### Abstract

Suppose that N,α,β∈ℕ$N,\alpha ,\beta \in \mathbb {N}$ are satisfying the conditions 2≤α≤N-1$2\le \alpha \le N-1$ and 2≤β≤N-1$2\le \beta \le N-1$. Let us fix p∈[1,min{NN-α+1,N-β+1})$p\in [1,\min \lbrace \frac{N}{N-\alpha +1},N-\beta +1\rbrace )$ and q∈[1,min{NN-β+1,N-α+1})$q\in [1,\min \lbrace \frac{N}{N-\beta +1},N-\alpha +1\rbrace )$. We construct a homeomorphism f:[0,1]N↦[0,1]N$f\colon [0,1]^N\mapsto [0,1]^N$ such that f∈W1,p([0,1]N,ℝN)$f\in W^{1,p}([0,1]^N, \mathbb {R}^N)$, f is the identity on the boundary, all minors of D f of the α-th order are zero almost everywhere, f-1∈W1,q([0,1]N,ℝN)$f^{-1}\in W^{1,q}([0,1]^N, \mathbb {R}^N)$ and all minors of Df-1$Df^{-1}$ of the β-th order are zero almost everywhere. A simplified version of our construction gives a homeomorphism f:[0,1]N↦[0,1]N$f\colon [0,1]^N\mapsto [0,1]^N$ such that f∈W1,p([0,1]N,ℝN)$f\in W^{1,p}([0,1]^N, \mathbb {R}^N)$, f is the identity on the boundary and all minors of D f of the α-th order are zero almost everywhere under a less restrictive assumption p∈[1,NN-α+1)$p\in [1,\frac{N}{N-\alpha +1})$.