Dissertation/Thesis Abstract

Asymptotic invariants of homotopy groups
by Manin, Fedor, Ph.D., The University of Chicago, 2015, 103; 3712650
Abstract (Summary)

We study the homotopy groups of a finite CW complex X via constraints on the geometry of representatives of their elements. For example, one can measure the “size” of α ∈ π n (X) by the optimal Lipschitz constant or volume of a representative. By comparing the geometrical structure thus obtained with the algebraic structure of the group, one can define functions such as growth and distortion in πn(X), analogously to the way that such functions are studied in asymptotic geometric group theory.

We provide a number of examples and techniques for studying these invariants, 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 volume distortion functions, and hence also their Lipschitz distortion functions, are linear.

Indexing (document details)
Advisor: Weinberger, Shmuel
Commitee: Calegari, Danny
School: The University of Chicago
Department: Mathematics
School Location: United States -- Illinois
Source: DAI-B 76/11(E), Dissertation Abstracts International
Source Type: DISSERTATION
Subjects: Mathematics
Keywords: Geometric group theory, Homotopy, Quantitative topology, Topology
Publication Number: 3712650
ISBN: 9781321896237
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