A converse of the Jordan-Brouwer theorem for quasi-regular immersions
It is known as the Jordan-Brouwer Theorem that the image of a topological embedding of \(S^{n - 1}\) into \(S^n\) separates \(S^n\) into exactly two connected components. In this paper, the author proves the converse of this theorem for quasi-regular immersions. A \(C^1\)-immersion \(f : M \to N\) into an \(n\)-dimensional manifold \(N\) is quasi-regular if the self-intersection locus \(B \subset f(M)\) is an immersed submanifold of \(N\) with the property that for each \(x \in B\) there exists a coordinate system for \(N\) valid in a neighborhood \(U\) of \(x\) so that \(x\) corresponds to \(0 \in \mathbb{R}^n\) and that the branches of \(f\) in \(U\) correspond to distinct linear subspaces of \(\mathbb{R}^n\); i.e., given a numbering \(y_1, \dots, y_m\) of the points of \(f^{-1} (x)\), there exist pairwise disjoint neighborhoods \(V_i \subset M\) around \(y_i\) so that \(U \cap f(M) = U \cap (\bigcup^m_{i = 1} f(V_i))\) is a union of \(m\) distinct linear subspaces of \(\mathbb{R}^n\). The main result of the paper is the following: Let \(f : M \to N\) be a quasi-regular immersion, where \(M\) is a closed connected \((n - 1)\)-dimensional manifold and \(N\) a connected \(n\)-dimensional manifold. Assume that \(H_1 (M; \mathbb{Z}_2) = 0\) and \(H_1 (N; \mathbb{Z}) = 0\). Then \(f\) is an embedding if and only if \(f(M)\) separates \(N\) into exactly two connected components. Note that an immersion with normal crossings are always quasi-regular and the above result generalizes the results of \textit{C. Biasi}, \textit{W. Motta} and the author [Topology Appl. 52, 81-87 (1993; Zbl 0791.57020); 61, 179-186 (1995; Zbl 0829.57017)] and of the author [Geom. Dedicata 57, 235-247 (1995; Zbl 0833.57007)] about the converse of the Jordan-Brouwer theorem for immersions with normal crossings. Note also that it has already been known that a proper codimension one immersion \(f : M \to N\) separates \(N\) if \(H_1 (N; \mathbb{Z}_2) = 0\) [\textit{M. E. Feighn}, Topology 27, 319-321 (1988; Zbl 0658.57019); \textit{M. D. Hirsch} and \textit{C. C. Pugh}, \(C^1\) immersed hypersurfaces separate \(\mathbb{R}^n\), to appear in Michigan Math. J.].
scientific article; zbMATH DE number 878731