For any abstract simplicial complex $K$ with the vertex set $K^{(0)}$ a Euclidean subset, its shadow, denoted $sh(K)$, is the union of the convex hulls of simplices of $K$.

We consider the homotopy properties of the shadow of Vietoris–Rips complexes $K=Rips_\beta(X)$ with vertices from $\mathbb{R}^N$, along with the canonical projection map $p\colon Rips_\beta (X) \to sh(Rips_\beta(X))$.

The study of the geometric/topological behavior of $p$ is a natural yet non-trivial problem. The map $p$ may have many “singularities”, which have been partially resolved only in low dimensions $N\leq 3$.

The obstacle naturally leads us to study systems of these complexes {$sh(Rips_{\beta}(S)) \mid \beta > 0, S\subset X$}. 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 “$\beta \to 0$ and $S \to X$”, the limit map “$\lim p$” behaves well with respect to homotopy/homology groups when $X$ is an ANR (Absolute Neighborhood Retract) and admits a metric that satisfies some regularity conditions. This results in limit theorems concerning the homotopy properties of systems of these complexes as the proximity scale parameter approaches zero and the sample set approaches the underlying space (e.g., a submanifold or Euclidean graph).

This is joint work with Kazuhiro Kawamura and Sushovan Majhi.

https://doi.org/10.48550/arXiv.2601.01359