Embedding 3-manifolds with circle actions in 4-space

Jonathan A. Hillman

Abstract

We give constraints on the Seifert invariants of closed orientable 3-manifolds M which admit fixed-point free circle actions and embed in 4-space. In particular, the generalized Euler invariant of the orbit fibration is determined up to sign by the base orbifold B unless H₁(M) is torsion free, in which case it can take at most one nonzero value (up to sign). No such manifold with base B=S²(a₁,…,a_r) with r odd and Euler invariant 0 embeds in 4-space. [This is an extensive revision of a 1998 Research Report.]

Keywords: embedding. Euler invariant. linking pairing. Seifert bundle.

AMS Subject Classification: Primary 57N10.