Diffeomorphisms in Dimension Four Through Corks, Pseudoisotopies, and Trisections

This dissertation examines diffeomorphisms of $4$-manifolds using pseudoisotopy and trisection theories. With the help of pseudoisotopy, we establish a cork theorem for exotic diffeomorphisms. Specifically, we prove that any diffeomorphism of a compact, simply-connected 4-manifold that can be isotoped to the identity after a single stabilization with $S^2 \times S^2$, can be smoothly isotoped to a diffeomorphism that is supported in a contractible submanifold. We also present a novel approach to study diffeomorphisms of $4$-manifolds up to isotopy using trisections. We show that every diffeomorphism of a closed trisected $4$-manifold that restricts to the identity on the spine of the trisection is pseudoisotopic to the identity relative to the spine. We also discuss some applications using a specific example, the $4$-dimensional Dehn twist along the neck of a connected sum.