License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2018.3
URN: urn:nbn:de:0030-drops-87162
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8716/
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Adamaszek, Michal ; Adams, Henry ; Gasparovic, Ellen ; Gommel, Maria ; Purvine, Emilie ; Sazdanovic, Radmila ; Wang, Bei ; Wang, Yusu ; Ziegelmeier, Lori

Vietoris-Rips and Cech Complexes of Metric Gluings

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LIPIcs-SoCG-2018-3.pdf (0.6 MB)

Abstract

We study Vietoris-Rips and Cech complexes of metric wedge sums and metric gluings. We show that the Vietoris-Rips (resp. Cech) complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris-Rips (resp. Cech) complexes. We also provide generalizations for certain metric gluings, i.e. 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. 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.

BibTeX - Entry

@InProceedings{adamaszek_et_al:LIPIcs:2018:8716,
  author =	{Michal Adamaszek and Henry Adams and Ellen Gasparovic and Maria Gommel and Emilie Purvine and Radmila Sazdanovic and Bei Wang and Yusu Wang and Lori Ziegelmeier},
  title =	{{Vietoris-Rips and Cech Complexes of Metric Gluings}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{3:1--3:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Bettina Speckmann and Csaba D. T{\'o}th},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8716},
  URN =		{urn:nbn:de:0030-drops-87162},
  doi =		{10.4230/LIPIcs.SoCG.2018.3},
  annote =	{Keywords: Vietoris-Rips and Cech complexes, metric space gluings and wedge sums, metric graphs, persistent homology}
}

Keywords: Vietoris-Rips and Cech complexes, metric space gluings and wedge sums, metric graphs, persistent homology
Collection: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 08.06.2018

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