24. May 2002
| || Look at the concept of a metric in elementary topology: For a set MMa metric is a real-valued map
MM × MM –––> ℝ ,
fulfilling the three axioms reflexivity, symmetry and the triangle unequality. But, there is missing something, something has been overlooked by the mathematician's community: Stop here and remember, how a metric leads to a topology on M, using systems of neighborhoods. Still no idea? ...Rethink... there is a concept in which metric is only the first of a series of topological concepts, the best notion of which would be finger tip topologies. Still no idea? ....
| Well – an area is a map
MM × MM × MM –––> ℝ,
fulfilling the four (?) axioms: Skew symmetry against permutation of adjacent entries, Jacobi identity, the tetraeder inequality and one (?) more axiom. To define a neighborhood one needs to construct the concept of a point of gravity, around which one can whirl a triangle of given area. The consecutive finger tip topology on MMin this line of argumentation is given by a volume-map
MM × MM × MM × MM –––> ℝ,
subject to five (?) axioms. Thus the category of topological spaces contains also area-, volume-, ... spaces besides the metric ones.
| Examples of the area-concept should by given by symplectic vector spaces ( not necessarily manifolds ), i.e. pairs of a set MMand a skew bilinear form
MM × MM –––> ℝ,
the area being constructed in terms of the symplectic form. This topologises MMentirely in terms of the symplectic form and the area.
Other examples should come from the 2nd cohomology of groups, i.e. short exact sequences
FF >–––> GG –––>> SS
of groups, where there exists a pair of maps Σ and Δ, fulfilling two axioms. There are two well-known special cases, the retract case and the split case, i.e. in group-theoretical language, almost- and semi-direct products ( which when both are fulfilled result in a direct product ). In the split case there is a skew map
SS × SS –––> FF,
usually called σ, which in the case of symplectic vector spaces is the symplectic bilinear form. In this case the resulting group GGis the Heisenberg group of quantum mechanics and the second example of an area map coincides with the first one.
| It would be nice to show that for all short exact sequences the third group SScarries an area map, constructed in terms of sigma and delta. In case the group is a Lie group, there is a 2nd cohomology of its Lie algebra, and there should be an induced area map.|| |
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| ||Literature with comments|
[...] any book on general and metric topology.