Embedding $3$-manifolds with circle actions
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- by J. A. Hillman
- Proc. Amer. Math. Soc. 137 (2009), 4287-4294
- DOI: https://doi.org/10.1090/S0002-9939-09-09985-7
- Published electronically: July 16, 2009
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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.References
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Bibliographic Information
- J. A. Hillman
- Affiliation: School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
- Email: jonh@maths.usyd.edu.au
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2538589