Complements of connected hypersurfaces in $S^4$

TL;DR

This paper investigates the topological properties of the complements of embedded 3-manifolds in 4-spheres, focusing on Euler characteristics and fundamental groups, using satellite constructions and Massey products.

Contribution

It introduces new methods to analyze the possible topological invariants of complements of 3-manifolds in $S^4$, including the use of satellite constructions and Massey products.

Findings

Characterizes possible Euler characteristics of complements.

Shows how to alter fundamental groups via satellite constructions.

Limits Euler characteristics using Massey products for certain bundles.

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.