Causal Order Complex and Magnitude Homotopy Type of Metric Spaces

Author:

Tajima Yu1,Yoshinaga Masahiko2

Affiliation:

1. Department of Mathematics, Graduate School of Science, Hokkaido University , North 10, West 8, Kita-ku, Sapporo 060-0810, Japan

2. Department of Mathematics, Graduate School of Science, Osaka University , Toyonaka 560-0043, Japan

Abstract

Abstract In this paper, we construct a pointed CW complex called the magnitude homotopy type for a given metric space $X$ and a real parameter $\ell \geq 0$. This space is roughly consisting of all paths of length $\ell $ 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 $X\times \mathbb{R}$ 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 $X\times \mathbb{R}$ 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.

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

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