On the asphericity of ribbon disc complements

Abstract

The complement of a ribbon n n -disc in the ( n + 2 ) (n + 2) -ball has a 2 2 -dimensional spine which shares some of the combinatorial properties of classical knot complement spines. It is an open question whether such 2 2 -complexes are always aspherical. To any ribbon disc we associate a labelled oriented tree, from which the homotopy type of the complement can be recovered, and we prove asphericity in certain special cases described by conditions on this tree. Our main result is that the complement is aspherical whenever the associated tree has diameter at most 3 3 .