Nested homotopy models of finite metric spaces and their spectral homology

Abstract

For over a decade, two theories have been actively developed: theory of magnitude and magnitude homology of metric spaces, and GLMY-theory of path homology of directed graphs. Recently Asao showed that for the case of directed graphs there is a unified approach to these theories via a spectral sequence which is now known as the magnitude-path spectral sequence. He also introduced a notion of r-homotopy for directed graphs and proved that the r+1-st page of the spectral sequence is r-homotopy invariant. We extend this theory to the general case of quasimetric spaces that include metric spaces and directed graphs. We show that for a real number r and a finite quasimetric space X there is a unique (up to isometry) r-homotopy equivalent quasimetric space of the minimal possible cardinality. It is called the r-minimal model of X. We use this to construct a decomposition of the magnitude-path spectral sequence of a digraph into a direct sum of spectral sequences with certain properties. We also construct an r-homotopy invariant of a quasimetric space X called spectral homology, that generalizes many other invariants: the pages of the magnitude-path spectral sequence, including path homology, magnitude homology, blurred magnitude homology and reachability homology.

Speaker Intro

Prof. Sergei Ivanov is a mathematician from St. Petersburg, Russia. His research interests include homological algebra, algebraic topology, group theory, simplicial homotopy theory, simplicial groups.