Vietoris thickenings and complexes are weakly homotopy equivalent

Abstract

Characterizing the homotopy types of the Vietoris–Rips complexes ${{\,\textrm{VR}\,}}(X;r)$ of a metric space X is in general a difficult problem. The Vietoris–Rips metric thickening ${{\,\textrm{VR}\,}}^m(X;r)$, a metric space analogue of ${{\,\textrm{VR}\,}}(X;r)$, was introduced as a potentially more amenable object of study with several advantageous properties, yet the relationship between its homotopy type and that of ${{\,\textrm{VR}\,}}(X;r)$ was not fully understood. We show that for any metric space X and threshold $r>0$, the natural bijection $|{{{\,\textrm{VR}\,}}(X;r)}|\rightarrow {{\,\textrm{VR}\,}}^m(X;r)$ between the (open) Vietoris–Rips complex and Vietoris–Rips metric thickening is a weak homotopy equivalence.