# Higher Order Corks

# Higher Order Corks

In 1960, Mazur constructed a contractible 4-manifold W with non-simply connected boundary whose product with an interval is the 5-ball. Thirty years later, Akbulut showed that ∂W supports an involution T that does not extend to a diffeomorphism of W, thus producing the first nontrivial *involutory cork *(W,T). Akbulut's proof was to embed W in a smooth 4-manifold so that the resulting *cork twist* (cut out W and reglue using T) changes the ambient smooth structure. It is now known by the *involutory cork theorem *of Curtis-Freedman-Hsiang-Stong and Matveyev that any two smooth structures on a closed, simply connected 4-manifold are related by a single involutory cork twist. The existence of higher order corks (where T is of order >2) whose twists by powers of T can produce distinct smooth structures was unknown before this year. In this talk I will describe the construction of such a cork (joint work with Auckly, Kim and Ruberman) and work in progress with Hannah Schwartz on a general *finite cork theorem* generalizing the involutory cork theorem.