Fibers of Monotone Maps of Finite Distortion

We study topologically monotone surjective W^1,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 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: 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.