2014/10/31 by Fedor Manin
Mathematics · #Algebraic Geometry and Number Theory #Computer science #Distortion (music) #Geometric and Algebraic Topology #Geometry #Homotopy #Homotopy and Cohomology in Algebraic Topology #Homotopy group #Homotopy sphere #Invariant (physics) #Inverse #Mathematical analysis #Mathematical physics #Mathematics #Pure mathematics #Rational function #Regular homotopy #Torsion (gastropod) #math.GT #n-connected
paper · pdf · doi:10.1007/s00039-016-0367-6
published as Geometric and Functional Analysis (GAFA), Vol. 26 Issue 2 (April 2016) pp 607--679 · 49 pages, 4 figures. Accepted for publication in Geometric and Functional Analysis (GAFA)
arxiv created 2016/03/14 · openalex publication_date 2016/04/01 · arxiv updated 2020/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Given a finite metric CW complex X and an element α∈ πn(X), what are the properties of a geometrically optimal representative of α? We study the optimal volume of kα as a function of k. Asymptotically, this function, whose inverse, for reasons of tradition, we call the volume distortion, turns out to be an invariant with respect to the rational homotopy of X. We provide a number of examples and techniques for studying this invariant, with a special focus on spaces with few rational homotopy groups. Our main theorem characterizes those X in which all non-torsion homotopy classes are undistorted, that is, their distortion functions are linear.