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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Notes

  1. 1.

    We thank Manuela Ana Cerdeiro for pointing this example out to us.

References

  1. 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)

  2. 2.

    Bogley, W.A., Pride, S.J.: Calculating generators of \(\pi _2\). In: Hog-Angeloni, C., Metzler, W., Sieradski, A.J. (eds.) Two-dimensional Homotopy and Combinatorial Group Theory. LMS Lecture Note Series 197, CUP (1993)

  3. 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

  4. 4.

    Gersten, S.M.: Branched coverings of 2-complexes and diagrammatic reducibility. Trans. AMS 303(2), 689–706 (1987)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Harlander, J., Rosebrock, S.: Generalized knot complements and some aspherical ribbon disc complements. J. Knot Theory Ramif. 12(7), 947–962 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Howie, J.: Some remarks on a problem of J.H.C. Whitehead. Topology 22, 475–485 (1983)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Howie, J.: On the asphericity of ribbon disc complements. Trans. AMS 289(1), 281–302 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  8. 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)

  9. 9.

    Huck, G., Rosebrock, S.: Aspherical labeled oriented trees and knots. Proc. Edinb. Math. Soc. 44, 285–294 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Kauffman, L.H.: Virtual knot theory. Eur. J. Comb. 20, 663–691 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Rosebrock, S.: The Whitehead conjecture—an overview. Sib. Electron. Math. Rep. 4, 440–449 (2007)

    MathSciNet  MATH  Google Scholar 

  12. 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)

  13. 13.

    Whitehead, J.H.C.: On the asphericity of regions in a 3-sphere. Fund. Math. 32, 149–166 (1939)

    MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Stephan Rosebrock.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Harlander, J., Rosebrock, S. Injective labeled oriented trees are aspherical. Math. Z. 287, 199–214 (2017). https://doi.org/10.1007/s00209-016-1823-6

Download citation

Keywords

  • Labeled oriented tree
  • Wirtinger presentation
  • 2-Complex
  • Asphericity

Mathematics Subject Classification

  • 57M20
  • 57M05
  • 20F05