Taking an open or closed subset on a simplicial complex or more generally a delta set produces new spaces. Not only the intrinsic topological spaces themselves, but interaction spaces. They are given and created by looking at pairs of points which intersect and consder them to be new points. There are point pairs for example where one is in K and one is in U, or then there are point pairs where both are in U and the intersection is in U. The set of all such pairs of some kind now form a new delta set with a cohomology that is still a topological invariant. We look here at an example of a zero knot in a 2-sphere. Eventually, we hope that we can use this cohomologies to distinguish classical knots in 3-space. For the code used to generate the tables on the board, see arxiv.org/abs/2406.17214
I did not get into the Gauss-Bonnet or Atiah-Singer part of the story. The topological index of a delta set is the sum of the curvatures, the analytic index is the Fredholm index dimker D - dimker D* if the Dirac operator D:E to F is considered as a map from the even to the odd dimensional forms. For any delta set and besides G,K,U, we also have KU,UK,UU have a Dirac operator and the general theory applies. I have called in 2017 the Atiah-Singer story arxiv.org/abs/1708.06070v1 blasphemy because in the simplicity in finite dimensions is an insult of the gods who created the theorem in the continuum.
The video footage at the beginning and end was shot on Friday noon. Today was a cloudy day.
4 авг 2024
