Embedding 3-manifolds with circle actions

Hillman, J.

doi:10.1090/S0002-9939-09-09985-7

Embedding $3$-manifolds with circle actions
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by J. A. Hillman PDF

Proc. Amer. Math. Soc. 137 (2009), 4287-4294 Request permission

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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