Math doesn't depend on speed. It is about deep.

# Four-dimensional space-time

We experience three spatial dimensions; and the description of physical processes can be somewhat simplified by treating time as a fourth dimension. [The latter is true even in Galileo's and Newton's models of physical reality; but it truly comes into its own once one is dealing with Maxwell's contributions.]

It is, of course, entirely possible that our universe has more than four indenpendant directions of displacement; just so long as space-time is wrapped up on itself, in these directions, on such a short scale that all familiar objects (including atoms and, probably, their nuclei) spread most of the way round – in the same sense that ants walking up and down a piece of thread shall consider it one-dimensional (because they cannot pass one another, along the thread, without interacting) even though tardigrades (which are much smaller) may perceive the same thread as a two-dimensional place (because two of them, on opposite sides of the thread, can pass without even being aware of one another's presence). However, the naïve interpretation of what we are in any position to observe sees only four dimensions.

As it happens, there are certain twists in the theory of many-dimensional spaces which single out particular dimensions as special, either as unique special cases or as boundaries between the low-dimensional special cases and the high-dimesional general case. Furthermore, four dimensional space is often either the start of the general case, the end of the special cases or the unique exception. I can't remember all the places I've seen this, but I'll try to record them here as I am reminded of them.

## Topologically but not smoothly equivalent manifolds

A topological space and a smooth atlas of it make up a smooth manifold. From a given topological space we may have several smooth atlases. If each of two atlases deems the charts of the other to be smooth, combining them with the topological space yields the same smooth manifold. If either involves a chart which the other doesn't accept as smooth, the manifolds obtained are different. However, even in this case, there is usually some topological auto-isomorphism of the space which serves as a smooth isomorphism mapping between the two smooth manifolds produced. But, apparently, not in dimension 4, at least not when flat …

## Inaudible topological differences

In the second chapter of his excellent book The sensual quadratic form, John Horton Conway examines the problem (originally posed by Mark Kac) of whether the harmonic structure of a surface (i.e. the set of frequencies at which it would resonate) suffices to determine the shape of the surface; and shows how to construct confusable pairs of surfaces at dimensions down to four. However, he reports that – when one knows the surface to be the quotient of a lattice – it has been established that the problem is indeed tractable in lower dimension than four.

## Constant curvature

I dimly remember a lecturer, possibly Dr. Horgan when talking about homotopy, mentioning the special case of surfaces of constant curvature. If I remember correctly, there is a general proof that, for any smooth manifold of dimension greater than four, constant curvature implies that the manifold is topologically spherical; while, for each dimension less than four, there is a specific proof that a constant-curvature manifold of that dimension must be the sphere; but, in the case of a four-dimensional manifold, there is a known counter-example. If your imagination can cope with the idea of a four-dimensional analogue of a circular piece of guttering (i.e. half of a torus), this last can be construed as the result of gluing 42 (Douglas Adams would be proud) of these together, each attached along one edge to the previous and along the other edge to the next, with the last regarding the first as next. However, I may be mis-remembering, possibly via a confused misunderstanding of some partial description of Poincaré's conjecture (see below).

## Poincaré

Poincaré had a conjecture which was proven for all dimensions from four upwards, and for each dimension below three, until the early twenty-first century, when Grigory Perelman gave a proof for the bounding case of three dimensions. The conjecture states that every compact topological manifold with the same homotopy as a sphere (of relevant dimension) is in fact homeomorphic to that sphere.

Written by Eddy.