Abstract
We give geometric formulae which enable us to detect (completely in some cases) the regular homotopy class of an immersion with trivial normal bundle of a closed oriented 3-manifold into 5-space. These are analogues of the geometric formulae for the Smale invariants due to Ekholm and the second author. As a corollary, we show that two embeddings into 5-space of a closed oriented 3-manifold with no 2-torsion in the second cohomology are regularly homotopic if and only if they have Seifert surfaces with the same signature. We also show that there exist two embeddings $F_0$ and of the 3-torus T 3 with the following properties: (1) is regularly homotopic to F 8 for some immersion , and (2) the immersion h as above cannot be chosen from a regular homotopy class containing an embedding.
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Received: 29 March 2001
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Saeki, O., Szűcs, A. & Takase, M. Regular homotopy classes of immersions of 3-manifolds into 5-space. Manuscripta Math. 108, 13–32 (2002). https://doi.org/10.1007/s002290200251
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DOI: https://doi.org/10.1007/s002290200251