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Proceedings of the London Mathematical Society 1998 76(3):685-710; doi:10.1112/S0024611598000379
© 1998 by London Mathematical Society
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© London Mathematical Society

Embedding Seifert Fibred 3-Manifolds and Sol3-Manifolds in 4-Space

JS Crisp and JA Hillman

School of Mathematics and Statistics, The University of Sydney Sydney, NSW 2006, Australia. E-mail: jsc{at}maths.soton.ac.uk, E-mail: jonh{at}maths.nsyd.edu.au

Received 11 June 1996. Revision received 22 April 1997.

We determine strong constraints on the generalized Euler invariants of Seifert bundles over non-orientable base orbifolds which may embed as topologically locally flat submanifolds of S4. In particular, a circle bundle over a non-orientable surface F embeds if and only if it embeds as the boundary of a regular neighbourhood of an embedding of F in S4, and we show that precisely thirteen geometric 3-manifolds with elementary amenable fundamental groups embed. With the exception of the Poincaré homology sphere, each member of the latter class may be obtained by 0-framed surgery on a link which is the union of two slice links, and so embeds smoothly in S4. 1991 Mathematics Subject Classification: 57N13.

Key Words: embedding • hypersurface • Seifert fibred 3-manifold • Sol3-manifold • slice link • surgery

Present address: Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, UK.


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