×

zbMATH — the first resource for mathematics

On the asphericity of ribbon disc complements. (English) Zbl 0572.57001
Im Zusammenhang mit Untersuchungen zur Whiteheadschen Asphärizitätsvermutung ist die Frage von Bedeutung, ob Komplemente von Bandscheiben (”ribbon disc complements”) asphärisch sind [s. J. Howie, Topology 22, 475-485 (1983; Zbl 0524.57002)]. In der vorliegenden Arbeit wird einem solchen Komplement ein beschrifteter orientierter Baum zugeordnet, der dessen Homotopietyp bestimmt. Wenn dieser Baum hinreichend einfach ist (”Durchmesser\(\leq 3'')\), wird die Asphärizität bewiesen. Die Untersuchungen werden gegenwärtig auch von S. Gersten (Utah) und A. J. Sieradski (Oregon) weitergeführt.
Reviewer: W.Metzler

MSC:
57M20 Two-dimensional complexes (manifolds) (MSC2010)
55P15 Classification of homotopy type
57M05 Fundamental group, presentations, free differential calculus
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. F. Adams, A new proof of a theorem of W. H. Cockcroft, J. London Math. Soc. 30 (1955), 482 – 488. · Zbl 0064.41503 · doi:10.1112/jlms/s1-30.4.482 · doi.org
[2] J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192 – 195. · Zbl 0131.38301
[3] Kouhei Asano, Yoshihiko Marumoto, and Takaaki Yanagawa, Ribbon knots and ribbon disks, Osaka J. Math. 18 (1981), no. 1, 161 – 174. · Zbl 0462.57011
[4] S. D. Brodskiĭ, Equations over groups and groups with one defining relation, Uspekhi Mat. Nauk 35 (1980), no. 4(214), 183 (Russian).
[5] Ian M. Chiswell, Donald J. Collins, and Johannes Huebschmann, Aspherical group presentations, Math. Z. 178 (1981), no. 1, 1 – 36. · Zbl 0443.20030 · doi:10.1007/BF01218369 · doi.org
[6] Tim Cochran, Ribbon knots in \?\(^{4}\), J. London Math. Soc. (2) 28 (1983), no. 3, 563 – 576. · Zbl 0501.57009 · doi:10.1112/jlms/s2-28.3.563 · doi.org
[7] S. M. Gersten, Conservative groups, indicability, and a conjecture of Howie, J. Pure Appl. Algebra 29 (1983), no. 1, 59 – 74. · Zbl 0513.20017 · doi:10.1016/0022-4049(83)90081-6 · doi.org
[8] C. McA. Gordon, Ribbon concordance of knots in the 3-sphere, Math. Ann. 257 (1981), no. 2, 157 – 170. · Zbl 0451.57001 · doi:10.1007/BF01458281 · doi.org
[9] Joel Hass, The geometry of the slice-ribbon problem, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 1, 101 – 108. · Zbl 0535.57004 · doi:10.1017/S030500410006093X · doi.org
[10] Graham Higman, A finitely generated infinite simple group, J. London Math. Soc. 26 (1951), 61 – 64. · Zbl 0042.02201 · doi:10.1112/jlms/s1-26.1.61 · doi.org
[11] James Howie, On locally indicable groups, Math. Z. 180 (1982), no. 4, 445 – 461. · Zbl 0471.20017 · doi:10.1007/BF01214717 · doi.org
[12] James Howie, Some remarks on a problem of J. H. C. Whitehead, Topology 22 (1983), no. 4, 475 – 485. · Zbl 0524.57002 · doi:10.1016/0040-9383(83)90038-1 · doi.org
[13] James Howie and Hans Rudolf Schneebeli, Homological and topological properties of locally indicable groups, Manuscripta Math. 44 (1983), no. 1-3, 71 – 93. · Zbl 0533.20022 · doi:10.1007/BF01166075 · doi.org
[14] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. · Zbl 0368.20023
[15] Stephen J. Pride, Some finitely presented groups of cohomological dimension two with property (FA), J. Pure Appl. Algebra 29 (1983), no. 2, 167 – 168. · Zbl 0513.20019 · doi:10.1016/0022-4049(83)90105-6 · doi.org
[16] Peter B. Shalen, Infinitely divisible elements in 3-manifold groups, Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), Princeton Univ. Press, Princeton, N.J., 1975, pp. 293 – 335. Ann. of Math. Studies, No. 84. · Zbl 0341.57002
[17] Allan J. Sieradski, Combinatorial isomorphisms and combinatorial homotopy equivalences, J. Pure Appl. Algebra 7 (1976), no. 1, 59 – 95. · Zbl 0345.20038 · doi:10.1016/0022-4049(76)90067-0 · doi.org
[18] John R. Stallings, Surfaces in three-manifolds and nonsingular equations in groups, Math. Z. 184 (1983), no. 1, 1 – 17. · Zbl 0496.57006 · doi:10.1007/BF01162003 · doi.org
[19] S. Young, Contractible \( 2\)-complexes, M. Sc. Dissertation, Cambridge, 1976.
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.