairs (3,10) and (3,12). Namely, we prove the following