Contraction of areas vs. topology of mappings

TL;DR

This paper investigates the relationship between the topological properties of maps between spheres and their geometric dilation, establishing bounds on how small the dilation can be for non-trivial maps.

Contribution

It constructs explicit examples of non-trivial maps with small dilation and proves lower bounds on dilation for such maps, linking topology and geometric distortion.

Findings

Explicit construction of non-trivial maps with arbitrarily small k-dilation for k > (m+1)/2

Proof that non-trivial maps cannot have arbitrarily small dilation for k ≤ (m+1)/2

Establishes a threshold relating topology and geometric distortion in sphere mappings

Abstract

We construct homotopically non-trivial maps from the unit m-sphere to the unit (m-1)-sphere with arbitrarily small k-dilation for each k greater than (m + 1)/2. We prove that homotopically non-trivial maps from the unit m-sphere to the unit (m-1)-sphere cannot have arbitrarily small k-dilation for k less than or equal to (m + 1)/2.