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The topological Tverberg theorem states that given any continuous map $fcolonDelta^{(d+1)(r-1)}tomathbb{R}^d$, there are pairwise disjoint faces $sigma_1,ldots,sigma_r$ of $Delta^{(d+1)(r-1)}$ such that $f(sigma_1)capcdotscap f(sigma_r) eemptyset$ whenever $r$ is a prime power. We generalize this theorem to a continuous map from a certain CW complex into a Euclidean space.
Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, in particular, for metric spaces. The purpose of this paper is to describe the magnitude homology group of a metric space in terms of order co
In previous work, we have defined---intrinsically, entirely within the digital setting---a fundamental group for digital images. Here, we show that this group is isomorphic to the edge group of the clique complex of the digital image considered as a
In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris-Rips, Cech and witness complexes) built on top of precompact spaces. Using recent developments in the theory of topological persistence
In this paper, we study Formans discrete Morse theory in the context of weighted homology. We develop weight
We present a detailed description of a fundamental group algorithm based on Formans combinatorial version of Morse theory. We use this algorithm in a classification problem of prime knots up to 14 crossings.