# Topology Seminar

**Abstract:** A cork is a contractible smooth $4$-manifold with an involution on its boundary that does not extend to a diffeomorphism of the entire manifold. Corks can be used to detect exotic structures; in fact any two smooth structures on a closed simply-connected $4$-manifold are related by a cork twist. Recently, Auckly-Kim-Melvin-Ruberman showed that for any finite subgroup $G$ of $SO(4)$ there exists a contractible $4$-manifold with an effective $G$-action on its boundary so that the twists associated to the non-trivial elements of $G$ do not extend to diffeomorphisms of the entire manifold. In this talk we will use Heegaard Floer techniques originating in work of Akbulut-Karakurt to give a different proof of this phenomenon.