Magnitude homology of metric spaces and order complexes

Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, in particular, metric spaces. The purpose of this paper is to describe group a space terms order complexes posets. In space, an interval (the set points between two chosen points) has natural poset structure, which called poset. Under additional assumptions on sizes $4$-cuts, we show that chain complex can be constructed using tensor products, direct sums degree shifts from We give several applications. First, vanishing higher convex subsets Euclidean space. Second, carry information about diameter hole. Third, construct finite graph whose $3$rd torsion.