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

Saeki, Osamu; Takase, Masamichi

doi:10.1090/S0002-9947-02-03070-2

Spin structures and codimension two embeddings of $3$-manifolds up to regular homotopy
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by Osamu Saeki and Masamichi Takase PDF

Trans. Amer. Math. Soc. 354 (2002), 5049-5061 Request permission

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.

References

T. Ekholm and A. Szűcs, Geometric formulas for Smale invariants of codimension two immersions, to appear in Topology.
Robert E. Gompf and András I. Stipsicz, $4$-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR 1707327, DOI 10.1090/gsm/020
John Forbes Hughes, Bordism and regular homotopy of low-dimensional immersions, Pacific J. Math. 156 (1992), no. 1, 155–184. MR 1182261
John F. Hughes and Paul M. Melvin, The Smale invariant of a knot, Comment. Math. Helv. 60 (1985), no. 4, 615–627. MR 826874, DOI 10.1007/BF02567435
Steve J. Kaplan, Constructing framed $4$-manifolds with given almost framed boundaries, Trans. Amer. Math. Soc. 254 (1979), 237–263. MR 539917, DOI 10.1090/S0002-9947-1979-0539917-X
Michel A. Kervaire, Sur le fibré normal à une sphère immergée dans un espace euclidien, Comment. Math. Helv. 33 (1959), 121–131 (French). MR 105118, DOI 10.1007/BF02565912
Robion C. Kirby, The topology of $4$-manifolds, Lecture Notes in Mathematics, vol. 1374, Springer-Verlag, Berlin, 1989. MR 1001966, DOI 10.1007/BFb0089031
Osamu Saeki, Theory of fibered 3-knots in $S^5$ and its applications, J. Math. Sci. Univ. Tokyo 6 (1999), no. 4, 691–756. MR 1742599
O. Saeki, A. Szűcs and M. Takase, Regular homotopy classes of immersions of $3$-manifolds into $5$-space, Manuscripta Math. 108 (2002), 13–32.
Masamichi Takase, Embeddings of $Z_2$-homology 3-spheres in $\textbf {R}^5$ up to regular homotopy, Pacific J. Math. 193 (2000), no. 1, 249–256. MR 1748191, DOI 10.2140/pjm.2000.193.249
Wu Wen-tsün, On the immersion of $C^{\infty }$-$3$-manifolds in a Euclidean space, Sci. Sinica 13 (1964), 335–336. MR 172302