Regular homotopy classes of immersions of 3-manifolds into 5-space

Details Regular homotopy classes of immersions of 3-manifolds into 5-space Regular homotopy classes of immersions of 3-manifolds into 5-space

2001-05-10

arxiv.org/abs/math/0105077v1

Manuscripta Mathematica 108 (2002) pp.13-32

Mathematics

Geometric Topology

16 pages, 1 figure

Scientific paper

10.1007/s002290200251

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 $F_8 : T^3 \hookrightarrow {\bf R}^5$ of the 3-torus $T^3$ with the following properties: (1) $F_0 \sharp h$ is regularly homotopic to $F_8$ for some immersion $h : S^3 \looparrowright {\bf R}^5$, and (2) the immersion $h$ as above cannot be chosen from a regular homotopy class containing an embedding.

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