Boundedness of the topological index of a~mapping as a quasiconformality condition (Q1963379)

Let \(U\) be an open set in \(\mathbb R^n\), \(n\geq 2\), and let \(f\: U\to \mathbb R^n\) be a continuous mapping, \(f\in W^1_{p,\text{loc}}(U)\). Suppose \(f\) is open, isolated, possesses the Luzin property and is differentiable almost everywhere. The author uses the following definitions for the global minimum distortion on the sphere of radius \(r\) and center \(x\), \[ l_f(x,r)=\min_{|\mu-x|=r}|f(\mu)-f(x)|, \] and for the local minimum distortion, \[ \lambda_1(x)=\min_{|\mu|=1,\mu\in\mathbb R^n} |f' (x)\mu|. \] In the article under review, topological properties of mappings are described by means of the global and local minimum distortions. Sharp estimates of the topological index are established. These estimates are applied to studying mappings with bounded distortion. The author introduces the notion of the topological quasiconformality coefficient \(K_T(f)\) and shows that, if \(K_T(f)<2\), then the branching set is empty.