Embedding $3$-manifolds with circle actions

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.