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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spin structures and codimension two embeddings of $3$-manifolds up to regular homotopy


Authors: Osamu Saeki and Masamichi Takase
Journal: Trans. Amer. Math. Soc. 354 (2002), 5049-5061
MSC (2000): Primary 57R42, 57M50; Secondary 57R40, 57M27
Published electronically: August 1, 2002
MathSciNet review: 1926849
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Abstract: We clarify the structure of the set of regular homotopy classes containing embeddings of a 3-manifold into $5$-space inside the set of all regular homotopy classes of immersions with trivial normal bundles. As a consequence, we show that for a large class of $3$-manifolds $M^3$, the following phenomenon occurs: there exists a codimension two immersion of the $3$-sphere whose double points cannot be eliminated by regular homotopy, but can be eliminated after taking the connected sum with a codimension two embedding of $M^3$. This involves introducing and studying an equivalence relation on the set of spin structures on $M^3$. Their associated $\mu$-invariants also play an important role.


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Additional Information

Osamu Saeki
Affiliation: Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
Address at time of publication: Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan
Email: saeki@math.sci.hiroshima-u.ac.jp, saeki@math.kyushu-u.ac.jp

Masamichi Takase
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
Email: takase@ms.u-tokyo.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03070-2
PII: S 0002-9947(02)03070-2
Received by editor(s): May 25, 2001
Published electronically: August 1, 2002
Additional Notes: The first author was partially supported by Grant-in-Aid for Scientific Research No. 13640076, Ministry of Education, Science and Culture, Japan.
Article copyright: © Copyright 2002 American Mathematical Society