# zbMATH — the first resource for mathematics

Spherical diagrams and labelled oriented trees. (English) Zbl 1134.57002
The main results of the paper under review are two asphericity criteria for the 2-complex $$K_P$$ associated to a presentation $$P = \langle x_1, \dots, x_n \,| \, R_1, \dots R_m\rangle$$, all of whose relators have the form $$x_ix_j = x_jx_k$$.
Such a presentation $$P$$ is called a LOG presentation, due to the fact that it can be encoded by a labelled oriented graph $$T_P$$, consisting of a vertex for each generator $$x_i$$ and an oriented edge labelled $$x_j$$ from $$x_i$$ to $$x_k$$, for each relator $$x_ix_j = x_jx_k$$. When $$T_P$$ is a tree, $$P$$ is called a LOT presentation. This special case is particularly interesting, since the corresponding LOT 2-complex $$K_P$$ is a spine for the complement of a ribbon disk in $$B^4$$ and its asphericity is asserted by the long standing ribbon disk conjecture.
The criteria proposed by the authors are based on the analysis of the spherical diagrams representing cellular maps $$\phi: S^2 \to K_P$$ (the spherical diagram of such a $$\phi$$ is the 1-skeleton of the cellular decomposition of $$S^2$$, whose 1-cells are labelled and oriented accordingly to $$\phi$$).
Conditions are proposed for a LOG 2-complex $$K_P$$, under which any spherical diagram is reducible, hence $$K_P$$ is aspherical. The key tool is a lemma of J. Stalling, asserting that any directed graph in $$S^2$$ has a source or a sink or bounds a consistently oriented region. If $$K_P$$ is a LOG 2-complex, then no spherical diagram of $$K_P$$ can bound consistently oriented regions. Under certain conditions, the authors show that all the sources and sinks can be eliminated from any reduced spherical diagram of $$K_P$$, by a suitable set of edge inversions, without creating consistently oriented regions. Then such a reduced diagram cannot exist.

##### MSC:
 57M20 Two-dimensional complexes (manifolds) (MSC2010) 57M15 Relations of low-dimensional topology with graph theory 05C05 Trees 05C10 Planar graphs; geometric and topological aspects of graph theory 05C20 Directed graphs (digraphs), tournaments
Full Text: