SMS scnews item created by Stephan Tillmann at Mon 21 Mar 2016 0932
Type: Seminar
Modified: Tue 22 Mar 2016 1635
Distribution: World
Expiry: 20 Jun 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

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.

ball Calendar (ICS file) download, for import into your favourite calendar application
ball UNCLUTTER for printing
ball AUTHENTICATE to mark the scnews item as read
School members may try to .