SMS scnews item created by Stephan Tillmann at Tue 26 Apr 2016 1107
Type: Seminar
Distribution: World
Expiry: 26 Jul 2016
Calendar1: 4 May 2016 1200-1300
CalLoc1: Carslaw 535A
CalTitle1: Complements of connected hypersurfaces in $$S^4$$
Auth: tillmann@p710.pc (assumed)

# Complements of connected hypersurfaces in $$S^4$$

### Jonathan Hillman (Sydney)

Wednesday 4 May 2016 from 12:00–13:00 in Carslaw 535A

Abstract: 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.