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

The 4-manifolds tag is for questions concerning 4-dimensional manifolds: smooth, PL or topological, or with any additional structure. See the Wikipedia page for a coarse summary of the area.

Many questions are answered in the affirmative or negative relatively easily for manifolds of dimension 5 or higher. Whereas analogous questions haven't been answered in lower dimensions. A particularly interesting case is dimension 4. As an example, exotic differentiable structures on the spheres. There is little known about the number of exotic differentiable structures on $S^4$. The very subject of four manifolds, studies questions such as these!

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

Many questions are answered in the affirmative or negative relatively easily for manifolds of dimension 5 or higher. Whereas analogous questions haven't been answered in lower dimensions. A particularly interesting case is dimension 4. As an example, exotic differentiable structures on the spheres. There is little known about the number of exotic differentiable structures on $S^4$. The very subject of four manifolds, studies questions such as these!