# zbMATH — the first resource for mathematics

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
Full Text: