Let me try to explain what I’m trying to do. [Especially not trying to re-invent the wheel when there are so many proven, useful models at hand].
I’m aiming at Combinatorial General Relativity, as Einstein mentioned at the end of some of his writings as an unsolved puzzle.
Yet “my way” is not the usual top-down –i.e. trying to find quantization-rules on the very top- but “bottom-up” that is investigating in a more or less systematic fashion the implicit and explicit combinatorial elements and structures on each level. Here “causal structure”, based on the work of Stephen Hawking, David Malament and others, is already an almost closed chapter (Q-Orders).
The current “target”: differential manifolds.
Usually a manifold is defined using local homeomorphism between some local neighbor-hoods in the manifold-topology and some local part of a Euclidean space. For the latter normally one uses its algebraic definition (module over some field), gets from there to the usual Euclidean metric and then obtains the topology.
Now there is a completely equivalent set of Axioms for Euclidean Geometries: the famous Hilbert Axioms. Why would one like to use this set? Well, it’s known that there are finite, discrete models that satisfy the first set of Hilbert-Axioms (incidence Axioms for points, lines and planes), yet do not satisfy the second set (Order-Axioms) nor the third (Congruence-Axioms) and where the forth (Parallel-Axiom) is optional.
Not only this, but as v. Staudte, Moufang, Veblen, v. Neumann and others have shown, already the first set allows the construction of coordinates and hence metric, i.e. coordinates and metric arise using incidence relations and nothing else. [And that’s what I understand Einstein had in mind with his remarks].
Incidence relations as “founding ground” have a second great advantage: topological homeomorphisms carry them over automatically, i.e. suitably defined the manifold inherits these relations automatically, while congruence-relations or algebraic properties do not.
The “missing link”: Hilbert takes “lines” and “planes” as primitives (as does synthetic Geometry). So in my next –failed- intent I was looking where to find “lines” and “planes” in the context of Riemann and Pseudo-Riemann manifolds, and voilá there they were: geodesics and totally normal neighborhoods, yet with a tremendous drawback: their definition already requires concepts of the very, very top like the metric-tensor and the exponential map.
So I did a step backward, asking myself: is it possible to introduce lines using ONLY topological concepts? The model at hand: balls and spheres as defined by the usual Euclidean metric.
Next step: is it possible to define balls and spheres without using metric? Well it is. One might use simply connected neighborhoods and simply connected surfaces and, as recently shown, we get nice spheres in all dimensions.
Problem: “simply connected” as concept requires Hausdorff-spaces and imports via Homotopy the whole “World of Reals”, incompatible with my aim to allow also discrete, finite models not talk about my suspicion that it is a far to restrictive assumption for the real World (there are holes almost everywhere). Therefore I replaced “simply connected” by just “locally path connected”.
Next step: given sets of surfaces of locally path connected neighborhoods, what would be a minimal set of axioms based only on incidence and topology that would convert this set locally into something similar to the Euclidean Balls and –Spheres we started with? There a point lies on a straight line between two points iff there are Spheres for each of them that touch each other exactly in that point. More over the Spheres made using Euclidean Distance form a complete order around each point. There are additional compatibility rules among the spheres of different points on the same straight line. Finally “completeness” respectively “coherence” is to be considered i.e. in Euclidean Geometry there is always a straight line that connects them. This again would provoke “continuous models only” already in three dimensions. So I weakened this requirement to: “it should be possible to go from anywhere to anywhere in straight steps” and “if there are intersecting spheres for two points, then there are touching spheres (and hence a line that connects them)”.
Done this, we get a set of definitions invariant under any topological homeomorphism. That means two things: they carry over “automatically” to the manifold (locally of course), but second they produce a natural gauge. (Not such a surprise: there quite many metrics already for the real line, which produce the same topology. For instance the max-norm reproduces the same topology yet has cubes as balls).
Once I’ve got lines, the remainder is almost straight forward, using well known properties from synthetic geometry for triangles and quadrilaterals as axioms required to be able to “construct” coordinates and metrics. Yet (a) they all use nothing but incidence relations (b) are satisfied by usual Euclidean Geometries and therefore (c) inherited –locally- by any manifold.
Adding the additional structure of lines, adds to the compatibility-conditions for an Atlas of “charts with lines”. While the topological part remains –coincidence of topology on intersections-, I’m not yet finished of what is required in my case. [Again not such a surprise, taking into account that a Differential Manifold imposes additional compatibility conditions and that –exotic spheres from 4 dimensions onward- there might be more but only one Differential Structure].
Compatibility-Conditions and polish-up may still take some weeks or months, among other because I’m not sure whether going from “simply connected” to “locally path connected” is to weak, that is at least something similar to “star-connected” might be required (yet this as concept already uses “straight lines” so … ).