Volume distortion in homotopy groups

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.

49 pages, 4 figures. Accepted for publication in Geometric and Functional Analysis (GAFA)

References in corpus (5)

• Generalized Dehn Functions I
• Lipschitz minimality of Hopf fibrations and Hopf vector fields
• Isoperimetric inequalities and rational homotopy invariants
• Directional isoperimetric inequalities and rational homotopy invariants
• Lipschitz minimality of the multiplication maps of unit complex, quaternion and octonion numbers

Cited by in corpus (2)

• Quantitative nullhomotopy and rational homotopy type
• Morse Spectra, Homology Measures and Parametric Packing Problems