Homotopically non-trivial maps with small k-dilation

2007/09/09 by Larry Guth, Guth, Larry
Mathematics · #53C23 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Combinatorics #Differential Geometry (math.DG) #Dilation (metric space) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Mathematics #math.DG #msc:53C23

paper · pdf · doi:10.48550/arxiv.0709.1241

7 pages

arxiv created 2007/09/09 · openalex publication_date 2007/09/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We construct homotopically non-trivial maps from S^m to Sⁿ with arbitrarily small 3-dilation for certain pairs (m,n). The simplest example is m=4, n=3. Other examples include arbitrarily large values of m and n. We show that a homotopy class in pi₇(S⁴) can be represented by maps with arbitrarily small 4-dilation if and only if the class is torsion.

Citations

• Metric Structures for Riemannian and Non-Riemannian Spaces
• On the groups J(X)—IV

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