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 F0 and F8 : T3 right arrow-hooked R5 of the 3-torus T3 with the following properties: (1) F0#h is regularly homotopic to F8 for some immersion h : S3 right arrow, looped R5, and (2) the immersion h as above cannot be chosen from a regular homotopy class containing an embedding.
!!!All Science Journal Classification (ASJC) codes
- 数学 (全般)