TITLE: What would the slice-ribbon conjecture imply?
QUESTION [26 upvotes]: What would the slice-ribbon conjecture imply for 4-dimensional topology?
I've heard people speak of the slice-ribbon conjecture as an approach to the 4-dimensional smooth Poincare conjecture, and to the classification of homology 3-spheres which bound homology 4-balls. But I've never understood what they were talking about.

REPLY [6 votes]: I have a question concerning Peter Teichner's answer. Aren't there candidate counterexamples, for instance in the following paper in `Topology and its Applications':
Some well-disguised ribbon knots
Robert E. Gompf, 1 and Katura Miyazakib, , 2
Abstract
For certain knots J in S1 × D2, the dual knot J* in S1 × D2 is defined. Let J(O) be the satellite knot of the unknot O with pattern J, and K be the satellite of J(O) with pattern J*. The knot K then bounds a smooth disk in a 4-ball, but is not obviously a ribbon knot. We show that K is, in fact, ribbon. We also show that the connected sum J(O) # J*(O) is a nonribbon knot for which all known algebraic obstructions to sliceness vanish.