Abstract
We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let \({\mathcal {U}}\) be a cover of a separable metric space X by open sets with a uniform diameter bound. The Vietoris complex \({\mathcal {V}}({\mathcal {U}})\) contains all simplices with vertex set contained in some \(U \in {\mathcal {U}}\), and the Vietoris metric thickening \({\mathcal {V}}^\textrm{m}({\mathcal {U}})\) is the space of probability measures with support in some \(U \in {\mathcal {U}}\), equipped with an optimal transport metric. We show that \({\mathcal {V}}^\textrm{m}({\mathcal {U}})\) and \({\mathcal {V}}({\mathcal {U}})\) have isomorphic homotopy groups in all dimensions. In particular, by choosing the cover \({\mathcal {U}}\) appropriately, we get isomorphisms between the homotopy groups of Vietoris–Rips metric thickenings and simplicial complexes \(\pi _k(\textrm{VR}^\textrm{m}(X;r))\cong \pi _k(\textrm{VR}(X;r))\) for all integers \(k\ge 0\), where both spaces are defined using the convention “diameter \(< r\)” (instead of \(\le r\)). Similarly, we get isomorphisms between the homotopy groups of Čech metric thickenings and simplicial complexes \(\pi _k(\check{\mathrm {{C}}}^\textrm{m}(X;r))\cong \pi _k(\check{\mathrm {{C}}}(X;r))\) for all integers \(k\ge 0\), where both spaces are defined using open balls (instead of closed balls).
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Notes
So long as one ignores whether an endpoint of a bar is open or closed.
For example, when \(r=0\), then \(\textrm{VR}_\le (X;0)\) is X equipped with the discrete topology, whereas \(\textrm{VR}^\textrm{m}_\le (X;0)\) is the metric space X equipped with its standard topology. A less trivial example is that if \(S^1\) is the geodesic circle of circumference \(2\pi \), then \(\textrm{VR}_\le (S^1;\frac{2\pi }{3})\simeq \bigvee ^\infty S^2\) is an uncountably infinite wedge sum of 2-dimensional spheres (Adamaszek and Adams 2017), whereas \(\textrm{VR}^\textrm{m}_\le (S^1;\frac{2\pi }{3})\simeq S^3\) obtains the expected homotopy type of a 3-sphere (Adamaszek et al. 2018; Adams et al. 2020). We say “expected” since we do have \(\textrm{VR}_\le (S^1;r)\simeq S^3\) for all \(\frac{2\pi }{3}<r<\frac{4\pi }{5}\). This entire footnote has analogues for Čech complexes and thickenings, as well.
To see that point-set topology assumptions are needed, consider a cover of a connected space X by two disjoint sets.
As any q-Wasserstein metric for \(1\le q< \infty \) induces the same topology, we make the choice \(q=1\) for convenience.
For example, if \(x,y\in X\) and \(U,U'\in {\mathcal {U}}\) satisfy \(x\in U\cap U'\) and \(y\in U'\setminus U\), then \(\delta _x\in M_U\). Any open ball in \(B_{{\mathcal {V}}^\textrm{m}({\mathcal {U}})}(\delta _x;\varepsilon )\) contains points of the form \((1-\varepsilon ')\delta _x + \varepsilon '\delta _y\notin M_U\) for \(\varepsilon '>0\) sufficiently small. This shows that \(M_U\) is not open.
One choice that suffices is to pick \(s' < \frac{s}{2}\) to furthermore satisfy \(s'<(1-p)\varepsilon \), since then \(\nu \in B_{{\mathcal {V}}^\textrm{m}({\mathcal {U}})}(\mu ;s')\) implies that some transport plan between \(\mu \) and \(\nu \) has cost less than \((1-p)\varepsilon \), which means that less than \(1-p\) of the mass in \(\nu \) can be outside of \(Y_1=\cup _i B_X(y_i;\varepsilon )\).
For example, define \(\phi (x)=\frac{d(x,Y_2^C)}{d(x,Y_1)+d(x,Y_2^C)}\).
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Acknowledgements
The authors would like to thank Robert Cardona for helpful conversations. This research was supported through the program “Research in Pairs” by the Mathematisches Forschungsinstitut Oberwolfach in 2019. The second author was supported by NSF Grant DMS 1855591, NSF CAREER Grant DMS 2042428, and a Sloan Research Fellowship. The third author was supported by Slovenian Research Agency Grants Nos. N1-0114 and P1-0292. The first and third authors would like to thank the Institute of Science and Technology Austria (ISTA) for hosting research visits.
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Adams, H., Frick, F. & Virk, Ž. Vietoris thickenings and complexes have isomorphic homotopy groups. J Appl. and Comput. Topology 7, 221–241 (2023). https://doi.org/10.1007/s41468-022-00106-5
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DOI: https://doi.org/10.1007/s41468-022-00106-5