Spin structures and codimension two embeddings of -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 | References | Similar Articles | Additional Information

Abstract: We clarify the structure of the set of regular homotopy classes containing embeddings of a 3-manifold into -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 -manifolds , the following phenomenon occurs: there exists a codimension two immersion of the -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 . This involves introducing and studying an equivalence relation on the set of spin structures on . Their associated -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

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