Complements of connected hypersurfaces in $S^4$

Abstract

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 $\chi(X)$ and $\chi(Y)$ when $M$ is the total space of an $S^1$-bundle with orientable base and Euler number 1.