Existence and uniqueness of additional structure

The interplay between successive geometrical structures is an active area of research with many unknowns. With each added structure we can ask several questions:

  • Can the higher structure always be defined on any object?
  • Can more than one non-equivalent structure be defined on a given object?
  • Does the structure provide new ways to distinguish between objects?

The answer to the first question can be negative: for example, there exist topological spaces that admit no Cartesian charts, and topological manifolds that admit no differential structure (in dimension 4 or higher). On the other hand, all differentiable manifolds admit a Riemannian metric. The answer to the second question is usually positive: for example, one can define non-equivalent metrics on a given differential manifold, and even non-equivalent differentiable structures on a given topological manifold (e.g. Milnor’s exotic 7-spheres). So we can conclude that in general, additional structure usually expands the possibilities, but also may eliminate spaces that are not “nice” enough.

The third question can also be positive, as seen in Donaldson theory and Seiberg-Witten theory, which use the structure of gauge theories in physics to create new ways of distinguishing 4-dimensional manifolds. These theories help show, among other things, that one can define non-equivalent differentiable structures on \({\mathbb{R}^{4}}\), a situation that is only true in four dimensions.

This brings us to a particularly interesting observation concerning geometrical structure, the fact that the answers to many questions often change or are most difficult to answer in dimension 3 and/or 4. These are of course the most common dimensionalities in physics, corresponding to space and spacetime. This suggests that there might be special features in these dimensions that could help explain their prominence in nature.

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