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An example of a five-sheeted exotic covering of \(\mathbb C^ 2\). (English. Russian original) Zbl 1033.32012
Math. Notes 71, No. 4, 486-499 (2002); translation from Mat. Zametki 71, No. 4, 532-547 (2002).
The example described in this paper emerged in connection with the well-known Jacobian conjecture: if a polynomial mapping of \(\mathbb C^2\) to itself is locally invertible, then it is globally invertible. An example of a five-sheeted covering which is a topological counterexample in the same sense as Vitushkin’s example is proposed: There exists a five-sheeted covering \((P,\psi)\) that consists of a connected four-dimensional real manifold \(P\) and its projection \(\psi\) on \(\mathbb C^2\) such that 1) the manifold \(P\) consists of three pairwise disjoint pieces: \(P^\ast\), homeomorphic to \(\mathbb R^4\), and \(F_1^\ast\) and \(F_2^\ast\), homeomorphic to \(\mathbb R^2\); 2) the map \(\psi\) is proper, locally homeomorphic on \(P^\ast\), and has a two-sheeted branching along \(F_1^\ast\) and \(F_2^\ast\); 3) the two-dimensional integer homology group with compact support is isomorphic to \(\mathbb Z^2\); as its generators, one can take two spheres \(S_1\) and \(S_2\) simplicially embedded without self-intersections such that \(S_i\) intersects \(F_i^\ast\) at a single point \((i = 1,2)\), \(S_i \cdot F_i^\ast =1\), and \(S_i\cap F_j^\ast =\varnothing\) for \(i\neq j\); in addition, \(S_1\cdot S_1=-3\), \(S_2\cdot S_2 = -15\), \(S_1\cdot S_2 = -3\), and the values of canonical classes are \(K(S_1)=K(S_2)=1\).
MSC:
32H99 Holomorphic mappings and correspondences
57M12 Low-dimensional topology of special (e.g., branched) coverings
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