Causal order complex and magnitude homotopy type of metric spaces

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 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. Contents