Surfaces in 4-manifolds and the surgery conjecture

Abstract

We give a survey of geometric approaches to the topological 4-dimensional surgery and 5-dimensional s-cobordism conjectures, with a focus on the study of surfaces in 4-manifolds. The geometric lemma underlying these conjectures is a statement about smooth immersions of disks and of certain 2-complexes, capped gropes, in a 4-manifold. We also mention a reformulation in terms of the A,B-slice problem, and the relation of this question to recent developments in the study of the classical knot concordance group.