# Embedding Seifert manifolds in the 4-sphere

## Author

**John S. Crisp**

## Status

Research Report 94-39

Date: 17 November 1994

## Abstract

The G-signature theorem is used to obtain a relative bound on the rational
Euler invariant of any Seifert manifold M, over a nonorientable closed surface,
which embeds in 4-space. This bound is sharp whenever the singular fibres
of M are all odd and occur in pairs of opposite sign, and in the case of genuine
circle bundles we obtain a complete classification of those which embed.
The torsion linking form is computed explicity for a Seifert manifold M
over a nonorientable closed surface, and conditions are found for this to be hyperbolic,
thereby giving extra necessary conditions for embedding M in the 4-sphere.

## Key phrases

Seifert manifold. embedding. 4-sphere. G-signature theorem. torsion linking.

## AMS Subject Classification (1991)

Primary: 57N10

Secondary: 57N35, 57N13

## Content

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Sydney Mathematics and Statistics