Indirect constructions of exotic surfaces in 4-manifolds

Two surfaces inside a 4-manifold are called exotic if they are topologically isotopic but not smoothly so. In this thesis, we provide examples of exotic surfaces in closed 4-manifolds with various properties. These are smoothly embedded surfaces that are topologically isotopic yet not smoothly so. For instance, we show that for any finitely presented group G, there is a simply connected closed 4-manifold containing an infinite family of pairwise exotic 2-links whose 2-link group is G. Moreover, if G satisfies the necessary topological conditions, these 2-links have nullhomotopic components. We show that some of these 2-links are pairwise Brunnianly exotic, that is, disregarding one component from each 2-link makes them smoothly equivalent. Using these 2-links, we show that there are exotic surfaces inside closed 4-manifolds that remain exotic after a specific internal stabilization. This yields examples of exotic nullhomologous tori whose fundamental groups of the complements can be chosen from an infinite set of examples. In a related direction, a Theorem of Baykur and Sunukjian states that two closed, oriented and simply connected 4-manifolds with the same Euler characteristic and signature are related via a sequence of torus surgeries i.e. by performing consecutive torus surgeries on one of them, we can obtain the other. We extend this result to closed, oriented 4-manifolds with arbitrary fundamental groups. We also investigate the sphere surgery analogue and show that the possible outcomes of sphere surgeries on a 4-manifold can differ even between homeomorphic 4-manifolds. In particular, We exhibit an example of two homeomorphic 4-manifolds such that a certain result of a sphere surgery on one of them cannot be obtained by sphere surgeries on the other.