2023/02/20 by Yu Tajima, Masahiko Yoshinaga, Tajima, Yu +1
Computer Science · Mathematics · #51F99 #55N35 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2302.09752
openalex publication_date 2023/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we construct a pointed CW complex called the magnitude homotopy type for a given metric space X and a real parameter ℓ ≥ 0. This space is roughly consisting of all paths of length ℓ and has the reduced homology group that is isomorphic to the magnitude homology group of X. To construct the magnitude homotopy type, we consider the poset structure on the spacetime X×ℝ defined by causal (time- or light-like) relations. The magnitude homotopy type is defined as the quotient of the order complex of an intervals on X×ℝ by a certain subcomplex. The magnitude homotopy type gives a covariant functor from the category of metric spaces with 1-Lipschitz maps to the category of pointed topological spaces. The magnitude homotopy type also has a ``path integral'' like expression for certain metric spaces. By applying discrete Morse theory to the magnitude homotopy type, we obtain a new proof of the Mayer-Vietoris type theorem and several new results including the invariance of the magnitude under sycamore twist of finite metric spaces.