The covering radius of a mapping and the multiplicity of a covering (Q5932636)

The article is a continuation of the author's study of a geometrical characteristic of a mapping from \(\mathbb R^n\) into \(\mathbb R^n\) [\textit{V.~I.~Semenov}, Sib. Math. J. 40, No. 4, 793-800 (1998); translation from Sib. Mat. Zh. 40, No. 4, 938-946 (1999; Zbl 0932.30018)], wherein, for a continuous a.e. in \(U\) differentiable mapping \(f:U\to \mathbb R^n\), \(f\in W_{p,\text{loc}}^1\), with \(U\subset\mathbb R^n\), the author studies the global least distortion \(l_f(a,r)\) of \(f\) on a sphere with radius \(r\) centered at \(a\) which is defined by the formula \[ l_f(a,r) = \min_{|\mu - a|= r}|f(\mu) - f(a)|. \] The number \(l_f(a,r)\) is called the covering radius of the mapping \(f\). In the above-mentioned article, the author established some estimates for the topological index of a mapping in the case \(p = n\). The aim of the article under review is to find conditions which guarantee the local injectivity property of \(f\) without any openness conditions on the mapping and to obtain an exact estimate for the topological index of open and isolated maps in the case \(p > n\). The proof is based on a theorem by Yu.~G.~Reshetnyak about the local structure of mappings with restricted distortion.