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A Counter-Example to Hausmann’s Conjecture
Foundations of Computational Mathematics ( IF 3 ) Pub Date : 2021-05-04 , DOI: 10.1007/s10208-021-09510-2
Žiga Virk

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.



中文翻译:

豪斯曼猜想的反例

1995年,让-克洛德·豪斯曼(Jean-Claude Hausmann)证明,对于较小的参数r,紧凑的黎曼流形X等于其Rips复数\ {{text {Rips}}(X,r)\)是同伦的。然后,他推测Rips络合物的连通性是r中的单调函数,到目前为止,所有已知示例都支持这种说法。在本文中,我们证明配备了一定黎曼度量的\(S ^ 3 \)是Hausmann猜想的反例。我们的证明结合了持久同源性的稳定性定理,Hausmann定理的持久版本以及Ferry和Okun的近似定理。

更新日期:2021-05-05
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