#### Volume 8, issue 4 (2008)

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Smooth surfaces with non-simply-connected complements

### Hee Jung Kim and Daniel Ruberman

Algebraic & Geometric Topology 8 (2008) 2263–2287
##### Abstract

We consider two constructions of surfaces in simply-connected $4$–manifolds with non simply-connected complements. One is an iteration of the twisted rim surgery introduced by the first author [Geom. Topol. 10 (2006) 27–56]. We also construct, for any group $G$ satisfying some simple conditions, a simply-connected symplectic manifold containing a symplectic surface whose complement has fundamental group $G$. In each case, we produce infinitely many smoothly inequivalent surfaces that are equivalent up to smooth $s$–cobordism and hence are topologically equivalent for good groups.

##### Keywords
knot, embedded surface, twist-spin, symplectic
Primary: 57R57
Secondary: 57N13