Causal order complex and magnitude homotopy type of metric spaces

TL;DR

This paper introduces the magnitude homotopy type, a topological space associated with metric spaces, linking it to magnitude homology and providing new tools for analyzing metric space invariants.

Contribution

It constructs the magnitude homotopy type as a CW complex from metric spaces, connecting it to magnitude homology and establishing new invariance results.

Findings

Constructed the magnitude homotopy type as a CW complex.

Established the homotopy type's relation to magnitude homology.

Proved invariance of magnitude under certain metric space transformations.

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.