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: | |
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Priority: | major | Milestone: | sage-9.9 |

Component: | manifolds | Keywords: | |

Cc: | gh-mjungmath, yzh, gh-kliem, egourgoulhon | Merged in: | |

Authors: | Reviewers: | ||

Report Upstream: | N/A | Work issues: | |

Branch: | Commit: | ||

Dependencies: | Stopgaps: |

### Description

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 E^{n+1} \ c to E^{n+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

Milestone: | sage-9.4 → sage-9.5 |
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### comment:2 Changed 14 months ago by

Milestone: | sage-9.5 → sage-9.6 |
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### comment:3 Changed 11 months ago by

Milestone: | sage-9.6 → sage-9.7 |
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### comment:4 Changed 5 months ago by

Milestone: | sage-9.7 → sage-9.8 |
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### comment:5 Changed 4 weeks ago by

Milestone: | sage-9.8 → sage-9.9 |
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