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September 29, 2025Open Access

Anti-Vietoris--Rips metric thickenings and Borsuk graphs

Key Points

The anti-vietoris-rips metric thickening exhibits different topological structures based on radius, showing contractibility for r=0.
A homotopy equivalence exists between anti-vietoris-rips thickenings and real projective space for 2π/3 < r ≤ π.
The covering dimension of anti-vietoris-rips thickenings relates to the packing number, affecting cohomology vanishing.
Borsuk graphs highlight limitations in graph homomorphisms for dimensions exceeding n, indicating chromatic number restrictions.

Abstract

For X a metric space and r 0, the anti-Vietoris-Rips metric thickening AVRᵐ (X;r) is the space of all finitely supported probability measures on X whose support has spread at least r, equipped with an optimal transport topology. We study the anti-Vietoris-Rips metric thickenings of spheres. We have a homeomorphism AVRᵐ (Sⁿ;r) Sⁿ for r >, a homotopy equivalence AVRᵐ (Sⁿ;r) RP^n for 23 n, no graph homomorphism Bor (Sᵏ;r) Bor (Sⁿ;) exists when > 23.

Anti-Vietoris--Rips metric thickenings and Borsuk graphs

Key Points

Abstract

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