Spherical diagrams and labelled oriented trees.

*(English)*Zbl 1134.57002The 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.

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.

Reviewer: Riccardo Piergallini (Camerino)