Volume distortion in homotopy groups

TL;DR

This paper investigates the volume distortion of elements in homotopy groups of finite metric CW complexes, revealing invariants related to rational homotopy and characterizing spaces with undistorted non-torsion classes.

Contribution

It introduces the concept of volume distortion as a rational homotopy invariant and characterizes spaces where all non-torsion classes are undistorted.

Findings

Volume distortion is an invariant under rational homotopy.

Spaces with few rational homotopy groups are analyzed.

Main theorem characterizes spaces with linear distortion functions for all non-torsion classes.

Abstract

Given a finite metric CW complex $X$ and an element $\alpha \in \pi_n(X)$, what are the properties of a geometrically optimal representative of $\alpha$? We study the optimal volume of $k\alpha$ 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.