The author gives a counterexample to a conjecture, posed by Vitushkin, connected with the Jacobian conjecture in \(\mathbb{R}^2\). Namely, he constructs a two-dimensional manifold \(M\) comprising one open two-dimensional cell \(N\) (homeomorphic to \(\mathbb{R}^2\)) and three open one-dimensional cells \(R\) (homeomorphic to \(\mathbb{R}\)) and a mapping \(F:M\to\mathbb{R}^2\) such that: 1. \(F\) is locally homeomorphic on \(N\); 2. in a neighbourhood of each \(P\in R\) there exists a coordinate system in which \(F\) is given by \(F(x,y)=(x,y^k)\) with \(k=1\) or \(k=2\); 3. \(F\) maps each \(R\) into a curve in \(\mathbb{R}^2\) that tends to infinity along the same asymptotic direction as a point of \(R\) tends to the ends of \(R\).