Geometric and Functional Analysis

, Volume 26, Issue 2, pp 607–679 | Cite as

Volume distortion in homotopy groups

  • Fedor Manin


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.


Universal Cover Homotopy Group Hyperbolic Group Distortion Function Euler Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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