Vietoris-Rips Metric Thickenings of the Circle

Vietoris-Rips metric thickenings have previously been proposed as an alternate approach to understanding Vietoris-Rips simplicial complexes and their persistent homology. Recent work has shown that for totally bounded metric spaces, Vietoris-Rips metric thickenings have persistent homology barcodes that agree with those of Vietoris-Rips simplicial complexes, ignoring whether endpoints of bars are open or closed. Combining this result with the known homotopy types and barcodes of the Vietoris-Rips simplicial complexes of the circle, the barcodes of the Vietoris-Rips metric thickenings of the circle can be deduced up to endpoints, and conjectures have been made about their homotopy types. We confirm these conjectures are correct, proving that the Vietoris-Rips metric thickenings of the circle are homotopy equivalent to odd-dimensional spheres at the expected scale parameters. Our approach is to find quotients of the metric thickenings that preserve homotopy type and show that the quotient spaces can be described as CW complexes. The quotient maps are also natural with respect to the scale parameter and thus provide a direct proof of the persistent homology of the metric thickenings.