# zbMATH — the first resource for mathematics

Injective labeled oriented trees are aspherical. (English) Zbl 1381.57002
A labeled oriented tree, or LOT, is an oriented graph without cycles, additionally equipped with a labeling map from its set of edges to its set of vertices. In this way, every edge of a LOT has a source, a target, and a label. Associated to every LOT $$\Gamma$$ there is a group presentation $$P(\Gamma)$$, and its corresponding standard 2-complex $$K_{\Gamma}$$, which is called a LOT complex. LOTs are combinatorial models of ribbon disc complements, generalizing the Wirtinger presentations for classical knots. Knot complements have been proved to be aspherical (see [C. D. Papakyriakopoulos, Ann. Math. (2) 66, 1–26 (1957; Zbl 0078.16402)]) and ribbon disc complements are conjectured to be aspherical as well. This problem is an important case of the Whitehead conjecture, which states that subcomplexes of aspherical 2-complexes are aspherical (see [J. Howie, Topology 22, 475–485 (1983; Zbl 0524.57002); S. Rosebrock, Sib. Èlektron. Mat. Izv. 4, 440–449 (2007; Zbl 1299.57005)]).
The authors introduce a relative version of the notion of vertex asphericity for 2-complexes, and prove that if a 2-complex $$K$$ is vertex aspherical relative to a subcomplex $$K_0$$ and $$K_0$$ is aspherical, then $$K$$ is aspherical. They also provide a test for relative vertex asphericity, based on a result of Stallings. Using these notions, the main result of the article is proved: injective LOTs are aspherical. A LOT is called injective if every vertex occurs at most once as an edge label. Injective LOTs are the analogues, in this context, to alternating knots. They also prove that an inclusion of injective LOTs induces an injective map on their fundamental groups. Finally the authors extend their main result to the non-injective case.

##### MSC:
 57M20 Two-dimensional complexes (manifolds) (MSC2010) 57M35 Dehn’s lemma, sphere theorem, loop theorem, asphericity (MSC2010) 57M05 Fundamental group, presentations, free differential calculus 20F05 Generators, relations, and presentations of groups
Full Text:
##### References:
 [1] Bogley, W.A.: J.H.C. Whitehead’s asphericity question. In: Hog-Angeloni, C., Metzler, W., Sieradski, A.J. (eds.) Two-dimensional Homotopy and Combinatorial Group Theory. LMS Lecture Note Series 197, CUP (1993) · Zbl 0811.57008 [2] Bogley, W.A., Pride, S.J.: Calculating generators of $$π _2$$. In: Hog-Angeloni, C., Metzler, W., Sieradski, A.J. (eds.) Two-dimensional Homotopy and Combinatorial Group Theory. LMS Lecture Note Series 197, CUP (1993) · Zbl 0811.57005 [3] Gersten, S.M.: Reducible diagrams and equations over groups. In: Gersten editor, S.M. (ed.) Essays in Group Theory, Mathematical Sciences Research Institute Publications. Springer, New York, pp. 15-73 · Zbl 0644.20024 [4] Gersten, SM, Branched coverings of 2-complexes and diagrammatic reducibility, Trans. AMS, 303, 689-706, (1987) · Zbl 0644.20023 [5] Harlander, J; Rosebrock, S, Generalized knot complements and some aspherical ribbon disc complements, J. Knot Theory Ramif., 12, 947-962, (2003) · Zbl 1053.57005 [6] Howie, J, Some remarks on a problem of J.H.C. Whitehead, Topology, 22, 475-485, (1983) · Zbl 0524.57002 [7] Howie, J, On the asphericity of ribbon disc complements, Trans. AMS, 289, 281-302, (1985) · Zbl 0572.57001 [8] Huck, G., Rosebrock, S.: Weight tests and hyperbolic groups. In: Duncan, A., Gilbert, N., Howie, J. (eds.) Combinatorial and Geometric Group Theory; Edinburgh 1993; Cambridge University Press; London Mathematical Society Lecture Notes Series 204, pp. 174-183 (1995) · Zbl 1299.57005 [9] Huck, G; Rosebrock, S, Aspherical labeled oriented trees and knots, Proc. Edinb. Math. Soc., 44, 285-294, (2001) · Zbl 0983.57003 [10] Kauffman, LH, Virtual knot theory, Eur. J. Comb., 20, 663-691, (1999) · Zbl 0938.57006 [11] Rosebrock, S, The Whitehead conjecture—an overview, Sib. Electron. Math. Rep., 4, 440-449, (2007) · Zbl 1299.57005 [12] Stallings, J.: A graph-theoretic lemma and group-embeddings. In: Gersten, J.R., Stallings, S.M. (eds.) Proceedings of the Alta Lodge 1984, Annals of Mathematical Studies, Princeton University Press, pp. 145-155 (1987) · JFM 65.1442.01 [13] Whitehead, JHC, On the asphericity of regions in a 3-sphere, Fund. Math., 32, 149-166, (1939) · Zbl 0021.16203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.