Exotic Dehn twists on 4-manifolds

We initiate the study of exotic Dehn twists along 3-manifolds embedded in 4-manifolds.This yields the first known examples of exotic diffeomorphisms of contractible 4-manifolds; more generally, it produces exotic diffeomorphisms of definite 4-manifolds, as well as exotic diffeomorphisms of 4-manifolds (whose boundary is not diffeomorphic to S 3 ) that persist after one stabilization.In particular, we obtain the first example of a diffeomorphism of a contractible 4-manifold that is not isotopic to the identity.We also construct the smallest closed 4-manifold currently known to support an exotic diffeomorphism.These exotic diffeomorphisms arise as Dehn twists along certain Seifert fibered 3-manifolds.As a consequence, we obtain loops of diffeomorphisms of 3-manifolds that extend topologically, but not smoothly, over some 4-manifold X .This implies that the map 1 .Diff.X // ! 1 .Homeo.X // is not surjective.Our method uses Seiberg-Witten theory for a 2-parameter family over RP 2 , whereas previous methods for detecting exotic diffeomorphisms relied on 1-parameter family gauge-theoretic invariants.Using a similar strategy, we also construct a new type of exotic diffeomorphism of 4-manifolds, realized as commutators of diffeomorphisms.