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Abstract. Diagrammatic geometry makes manifold theory a joyous exercise of diagram-drawing. It has far-reaching consequences for how understanding of the interaction between higher algebra, combinatorics, and differential geometry.
Context
This note is a stub to record some note-worthy idea in diagrammatic geometry. (Todo: add some color and references.)
Intro
Diagrammatic geometry is the study of knots, manifolds, and their singularities through manifold diagrams and tame tangles.
Frontiers of knowledge
Classification of space
- Homoeomorphism, PL homeomorphism, and diffeomorphism are undecidable
- Homotopy equivalence is undecidable
- Simply connected homotopy equivalence is decidable
- Corbordism in high dimensions is decidable (This uses Thom-Pontryagin; see the excellent “An introduction to cobordism theory” by Tom Weston)
Classification of manifolds
- Sphere recognition is undecidable
- Manifold recognition is undecidable (“is this space a manifold”)
- (?) Manifold equivalence classes cannot be enumerated (even though, say in the compact case, there are only countably many of them)
- Top and PL are not the same (e.g., the double suspension of a PL homology sphere is not a PL manifold)
- PL and smooth manifolds are not the same (e.g., the cone of an exotic smooth 7-sphere is not a smooth manifold)
The story can be told in TOP, PL, or DIFF, each with their approach notion of equivalence.
Tame facts
- PL structures of manifold strata in manifold diagrams are canonically determined.
- Moreover, they are framed, so they also have canonical smooth structure
Tame conjectures
- Any compact smooth manifold can be represented in a manifold diagrams.
- Diagrammatic structure is a faithful invariant for smooth structure.
- Isotopies (“higher braids”) are classifiable.
- This continues the sequence of Braid, Yang-Baxterator, Zamolodchikov equation, etc.
- Singularities are classifiable and are closely linked to the classical ADE classification of singularities.
Applications
- If we classify generating singularities, then we can enumerate smooth manifolds as tame tangles.
For fun
I once drew a Mazur manifold as a tame tangle.
