# Paper: Manifold diagrams and tame tangles

first published

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## Idea

Roughly (and incompletely), the paper contains:

• Sec. 1: A concise recollection of trusses, meshes and tame stratifications (Ch. 2, Ch. 4, Ch. 5, and App. B in the FCT book)
• Sec. 2: A topological definition of manifold diagrams by a simple generalization of conical stratifications to framed spaces; a discussion of PL and smooth properties of manifold diagrams; a fully combinatorial classification of framed homeomorphisms classes of manifold diagrams.
• Sec. 3: A topological definition of tangles; a discussion of their relation to manifold diagrams; a combinatorial classification; work towards using tangles as a foundation for singularity theory (and a empiric discussion of the relation to classical singularity theory); conjectures about how tangles “combinatorialize” smooth strctures.

It does not contain:

• Actual “Geometric” Higher Category Theory (i.e. higher category theory based on framed combinatorial-topological structures)