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

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

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

Published electronically: August 1, 2002

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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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