The focus of this thesis is the study of smooth 4-dimensional manifolds. We examine two problems relating to 4-manifolds, the first pertaining to pseudo-isotopies and diffeomorphisms of 4-manifolds, and the second pertaining to embedded surfaces in 4-manifolds. We summarise our key results below.

A diffeomorphism $f$ of a compact manifold $X$ is pseudo-isotopic to the identity if there is a diffeomorphism $F$ of $X\times I$ which restricts to $f$ on $X\times 1$, and which restricts to the identity on $X\times 0$ and $\partial X\times I$. We construct examples of diffeomorphisms of 4-manifolds which are pseudo-isotopic but not isotopic to the identity. To do so, we further understanding of which elements of the “second pseudo-isotopy obstruction”, defined by Hatcher and Wagoner, can be realised by pseudo-isotopies of 4-manifolds. We also prove that all elements of the first and second pseudo-isotopy obstructions can be realised after connected sums with copies of $S^2\times S^2$.

If $\Sigma$ and $\Sigma’$ are homotopic embedded surfaces in a $4$-manifold then they may be related by a regular homotopy (at the expense of introducing double points) or by a sequence of stabilisations and destabilisations (at the expense of adding genus). This naturally gives rise to two integer-valued notions of distance between the embeddings: the singularity distance $d_{\sing}(\Sigma,\Sigma’)$ and the stabilisation distance $d_{\st}(\Sigma,\Sigma’)$. We use techniques similar to those used by Gabai in his proof of the 4-dimensional light-bulb theorem, to prove that $d_{\st}(\Sigma,\Sigma’)\leq d_{\sing}(\Sigma,\Sigma’)+1$.