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

DOI:
https://doi.org/10.1090/S0002-9947-02-03070-2

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.

**1.**T. Ekholm and A. Szucs,*Geometric formulas for Smale invariants of codimension two immersions*, to appear in Topology.**2.**R. E. Gompf and A. I. Stipsicz,*-manifolds and Kirby calculus*, Graduate Studies in Mathematics**20**, American Mathematical Society, Providence, RI, 1999. MR**2000h:57038****3.**J. Hughes,*Bordism and regular homotopy of low-dimensional immersions*, Pacific J. Math.**156**(1992), 155-184. MR**93j:57019****4.**J. Hughes and P. Melvin,*The Smale invariant of a knot*, Comment. Math. Helv.**60**(1985), 615-627. MR**87g:57045****5.**S. J. Kaplan,*Constructing framed**-manifolds with given almost framed boundaries*, Trans. Amer. Math. Soc.**254**(1979), 237-263. MR**82h:57015****6.**M. A. Kervaire,*Sur le fibré normal à une sphère immergée dans un espace euclidien*, Comment. Math. Helv.**33**(1959), 121-131. MR**21:3863****7.**R. Kirby,*The topology of**-manifolds*, Lecture Notes in Math.**1374**, Springer-Verlag, 1989. MR**90j:57012****8.**O. Saeki,*The theory of fibered**-knots in**and its applications*, J. Math. Sci. Univ. Tokyo**6**(1999), 691-756. MR**2001b:57058****9.**O. Saeki, A. Szucs and M. Takase,*Regular homotopy classes of immersions of**-manifolds into**-space*, Manuscripta Math.**108**(2002), 13-32.**10.**M. Takase,*Embeddings of**-homology**-spheres in**up to regular homotopy*, Pacific J. Math.**193**(2000), 249-256. MR**2001c:57030****11.**Wen-Tsün Wu,*On the immersion of**-**-manifolds in a Euclidean space*, Scientia Sinica**13**(1964), 335-336. MR**30:2521**

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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:
https://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