Questions specifically about $4$-dimensional manifolds

In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).

4-manifolds are important in physics because in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold.

It can also be used for constructions specific or typical for $4$-manifolds, e.g. the signature, Kirby diagrams, Akbulut diagrams, exotic $\mathbb{R}^4$s, etc.