On homotopy types of Vietoris–Rips complexes of metric gluings
On homotopy types of Vietoris–Rips complexes of metric gluings
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DOI：
10.1007/s41468-020-00054-y
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发表时间：
2017-12
期刊：
影响因子：
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通讯作者：
Michal Adamaszek;Henry Adams;Ellen Gasparovic;Maria Gommel;Emilie Purvine;R. Sazdanovic;Bei Wang;
中科院分区：
文献类型：
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作者：
Michal Adamaszek;Henry Adams;Ellen Gasparovic;Maria Gommel;Emilie Purvine;R. Sazdanovic;Bei Wang;
研究了度量楔形和的Vietoris-Rips复形和度量胶合。我们证明了具有自然度量的楔形和的Vietoris-Rips复形与Vietoris-Rips复形的楔形和是同伦等价的。我们还提供了当两个度量空间沿着公共等距子集粘合在一起时的推广。作为我们的主要示例，我们推导出沿着足够短的路径粘合在一起的两个度量图的Vietoris-Rips复合体的同伦类型（与输入图中某些循环的长度相比）。因此，我们可以在所有的同调维上描述一类广度量图的Vietoris-Rips复形的持续同调。
We study Vietoris–Rips complexes of metric wedge sums and metric gluings. We show that the Vietoris–Rips complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris–Rips complexes. We also provide generalizations for when two metric spaces are glued together along a common isometric subset. As our main example, we deduce the homotopy type of the Vietoris–Rips complex of two metric graphs glued together along a sufficiently short path (when compared to lengths of certain loops in the input graphs). As a result, we can describe the persistent homology, in all homological dimensions, of the Vietoris–Rips complexes of a wide class of metric graphs.