| Abstract: |
| The Vietoris-Rips (VR) complex of a metric space at a certain scale is an abstract simplicial complex where each k-simplex corresponds to (k+1)-point sets with diameter less than that scale. The pioneering work by Hausmann established that any closed Riemannian manifold is homotopy equivalent to its Vietoris-Rips complex for sufficiently small scales. This fundamental result naturally motivated the finite reconstruction problem of a manifold with a finite Hausdorff close sample, which was answered by Latschev.
For any abstract simplicial complex whose vertex set is a Euclidean subset, its shadow is the union of the convex hulls of its simplices. We consider the homotopy properties of the shadow of VR complexes, along with the canonical projection map from the VR complex to its shadow. The study of the geometric/topological behavior of this projection map is a natural yet non-trivial problem, and the map may have many ``singularities`` which have been partially resolved only in dimensions up to 3. We address the challenge posed by singularities in the shadow projection map by studying systems of the shadow complex using inverse system techniques from shape theory, showing that the limit map exhibits favorable homotopy-theoretic properties. More specifically, leveraging ideas and frameworks from shape theory, we show that in the limit when the scale approaches zero and the sample becomes increasingly dense, the limit map behaves well with respect to homotopy/homology groups when the space is an ANR (absolute neighborhood retract) and admits a metric that satisfies some regularity conditions. This results in Hausmann-type and Latschev-type reconstruction theorems in the limit. |
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