## Abstract

We study topologically monotone surjective W^{1}^{,}^{n}-maps of finite distortion f: Ω → Ω ^{′}, where Ω , Ω ^{′} are domains in R^{n}, 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 K_{f} 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 H_{k}(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: R^{3}→ R^{3} 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 language | English (US) |
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Article number | 299 |

Journal | Journal of Geometric Analysis |

Volume | 32 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2022 |

## Keywords

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

## ASJC Scopus subject areas

- Geometry and Topology