Vietoris-Rips complexes play an important role in geometric topology, geometric group theory, and topological data analysis. For a given scale parameter $r>0$ and a metric space $X$, a Vietoris-Rips complex, $\mathrm{VR}(X,r)$, is defined as a simplicial complex with the vertex set $X$, and so that the simplices are finite collections of points from $X$ of diameter $<r$. One of the main difficulties in working with Vietoris-Rips complexes is that $\mathrm{VR}(X,r)$ is not metrizable unless $X$ is discrete. Moreover, the space $X$, in general, cannot be viewed as naturally embedded in $\mathrm{VR}(X,r)$. To remedy these problems, one can consider so-called metric thickening ${\mathrm{VR}}^m(X,r)$ of $\mathrm{VR}(X,r).$ To this end, we can view $\mathrm{VR}(X,r)$ as a set of all finitely supported measures with diameter of support $<r$, and endow it with the Wasserstein metric. The main focus of this talk will be on the relation between the homotopy types of $\mathrm{VR}(X,r)$ and ${\mathrm{VR}}^m(X,r).$ A recent result by Gillespie implies that $\mathrm{VR}(x,r)$ and ${\mathrm{VR}}^m(X,r)$ are weekly homotopy equivalent. Therefore, to conclude that they are homotopy equivalent it is sufficient to show that ${\mathrm{VR}}^m(X,r)$ is an ANR. It has been previously demonstrated by Adams, Frick, and Virk that ${\mathrm{VR}}^m(X,r)$ is locally contractible. Using different method, we prove that ${\mathrm{VR}}^m(X,r)$ is strongly locally contractible for a compact metric space $X.$ We also show that if such X is finite-dimensional then ${\mathrm{VR}}^m(X,r)$ is an ANR. (Note: this is a joint work with Henry Adams and Ziga Virk).

Co-authors: Henry Adams and Ziga Virk