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

Jonathan Hillman (Sydney)


If \(M\) is a closed hypersurface in \(S^4=X\cup_MY\) and \(\beta=\beta_1(M)\) then elementary arguments using Mayer-Vietoris and duality show that \(\chi(X)+\chi(Y)=2\), \(1-\beta\leq\chi(X)\leq1+\beta\) and \(\chi(X)\equiv1-\beta\quad{mod}~(2)\). We shall give examples where these values are all realized, and where some or most are not realizable. If one of the complementary regions \(X\), say, is not simply-connected (e.g., if \(\beta>0\)) then there are infinitely many embeddings with a complementary region having Euler characteristic \(\chi(X)\) but distinct fundamental group. The constructions are in terms of framed link presentations for \(M\) (and 2-knot surgery for the result on \(\pi_1(X)\)); the obstructions are related to the lower central series of \(\pi_1(M)\) variously through an old theorem of Stallings or via the dual notion of Massey product.