Fibers of Monotone Maps of Finite Distortion

Ilmari Kangasniemi, Jani Onninen

Research output: Contribution to journalArticlepeer-review

Abstract

We study topologically monotone surjective W1,n-maps of finite distortion f: Ω → Ω , where Ω , Ω are domains in Rn, n≥ 2. If the outer distortion function Kf∈Llocp(Ω) with p≥ n- 1 , then any such map f is known to be homeomorphic, and hence the fibers f- 1{ y} are singletons. We show that as the exponent of integrability p of the distortion function Kf increases in the range 1 / (n- 1) ≤ p< n- 1 , then for increasingly many k∈ { 0 , ⋯ , n} depending on p, the k:th rational homology group Hk(f- 1{ y} ; Q) of any reasonably tame fiber f- 1{ y} of f is equal to that of a point. In particular, if p≥ (n- 2) / 2 then this is true for all k∈ { 0 , ⋯ , n}. We also formulate a Sobolev realization of a topological example by Bing of a monotone f: R3→ R3 with homologically non-trivial fibers. This example has Kf∈Lloc1/2-ε(R3) for all ε> 0 , which shows that our result is sharp in the case n= 3.

Original languageEnglish (US)
Article number299
JournalJournal of Geometric Analysis
Volume32
Issue number12
DOIs
StatePublished - Dec 2022

Keywords

  • Conformal cohomology
  • Fiber
  • Homology
  • Mappings of finite distortion
  • MFD
  • Monotone

ASJC Scopus subject areas

  • Geometry and Topology

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