Opened 22 months ago

Last modified 4 weeks ago

#31728 new enhancement

manifolds.Sphere: Make relation to simplicial spheres more concrete

Reported by: mkoeppe Owned by:
Priority: major Milestone: sage-9.9
Component: manifolds Keywords:
Cc: gh-mjungmath, yzh, gh-kliem, egourgoulhon Merged in:
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Report Upstream: N/A Work issues:
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The example manifolds.Sphere defines a method minimal_triangulation, which returns a simplicial sphere (the boundary of the (n+1)-simplex) as an abstract simplicial complex.

We propose to make the relationship between the two objects more concrete by introducing intermediate objects and maps as follows.

Define a geometric realization of the simplicial complex as the polyhedral complex that is the boundary of a geometric (n+1)-simplex (or of any full-dimensional polyhedron).

Define the face_manifold_poset (#31660) of the polyhedron. The poset elements are images of embedded submanifolds (and subsets) of the Euclidean space.

Let c be a point in the interior of the polyhedron.

Define the differentiable map sending En+1 \ c to En+1 \ 0, defined in Cartesian coordinates as x ⟼ (x-c)/|x-c|. It pulls back to differentiable maps on the embedded submanifolds, defining differentiable embeddings into the sphere; and to a continuous map on their union, defining a continuous embedding into the sphere.

Change History (5)

comment:1 Changed 19 months ago by mkoeppe

Milestone: sage-9.4sage-9.5

comment:2 Changed 14 months ago by mkoeppe

Milestone: sage-9.5sage-9.6

comment:3 Changed 11 months ago by mkoeppe

Milestone: sage-9.6sage-9.7

comment:4 Changed 5 months ago by mkoeppe

Milestone: sage-9.7sage-9.8

comment:5 Changed 4 weeks ago by mkoeppe

Milestone: sage-9.8sage-9.9
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