Complements of connected hypersurfaces in \(S^4\)

Jonathan A. Hillman

Abstract

Let \(X\) and \(Y\) be the complementary regions of a closed hypersurface \(M\) in \(S^4=X\cup_MY\). We use the Massey product structure in \(H^*(M;\mathbb{Z})\) to limit the possibilities for \(\chi(X)\) and \(\chi(Y)\). We show also that if \(\pi_1(X)\not=1\) then it may be modified by a 2-knot satellite construction, while if \(\chi(X)\leq1\) and \(\pi_1(X)\) is abelian then \(\beta_1(M)\leq4\) or \(\beta_1(M)=6\). Finally we use TOP surgery to propose a characterization of the simplest embeddings of \(F\times{S^1}\).

Keywords: embedding, Euler characteristic, lower central series, Massey product, satellite, Seifert manifold, surgery.

AMS Subject Classification: Primary 57N13.