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Embedding $ 3$-manifolds with circle actions


Author: J. A. Hillman
Journal: Proc. Amer. Math. Soc. 137 (2009), 4287-4294
MSC (2000): Primary 57N10; Secondary 57N13
DOI: https://doi.org/10.1090/S0002-9939-09-09985-7
Published electronically: July 16, 2009
MathSciNet review: 2538589
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Abstract: Constraints on the Seifert invariants of orientable 3-manifolds $ M$ which admit fixed-point free $ S^1$-actions and embed in $ \mathbb{R}^4$ are given. In particular, the generalized Euler invariant of the orbit fibration is determined up to sign by the base orbifold $ B$ unless $ H_1(M;\mathbb{Z})$ is torsion free, in which case it can take at most one nonzero value (up to sign). An $ \mathbb{H}^2\times\mathbb{E}^1$-manifold $ M$ with base orbifold $ B=S^2(\alpha_1,\dots,\alpha_r)$ where all cone point orders are odd embeds in $ \mathbb{R}^4$ if and only if its Seifert data $ S$ is skew-symmetric.


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Additional Information

J. A. Hillman
Affiliation: School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
Email: jonh@maths.usyd.edu.au

DOI: https://doi.org/10.1090/S0002-9939-09-09985-7
Keywords: Embedding, Euler invariant, linking pairing, Seifert bundle
Received by editor(s): January 19, 2009
Received by editor(s) in revised form: April 2, 2009
Published electronically: July 16, 2009
Additional Notes: This paper began as a 1998 University of Sydney Research Report, but the main result was obtained while the author was visiting the University of Durham as the Grey College Mathematics Fellow for Michaelmas Term of 2008.
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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