A reduced spherical diagram into a ribbon-disk complement and related examples

Abstract

The Whitehead conjecture states, that subcomplexes of aspherical 2-complexes are aspherical. Howie pointed out in [7], that "most of it" would be proved, if the asphericity of ribbon-disk complements were shown, but each non-aspherical ribbon-disk complement is already a counterexample to the Whitehead conjecture. In [8] and [4] various subclasses of ribbon-disk complements are shown to be aspherical. In the last years different notions of asphericity such as "diagrammatically reducible" (DR) or "diagrammatically aspherical" (DA) were defined for 2-complexes and became more and more important for instance in the theory of equations over groups. In this paper I will exhibit examples of non DR and non DA ribbon-disk complements, which answers in the negative the question raised in section 6.20 of [4]. Furthermore the existence of combinatorial reduced surface mappings into ribbon-disk complements is proved. I am grateful to Wolfgang Metzler, who had the "link idea" and others from our seminar, as Cynthia Hog-Angeloni and Günther Huck who inspired me in my work.