Quantitative nullhomotopy and the Hopf Invariant

TL;DR

This paper investigates the relationship between the Hopf invariant and the dilations of null-homotopies, demonstrating that nontrivial maps require large dilation unless specific constructions with small dilation are used.

Contribution

It establishes quantitative bounds linking the Hopf invariant to dilation in null-homotopies and constructs explicit examples with small dilation, advancing understanding of homotopy complexity.

Findings

Null-homotopies of maps with nonzero Hopf invariant must have large dilation.

Explicit smooth null-homotopies with arbitrarily small dilation are constructed.

Quantitative bounds connect topological invariants to geometric deformation measures.

Abstract

Let $G: S^{4n-1} \rightarrow S^{2n}$ be a map with nonzero Hopf Invariant. Using the generalized Hopf invariant introduced by Haj\l{}asz, Schikorra and Tyson, we show that any null-homotopy $F: B^{4n} \rightarrow B^{2n+1}$ of $G$ with small $(2n+1)$-dilation must have large $(2n)$-dilation. Conversely, we show that these results are sharp by constructing smooth null-homotopies with arbitrarily small $(2n+1)$-dilation.