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Abstract
In this dissertation, we discuss smooth 4- and 5-manifolds, their interactions, and contractible smooth high-dimensional manifolds. First, we study 5-dimensional cobordisms with 2- and 3-handles, 5-dimensional 3-handlebodies, and closed, orientable 5-manifolds via (5-dimensional) Heegaard diagrams. We show that every such smooth 5- manifold can be represented by a Heegaard diagram, and two Heegaard diagrams represent diffeomorphic 5-manifolds if and only if they are related by certain moves. As an application, we construct Heegaard diagrams for 5-dimensional cobordisms from the standard 4-sphere to the Gluck twists along knotted 2-spheres. This provides some equivalent statements regarding the Gluck twists being diffeomorphic to the standard 4-sphere. Second, for any integer n ≥ 2, we construct a contractible, compact, smooth (n + 3)-manifold which is not homeomorphic to the standard (n + 3)-ball, using a 0-handle, an n-handle, and an (n + 1)-handle. The key step is the construction of an interesting knotted n-sphere in $S^n\times S^2$ generalizing the Mazur pattern. As a corollary, for any integer n ≥ 2, there exists a smooth involution of $S^{n+3}$ whose fixed point set is a non-simply connected homology (n + 2)-sphere.