TY - JOUR

T1 - Regular homotopy classes of immersions of 3-manifolds into 5-space

AU - Saeki, Osamu

AU - Szucs, András

AU - Takase, Masamichi

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

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U2 - 10.1007/s002290200251

DO - 10.1007/s002290200251

M3 - Article

AN - SCOPUS:0036999331

SN - 0025-2611

VL - 108

SP - 13

EP - 32

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

IS - 1

ER -