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Algebraic versus geometric complexity in homotopy groups

Banff International Research Station for Mathematical Innovation and Discovery

Given a metric space X and an element α ∈ πn(X), how does the minimal geometric complexity of a representative of kα grow as a function of k? If we can find a representative simpler than the ”obvious” one, as quantified by measures such as Lipschitz constant or volume, we say, by analogy to the π1 case, that α is distorted. Asymptotically, distortion functions are topological invariants of nice compact spaces. For such spaces, we discuss various ways in which distortion can arise and establish conditions on X under which homotopy elements are or are not distorted. Methods include rational homotopy theory, filling functions, and L∞ cohomology.

Mathematics; Differential geometry; Manifolds and cell complexes

video/mp4

Author affiliation: University of Chicago

2014-08-06

Attribution-NonCommercial-NoDerivs 2.5 Canada

http://hdl.handle.net/2429/49421

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/2.5/ca/