Nonexistence of twists and surgeries generating exotic 4-manifolds

Abstract

It is well known that for any exotic pair of simply connected closed oriented 4-manifolds, one is obtained from the other by twisting a compact contractible submanifold via an involution on the boundary. By contrast, here we show that for each positive integer n n , there exists a simply connected closed oriented 4-manifold X X such that for any compact (not necessarily connected) codimension zero submanifold W W with b 1 ( ∂ W ) > n b_1(\partial W)>n , the set of all smooth structures on X X cannot be generated from X X by twisting W W and varying the gluing map. As a corollary, we show that there exists no “universal” compact 4-manifold W W such that for any simply connected closed 4-manifold X X , the set of all smooth structures on X X is generated from a 4-manifold by twisting a fixed embedded copy of W W and varying the gluing map. Moreover, we give similar results for surgeries.