Vietoris-Rips complexes and Borsuk-Ulam theorems

Abstract: Given a metric space X and a scale parameter r, the Vietoris-Rips simplicial complex VR(X;r) has X as its vertex set, and contains a finite subset as a simplex if its diameter is at most r. Vietoris-Rips complexes were invented by Vietoris in order to define a (co)homology theory for metric spaces, and by Rips for use in geometric group theory. More recently, they have found applications in computational topology for approximating of the shape of a dataset. I will explain how the Vietoris-Rips complexes of the circle, as the scale parameter r increases, obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until they are finally contractible. Only very little is understood about the homotopy types of the Vietoris-Rips complexes of the n-sphere. Knowing the homotopy connectivities of Vietoris-Rips complexes of spheres allows one to prove generalizations of the Borsuk-Ulam theorem for maps from the n-sphere into k-dimensional Euclidean space with k > n. Joint work with John Bush and Florian Frick.

algebraic topology

Audience: researchers in the topic

Online algebraic topology seminar

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