What would the sliceribbon conjecture imply for 4dimensional topology?
I've heard people speak of the sliceribbon conjecture as an approach to the 4dimensional smooth Poincare conjecture, and to the classification of homology 3spheres which bound homology 4balls. But I've never understood what they were talking about.



I think of the ribbonslice conjecture as a wish that would simplify certain 4D questions. Let me explain this in 3 examples.
It is amazing that there are no proposed counter examples to this conjecture, not even for links. 


I don't know a genuine link to the smooth Poincare conjecture but the link to cobordism for homology spheres is simple. Given a slice disc, construct a branched cover of $D^4$, branched over the slice disc. That gives you a 4manifold bounding the associated branched cover of the knot in $S^3$. I wouldn't describe it as an approach to determining which homology 3spheres bound homology 4balls but it's a natural source of examples, and a linkage. If anything the information seems to flow mostly the other direction. For example, Paolo Lisca's recent paper where he determines precisely which connectsums of lens spaces bound rational homology balls. As a corollary he deduces the order of 2bridge knots in the concordance group of knots in $S^3$. EDIT: Not exactly addressing your question, I think of the sliceribbon conjecture as a primitive 4dimensional knotting problem. Given a slice disc you could ask if it's isotopic to a ribbon disc (if the height function on $D^4$ when restricted to the slice disc has only 1handle and 2handle attachments, in that order). You can mess up a ribbon disc by taking connectsums with 2knots. So modulo connect sums with 2knots is every slice disc isotopic to a ribbon disc? Perhaps that's too much to ask too, so you can ask the sliceribbon problem. 2nd edit: As far as I know, the sliceribbon conjecture has no major consequences. As I describe above, it's more of an ''outermarker'' type of conjecture. It's a measure of how well we understand knotting of 2dimensional things in 4dimensional things. 3rd edit: Here is a type of mild consequence that was pointed out to me recently. In my arXiv preprint on embeddings of 3manifolds in $S^4$ there's Construction 2.9 which creates embeddings of certain 3manifolds $M$ in homotopy 4spheres. The first step is to find a contractible $4$manifold $W$ that bounds the 3manifold $M$, then you double $W$ to get a homotopy $S^4$. If the link used in the construction is a ribbon link, the contractible manifold $W$ admits a handle decomposition with one 0handle, $n$ 1handles and $n$ 2handles (for some $n$) and no higher dimensional handles. So the homotopy $S^4$ constructed that contains $M$ is diffeomorphic to $S^4$ provided the corresponding presentation of $\pi_1 W$ is trivializable by AndrewsCurtis moves (handle slides for the handle presentation). This argument will appear in the next draft of the paper, which should appear before January. 


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 welldisguised 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 4ball, 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. 

