Abstract
In 1995 Jean-Claude Hausmann proved that a compact Riemannian manifold X is homotopy equivalent to its Rips complex \({\text {Rips}}(X,r)\) for small values of parameter r. He then conjectured that the connectivity of Rips complexes is a monotone function in r, a statement which has been supported by all known examples up to present. In this paper, we prove that \(S^3\) equipped with a certain Riemannian metric is a counter-example to Hausmann’s conjecture. Our proof combines the Stability Theorem of persistent homology, a persistent version of Hausmann’s Theorem, and an approximation theorem of Ferry and Okun.
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The author would like to thank the referees for very good suggestions that improved the overall quality of the paper.
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Communicated by Shmuel Weinberger.
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Research was supported by Slovenian Research Agency Grant No. N1-0114.
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Virk, Ž. A Counter-Example to Hausmann’s Conjecture. Found Comput Math 22, 469–475 (2022). https://doi.org/10.1007/s10208-021-09510-2
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DOI: https://doi.org/10.1007/s10208-021-09510-2