Complements of connected hypersurfaces in $S^4$

We consider the possible Euler characteristics and fundamental groups of the complementary components $X$ and $Y$ of an embedding of a connected closed 3-manifold $M$ in $S^4$. We use a 2-knot satellite construction to change the fundamental groups, and Massey products to limit the values of $χ(X)$ and $χ(Y)$ when $M$ is the total space of an $S^1$-bundle with orientable base and Euler number 1.

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