Paper detail

Volume distortion in homotopy groups

Given a finite metric CW complex $X$ and an element $α\in π_n(X)$, 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.