Smooth surfaces with non-simply-connected complements

TL;DR

This paper presents methods to construct surfaces in simply-connected 4-manifolds with non-simply-connected complements, including infinitely many inequivalent surfaces with prescribed fundamental groups of their complements.

Contribution

It introduces two new constructions of such surfaces, extending twisted rim surgery and providing examples with arbitrary fundamental groups under certain conditions.

Findings

Constructed surfaces with non-simply-connected complements in simply-connected 4-manifolds.

Produced infinitely many smoothly inequivalent but topologically equivalent surfaces.

Extended the class of fundamental groups realizable as complements of surfaces.

Abstract

We give 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. 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.