Vietoris thickenings and complexes of manifolds are homotopy equivalent

Henry Adams, Alexandre Karassev, Ziga Virk

We show that if $X$ is a finite-dimensional Polish metric space, then the natural bijection $\mathrm{VR}(X;r)\to \mathrm{V}{\mathrm{R}}^{\mathrm{m}}(X;r)$ from the (open) Vietoris-Rips complex to the Vi…