×

A note on separation properties of codimension-1 immersions with normal crossings. (English) Zbl 0791.57020

Summary: Let \(f: M^{n-1} \to N^ n\) be an immersion with normal crossings between closed connected manifolds. The article is concerned with the problem of separation of \(N\) by \(f(M)\). The main result of this paper is a converse of the Jordan-Brouwer Theorem, under the hypothesis that \(M\) is oriented and \(H_ 1(N;\mathbb{Z}_ 2) = 0\). More precisely, with the above hypothesis, \(f\) is an embedding if and only if \(N - f(M)\) has two connected components.

MSC:

57R40 Embeddings in differential topology
57R42 Immersions in differential topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ballesteros, J. J.N., Sobre la función de bitangencia asociada a una curva genérica en \(R^3\), (Thesis (1991), University of Valencia)
[2] Ballesteros, J. J.N.; Fuster, M. C.R., Separation properties of continuous maps in codimension-1 and geometrical applications, Topology Appl., 46, 107-111 (1992) · Zbl 0787.57012
[3] Biasi, C.; Fuster, M. C.R., A converse of the Jordan-Brouwer theorem, Illinois J. Math., 36, 500-504 (1992) · Zbl 0739.57020
[4] Feighn, M. E., Separation properties of codimension-1 immersions, Topology, 27, 319-321 (1988) · Zbl 0658.57019
[5] Herbert, R. J., Multiple points of immersed manifolds, Mem. Amer. Math. Soc., 34, 250 (1981) · Zbl 0493.57012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.