Injective labeled oriented trees are aspherical

Abstract

A labeled oriented tree is called injective, if each vertex occurs at most once as an edge label. We show that injective labeled oriented trees are aspherical. The proof uses a new relative asphericity test based on a lemma of Stallings.MSC: 57M20, 57M05, 20F05

An answer to the Whitehead asphericity question

“…There is a large corpus of results which are related to ours and is mostly contained in [4], [5], [8], [10], [14], [18], [19], [20], [22], [23], [24], [25], [26], [30], [32], [16] and [43].…”

Aspherical word labeled oriented graphs and cyclically presented groups

“…See Bogley [1] and Rosebrock [9] for surveys on the Whitehead Conjecture. Recently the authors have shown that injective labeled oriented trees are aspherical [4]. Here we show that certain injective word labeled oriented graphs are aspherical (in fact, diagrammatically reducible, a strong combinatorial version of asphericity).…”

Relative Vertex Asphericity

“…Relative vertex asphericity for pairs of 2-complexes K ⊆ L already appeared in a previous article [9] by the authors, where it was used to establish asphericity of injective LOT-complexes. Other and related notions of relative combinatorial asphericity are in the literature.…”

“…In [12] Huck and Rosebrock proved that prime injective labeled oriented trees satisfy Sieradski's coloring test. In [9] the authors showed that injective labeled oriented trees are aspherical. We strengthen this result here by showing that injective labeled oriented trees satisfy a relative weight test with weights 0 and 1 and hence are VA.…”