On manifold spines and cyclic presentations of groups

Alberto Cavicchioli; Friedrich Hegenbarth; Dušan Repovš

Banach Center Publications (1998)

  • Volume: 42, Issue: 1, page 49-56
  • ISSN: 0137-6934

Abstract

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This is a survey of results and open problems on compact 3-manifolds which admit spines corresponding to cyclic presentations of groups. We also discuss questions concerning spines of knot manifolds and regular neighborhoods of homotopically PL embedded compacta in 3-manifolds.

How to cite

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Cavicchioli, Alberto, Hegenbarth, Friedrich, and Repovš, Dušan. "On manifold spines and cyclic presentations of groups." Banach Center Publications 42.1 (1998): 49-56. <http://eudml.org/doc/208824>.

@article{Cavicchioli1998,
abstract = {This is a survey of results and open problems on compact 3-manifolds which admit spines corresponding to cyclic presentations of groups. We also discuss questions concerning spines of knot manifolds and regular neighborhoods of homotopically PL embedded compacta in 3-manifolds.},
author = {Cavicchioli, Alberto, Hegenbarth, Friedrich, Repovš, Dušan},
journal = {Banach Center Publications},
keywords = {group presentation},
language = {eng},
number = {1},
pages = {49-56},
title = {On manifold spines and cyclic presentations of groups},
url = {http://eudml.org/doc/208824},
volume = {42},
year = {1998},
}

TY - JOUR
AU - Cavicchioli, Alberto
AU - Hegenbarth, Friedrich
AU - Repovš, Dušan
TI - On manifold spines and cyclic presentations of groups
JO - Banach Center Publications
PY - 1998
VL - 42
IS - 1
SP - 49
EP - 56
AB - This is a survey of results and open problems on compact 3-manifolds which admit spines corresponding to cyclic presentations of groups. We also discuss questions concerning spines of knot manifolds and regular neighborhoods of homotopically PL embedded compacta in 3-manifolds.
LA - eng
KW - group presentation
UR - http://eudml.org/doc/208824
ER -

References

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  1. [1] B. G. Casler, An embedding theorem for connected 3-manifolds with boundary, Proc. Amer. Math. Soc. 16 (1965), 559-566. Zbl0129.15801
  2. [2] A. Cavicchioli, Imbeddings of polyhedra in 3-manifolds, Annali di Mat. Pura ed Appl. 162 (1992), 157-177. Zbl0777.57003
  3. [3] A. Cavicchioli, Neuwirth manifolds and colourings of graphs, Aequationes Math. 44 (1992), 168-187. Zbl0772.57021
  4. [4] A. Cavicchioli and F. Hegenbarth, Knot manifolds with isomorphic spines, Fund. Math. 145 (1994), 79-89. Zbl0832.57007
  5. [5] A. Cavicchioli, F. Hegenbarth and A. C. Kim, A geometric study of Sieradski groups, Algebra Colloq. 5 (1998), to appear. Zbl0902.57023
  6. [6] A. Cavicchioli, W. B. R. Lickorish and D. Repovš, On the equivalent spines problem, Boll. Un. Mat. Ital., to appear. Zbl0893.57013
  7. [7] A. Cavicchioli and D. Repovš, Peripheral acyclicity and homology manifolds, Annali di Mat. Pura ed Appl. 172 (1997), 5-24. Zbl0931.57019
  8. [8] A. Cavicchioli and F. Spaggiari, The classification of 3-manifolds with spines related to Fibonacci groups, in 'Algebraic Topology-Homotopy and Group Cohomology', Lect. Notes in Math., Springer Verlag 1509 (1992), 50-78. Zbl0752.57007
  9. [9] H. Helling, A. C. Kim and J. L. Mennicke, A geometric study of Fibonacci groups, Preprint Universitat Bielefeld 343 (1990). Zbl0896.20026
  10. [10] H. M. Hilden, M. T. Lozano and J. M. Montesinos, The arithmeticity of the figure eight knot orbifolds, in 'Topology '90', Walter de Gruyter Ed., Berlin - New York (1992), 169-183. Zbl0767.57004
  11. [11] C. Hog-Angeloni, W. Metzler and A. J. Sieradski, Two-dimensional homotopy and combinatorial group theory, London Math. Soc. Lect. Note Ser. 197, Cambridge Univ. Press, Cambridge, 1993. Zbl0788.00031
  12. [12] J. Howie, Cyclic presentations and (2,2k+1) torus knots, unpublished. 
  13. [13] A. C. Kim, On the Fibonacci group and related topics, Contemporary Math. 184 (1995), 231-235. Zbl0839.20046
  14. [14] D. L. Johnson and W. K. Odoni, Some results on symmetrically presented groups, Proceed. Edinburgh Math. Soc. 37 (1994), 227-237. Zbl0835.20044
  15. [15] A. Mednykh and A. Vesnin, Hyperbolic volumes of Fibonacci manifolds, Siberian Math. J. 36 (1995), 235-245. Zbl0865.57012
  16. [16] J. L. Mennicke, On Fibonacci groups and some other groups, Proceed. of Groups-Korea 1988, Pusan, August 1988, 117-123. 
  17. [17] J. Milnor, On the 3-dimensional Brieskorn manifolds M(p,q,r), in 'Knots, Groups and 3-Manifolds' (L. P. Neuwirth Ed.), Ann. of Math. Studies 84, Princeton Univ. Press, Princeton, N. J., 1975, 175-225. 
  18. [18] W. J. R. Mitchell, J. H. Przytycki and D. Repovš, On spines of knot spaces, Bull. Polish Acad. Sci. 37 (1989), 563-565. Zbl0758.57006
  19. [19] L. Neuwirth, An algorithm for the construction of 3-manifolds from 2-complexes, Proc. Camb. Phil. Soc. 64 (1968), 603-613. Zbl0162.27603
  20. [20] D. Repovš, Regular neighbourhoods of homotopically PL embedded compacta in 3-manifolds, Suppl. Rend. Circ. Mat. Palermo 18 (1988), 415-422. Zbl0649.57020
  21. [21] P. Scott, The Geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401-487. Zbl0561.57001
  22. [22] A. J. Sieradski, Combinatorial squashings, 3-manifolds, and the third homotopy of groups, Invent. Math. 84 (1986), 121-139. Zbl0604.57001
  23. [23] R. Thomas, On a question of Kim concerning certain group presentations, Bull. Korean Math. Soc. 28 (1991), 219-244. Zbl0752.20013

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