Complements of connected hypersurfaces in $S^4$

Jonathan Hillman (Sydney)

Abstract

If $M$ is a closed hypersurface in $S^4=X{\cup }_{M}Y$ and $\beta ={\beta }_1(M)$ then elementary arguments using Mayer-Vietoris and duality show that $\chi (X)+\chi (Y)=2$, $1-\beta \le \chi (X)\le 1+\beta$ and $\chi (X)\equiv 1-\beta mod(2)$. 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.