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Surfaces in 3-space that do not lift to embeddings in 4-space. (English) Zbl 0910.57009
Jones, Vaughan F. R. (ed.) et al., Knot theory. Proceedings of the mini-semester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42, 29-47 (1998).
The main result of this paper is a necessary and sufficient condition for an immersion of a surface in 3-space to lift to an embedding in 4-space, in terms of the pre-images of the double and triple points. Several examples and applications are given, including an easy proof of the result of Giller, that the double cover of Boy’s surface cannot be lifted to an embedding. For orientable surfaces an alternative criterion is proved, also in terms of the double point curves. The latter result applies also to generic surfaces, ie ones having branch points in addition to double and triple points. Finally, a construction of unliftable surfaces is given by using surface braids.
For the entire collection see [Zbl 0890.00048].
Reviewer: C.Kearton (Durham)

MSC:
57Q35 Embeddings and immersions in PL-topology
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57N35 Embeddings and immersions in topological manifolds
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