**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)

### Geometry & Topology

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

### Jonathan Hillman (Sydney)

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

Please join us for lunch after the talk!

**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.