  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.
