Abstract
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.
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Notes
For every vertex, the lengths of the edges incident to that vertex are bounded away from zero (Bridson and Haefliger 1999, Section 1.9).
Not every persistence module with a real-valued filtration parameter has a corresponding persistence diagram, but isomorphic persistence modules have identical persistence diagrams when they are defined. It follows from Chazal et al. (2013), Proposition 5.1) that if X is a totally bounded metric space, one can define a persistence diagram for \({{\,\mathrm{PH}\,}}_i(\mathrm {VR}(X;r);k)\) and for \({{\,\mathrm{PH}\,}}_i(\mathrm {{\check{C}}ech}(X,X';r);k)\) (where \(X\subseteq X'\)).
One might also call this a pseudosemimetric.
The concatenation of these paths may not be injective, but it is not hard to modify the non-injective loop to obtain an (injective) cycle.
This shortest path is unique since there are no cycles of length at most \(2\alpha \le 3\alpha < \ell \).
We make this assumption for simplicity’s sake, even though it can be relaxed.
The case of \(C_3\), a cyclic graph with three unit-length edges, is instructive. Since \(C_3\) is dismantlable we have that \(\mathrm {VR}(V(C_3);r)\) is contractible for any \(r\ge 1\). But since the metric graph \(C_3\) is isometric to a circle of circumference 3, it follows from Adamaszek and Adams (2017) that \(\mathrm {VR}(C_3;r)\) is not contractible for \(0<r<\frac{3}{2}\).
We have switched the roles of L and K for notational convenience.
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Acknowledgements
We are grateful for the Women in Computational Topology (WinCompTop) workshop for initiating our research collaboration. In particular, participant travel support was made possible through the grant NSF-DMS-1619908. This collaborative group was also supported by the American Institute of Mathematics (AIM) Structured Quartet Research Ensembles (SQuaRE) program. EP was supported by the High Performance Data Analytics (HPDA) program at Pacific Northwest National Laboratory. RS is partially supported by the SIMONS Collaboration Grant 318086 and NSF Grant DMS-1854705. BW is partially supported by the NSF Grants DBI-1661375 and IIS-1513616. YW is partially supported by National Science Foundation (NSF) via grants CCF-1526513, CCF-1618247, CCF-1740761, and DMS-1547357. LZ is partially supported by the NSF Grant CDS&E-MSS-1854703.
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Counterexample to Adamaszek et al. (2018, Corollary 9)
Counterexample to Adamaszek et al. (2018, Corollary 9)
We thank Wojciech Chachólski, Alvin Jin, Martina Scolamiero, and Francesca Tombari for the following counterexample to Adamaszek et al. (2018, Corollary 9) in the conference version of this paper. We would also like to mention their forthcoming work on homotopical decompositions of Vietoris–Rips complexes (Chachólski et al. 2020).
As shown in Fig. 8, suppose X consists of 4 points in the shape of a quadrilateral of side lengths 0.5, 0.6, 0.5, and 0.6, and with diagonals of length 1.1. Also, let Y consist of 3 points in the shape of a triangle of side lengths 0.5, 0.5, and 0.6. Their intersection \(A=X\cap Y\) is two points at a distance of 0.6. For \(r=1\) the gluing of the corresponding Vietoris–Rips complexes is homotopy equivalent to a circle since \(\mathrm {VR}(X;r) \simeq S^1\) and \(\mathrm {VR}(Y;r)\) is contractible. However, the Vietoris–Rips complex of the gluing is in fact contractible (a cone with apex the single point in \(Y{\setminus } A\)), and hence not homotopy equivalent to the gluing of the Vietoris–Rips complexes.
A counterexample to Adamaszek et al. (2018, Corollary 9), where metric spaces X, Y, and \(A = X \cap Y\) are denoted in blue, red, and purple, respectively. Distances between each pair of points are indicated on the dashed lines
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Adamaszek, M., Adams, H., Gasparovic, E. et al. On homotopy types of Vietoris–Rips complexes of metric gluings. J Appl. and Comput. Topology 4, 425–454 (2020). https://doi.org/10.1007/s41468-020-00054-y
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DOI: https://doi.org/10.1007/s41468-020-00054-y