Skip to content
Open
Show file tree
Hide file tree
Changes from all commits
Commits
File filter

Filter by extension

Filter by extension

Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
1 change: 1 addition & 0 deletions TauCetiRoadmap/ConformalMapping/README.md
Original file line number Diff line number Diff line change
Expand Up @@ -137,6 +137,7 @@ deliberately scoped to the conformal-mapping spine that none of them build:

- **Below — `ContourIntegration` (PR #35).** Provides residues, winding numbers, the
argument principle, and the global Cauchy theorem (Dixon). **L0 consumes it.**
- **Beside — `PlanarTopology`** Owns the plane topology of Jordan curves: separation, the crosscut theorem, accessibility, invariance of domain in dimension two, and the Schoenflies theorem. L2 provides that the inside of a Jordan curve is a Jordan domain
- **Above — `ModularForms` (PR #47).** A 13-layer entry targeting the complex modular curves
`Y(Γ)(ℂ) ≅ Γ\ℍ` (the complex-analytic versions, not the moduli-space framing). The
**modular & elliptic uniformization** — including the
Expand Down
6 changes: 3 additions & 3 deletions TauCetiRoadmap/GeometricTopology/README.md
Original file line number Diff line number Diff line change
Expand Up @@ -745,7 +745,7 @@ the disproof needs the genus and the rank side by side.
**From Mathlib / earlier layers.** Layer 1's handlebodies (a handlebody is a boundary-sum
of solid tori, built from layer 1's gluing) and boundary-gluing; `FundamentalGroup`
(`Mathlib/AlgebraicTopology/FundamentalGroupoid/`) and `Group.rank` for the rank side; the
surface (closed orientable 2-manifold) classification for the splitting surface.
surface (closed orientable 2-manifold) classification for the splitting surface from [Surface Topology](../SurfaceTopology/README.md).

**What to build.**
- **Handlebodies** of each genus and the **Heegaard splitting** of a closed 3-manifold as
Expand Down Expand Up @@ -845,15 +845,15 @@ groupoid.
- **Triangulation of a space** in the general sense: `IsTriangulable M := ∃ K, Nonempty (|K| ≃ₜ M)`.
This is the notion in Manolescu's theorem, and it is *weaker* than asking for a
combinatorial or PL structure. (Moise, *Geometric Topology in Dimensions 2 and 3*, treats
the dimension-3 existence, [extract](references/moise-triangulation-dim3.md).)
the dimension-3 existence, [extract](references/moise-triangulation-dim3.md).) The dimension 2 case, Rado's theorem, is handled by the [Planar Topology](../PlanarGeometry/README.md) roadmap.
- **Combinatorial manifolds via the link condition** (Kim's point, the simplicial side of
PL topology). Define, recursively in dimension, a **combinatorial `n`-sphere** and
**combinatorial `n`-ball** (a complex PL-homeomorphic, after subdivision, to the boundary
of the standard `(n+1)`-simplex, respectively the standard `n`-simplex), bottoming out at
the combinatorial `0`-sphere (two points) and `0`-ball (one point). Then `K` is a
**combinatorial `n`-manifold** if the link of every vertex is a combinatorial
`(n-1)`-sphere (interior point) or `(n-1)`-ball (boundary point). This is the definition
that "has the correct links to be a manifold".
that "has the correct links to be a manifold". The generalized maps of the [Surface Topology](../SurfaceTopology/README.md) roadmap give a complementary, dart-based encoding of local manifold structure. In dimension 2 the condition is decidable and recursion is unecessary.
- **The reconciliation with layer 1.** Prove that the polyhedron of a combinatorial
`n`-manifold is a PL `n`-manifold in the sense of layer 1's `PLGroupoid`, and conversely
that every PL manifold admits a combinatorial triangulation (Whitehead). This is the
Expand Down
Loading
Loading