Homotopically nontrivial area contracting maps

Abstract

In this thesis, we investigate the existence of area contracting maps in a given ho-motopy class. In particular, we give various generalizations of Guth’s h-principle for k-dilation of maps between spheres to other manifolds. This includes all maps from highly connected domains and maps to highly connected targets with a homology vanishing condition. We give necessary and sufficient conditions for a homotopy class of maps 𝑋ᵐ → 𝑌ⁿ between closed oriented manifolds to have representatives with small 𝑘-dilation for 𝑘 = 𝑛, 𝑛 − 1 when 𝑘 > (𝑚 + 1)/2.