Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Volumes of Nullhomotopies in Nilpotent Spaces

Kyle Hansen

TL;DR

This work advances quantitative topology by generalizing the Shadowing Principle to nilpotent spaces through towers of principal $K(G,n)$ fibrations, enabling controlled nullhomotopies of Lipschitz maps into nilpotent targets. By developing nilpotent shadowing and embedding these into Sullivan-minimal models, the authors derive upper bounds on the volumes of nullhomotopies, quantified via the higher-order Dehn function $\mathrm{V\delta}^{n}_{Y}(L)$, with sharp, dimension-specific refinements: $O(L^{2n})$ for simply connected and simple targets, $O(L^{(4c-1)n})$ for $c$-step nilpotent targets, and $O(L^{(c-1)n})$ when the target is coformal and $c$-step nilpotent. The results are complemented by a more precise analysis in the coformal setting, yielding near-sharp bounds and a family of examples that approach these limits, while lower bounds are established in the simply connected case and remain open in broader nilpotent regimes. The paper leverages Sullivan localization, the Shadowing Principle, and a careful study of fibrations to translate bounded algebraic data into bounded geometric maps, ultimately linking quantitative topology with higher-order Dehn-type questions in nilpotent spaces.

Abstract

The Shadowing Principle of Manin has proved a valuable tool for addressing questions of quantitative topology raised by Gromov in the late 1900s. The principle informally provides a way for bounded algebraic maps between differential graded algebras to be translated into nearby genuine maps between their geometric realizations. We extend this principle to finite towers of principal $K(G,n)$ fibrations, and in particular apply this construction to nilpotent spaces. As a specific application of the extended principle, we provide upper bounds on the asymptotic behavior of volumes of nullhomotopies of Lipschitz maps into nilpotent spaces. We further refine these bounds in the case when $c = 1$ to nearly meet those of the simply connected setting. We similarly refine these bounds in the event the target space is coformal, and demonstrate that the bounds in this setting are nearly sharp.

Volumes of Nullhomotopies in Nilpotent Spaces

TL;DR

This work advances quantitative topology by generalizing the Shadowing Principle to nilpotent spaces through towers of principal fibrations, enabling controlled nullhomotopies of Lipschitz maps into nilpotent targets. By developing nilpotent shadowing and embedding these into Sullivan-minimal models, the authors derive upper bounds on the volumes of nullhomotopies, quantified via the higher-order Dehn function , with sharp, dimension-specific refinements: for simply connected and simple targets, for -step nilpotent targets, and when the target is coformal and -step nilpotent. The results are complemented by a more precise analysis in the coformal setting, yielding near-sharp bounds and a family of examples that approach these limits, while lower bounds are established in the simply connected case and remain open in broader nilpotent regimes. The paper leverages Sullivan localization, the Shadowing Principle, and a careful study of fibrations to translate bounded algebraic data into bounded geometric maps, ultimately linking quantitative topology with higher-order Dehn-type questions in nilpotent spaces.

Abstract

The Shadowing Principle of Manin has proved a valuable tool for addressing questions of quantitative topology raised by Gromov in the late 1900s. The principle informally provides a way for bounded algebraic maps between differential graded algebras to be translated into nearby genuine maps between their geometric realizations. We extend this principle to finite towers of principal fibrations, and in particular apply this construction to nilpotent spaces. As a specific application of the extended principle, we provide upper bounds on the asymptotic behavior of volumes of nullhomotopies of Lipschitz maps into nilpotent spaces. We further refine these bounds in the case when to nearly meet those of the simply connected setting. We similarly refine these bounds in the event the target space is coformal, and demonstrate that the bounds in this setting are nearly sharp.

Paper Structure

This paper contains 6 sections, 2 theorems, 4 equations, 1 table.

Table of Contents

  1. Introduction
  2. Summary of Results
  3. Structure of Paper
  4. Acknowledgements
  5. Background and Preliminaries
  6. Nilpotent Spaces

Key Result

theorem 1

If $Y$ is simply connected and $f \colon \mathbf{S}^{n} \to Y$ is $L$-Lipschitz, then one can always find a nullhomotopy whose volume is at most $O(L^{2n})$.

Theorems & Definitions (4)

  • theorem 1: Manin_2019
  • theorem 2: Chambers_MW_2018_quantitative_nullhomotopy
  • definition 2.1
  • definition 2.2