PL structures on combinatorial manifolds

The weak realization of a combinatorial (n + 1)-manifold has a PL atlas modelled on the half-space in (Fin n → ℝ) × ℝ. If every vertex link is a combinatorial n-sphere, it also has a PL atlas without boundary, modelled on ℝ^(n + 1). Both atlases are assembled from charts on the open vertex stars. In the half-space atlas, vertices with sphere links get charts into the open half-space, and vertices with ball links get charts sending them to the boundary. PL compatibility follows by composing the ambient forward and inverse barycentric formulas of the charts on overlapping stars.

No finiteness or countability of the ambient complex is required. Each link is finite, so its closed star supplies the local simplex model.

References

• C. P. Rourke, B. J. Sanderson, Introduction to Piecewise-Linear Topology, Springer (1972), Chapter 3 (the PL structure of a combinatorial manifold).

theorem AbstractSimplicialComplex.exists_chartedSpace_hasGroupoid_of_isCombinatorialSphere_link {ι : Type u_1} [DecidableEq ι] (K : AbstractSimplicialComplex ι) {n : ℕ} (h : ∀ (v : ι), (K.link {v}).IsCombinatorialSphere n) : ∃ (C : ChartedSpace (Fin (n + 1) → ℝ) K.Realization), have this := C; HasGroupoid K.Realization (TauCeti.PLGroupoid (modelWithCornersSelf ℝ (Fin (n + 1) → ℝ)))

Sphere links at every vertex give the weak realization a PL structure without boundary. The atlas is assembled on the existing weak topology, with no global finiteness assumption.

theorem PreAbstractSimplicialComplex.IsCombinatorialManifold.exists_chartedSpace_hasGroupoid_PLGroupoid {ι : Type u_1} [DecidableEq ι] {K : AbstractSimplicialComplex ι} {n : ℕ} (h : K.IsCombinatorialManifold (n + 1)) : ∃ (C : ChartedSpace (↑{q : (Fin n → ℝ) × ℝ | 0 ≤ q.2}) K.Realization), have this := C; HasGroupoid K.Realization (TauCeti.PLGroupoid (TauCeti.modelWithCornersHalfSpace (Fin n → ℝ)))

The weak realization of a positive-dimensional combinatorial manifold is a PL manifold, possibly with boundary: it has an atlas modelled on the half-space in (Fin n → ℝ) × ℝ whose chart transitions lie in the PL groupoid. No finiteness or countability of the complex is assumed.