• Otera, Daniele Ettore (Mathematics and Informatics Institute Vilnius University)
  • Received : 2016.04.01
  • Published : 2017.05.01


An open smooth manifold is said of finite geometric rank if it admits a handlebody decomposition with a finite number of 1-handles. We prove that, if there exists a proper submanifold $W^{n+3}$ of finite geometric rank between an open 3-manifold $V^3$ and its stabilization $V^3{\times}B^n$(where $B^n$ denotes the standard n-ball), then the manifold $V^3$ has the Tucker property. This means that for any compact submanifold $C{\subset}V^3$, the fundamental group ${\pi}_1(V^3-C)$ is finitely generated. In the irreducible case this implies that $V^3$ has a well-behaved compactification.


handlebody decomposition;singularities;triangulations;Tucker property


Supported by : Research Council of Lithuania


  1. L. Funar, On proper homotopy type and the simple connectivity at infinity of open 3-manifolds, Atti Sem. Mat. Fis. Univ. Modena 49 (2001), 15-29.
  2. L. Funar and S. Gadgil, On the geometric simple connectivity of open manifolds, Int. Math. Res. Not. 2004 (2004), no. 24, 1193-1248.
  3. L. Funar and D. E. Otera, On the wgsc and qsf tameness conditions for finitely presented groups, Groups Geom. Dyn. 4 (2010), no. 3, 549-596.
  4. R. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Princeton University Press, 1977.
  5. D. E. Otera, On the proper homotopy invariance of the Tucker property, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 3, 571-576.
  6. D. E. Otera, Topological tameness conditions of spaces and groups: results and developments, Lith. Math. J. 56 (2016), no. 3, 357-376.
  7. D. E. Otera, An application of Poenaru's zipping theory, Indag. Math. (N.S.) 27 (2016), no. 4, 1003-1012.
  8. D. E. Otera and V. Poenaru, "Easy" representations and the qsf property for groups, Bull. Belg. Math. Soc. Simon Stevin 19 (2012), no. 3, 385-398.
  9. D. E. Otera and V. Poenaru, Tame combings and easy groups, Forum Math., (to appear).
  10. D. E. Otera and V. Poenaru, Finitely presented groups and the Whitehead nightmare, Groups, Geom., Dyn. (to appear).
  11. D. E. Otera, V. Poenaru, and C. Tanasi, On geometric simple connectivity, Bull. Math. Soc. Sci. Math. Roumanie 53(101) (2010), no. 2, 157-176.
  12. D. E. Otera and F. G. Russo, On the wgsc property in some classes of groups, Mediterr. J. Math. 6 (2009), no. 4, 501-508.
  13. D. E. Otera and F. G. Russo, On topological filtrations of groups, Period. Math. Hungar. 72 (2016), no. 2 (2016), 218-223.
  14. V. Poenaru, On the equivalence relation forced by the singularities of a non-degenerate simplicial map, Duke Math. J. 63 (1991), no. 2, 421-429.
  15. V. Poenaru, Killing handles of index one stably and ${\pi}^{\infty}_1$, Duke Math. J. 63 (1991), no. 2, 431-447.
  16. V. Poenaru, Almost convex groups, Lipschitz combing, and ${\pi}^{\infty}_1$ for universal covering spaces of closed 3-manifolds, J. Differential Geom. 35 (1992), no. 1, 103-130.
  17. V. Poenaru, Equivariant, locally finite inverse representations with uniformly bounded zipping length, for arbitrary finitely presented groups, Geom. Dedicata 167 (2013), 91-121.
  18. V. Poenaru, Geometric simple connectivity and finitely presented groups, Preprint (2014), arXiv:1404.4283 [math.GT].
  19. V. Poenaru and C. Tanasi, Hausdorff combing of groups and ${\pi}^{\infty}_1$ for universal covering spaces of closed 3-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), no. 3, 387-414.
  20. V. Poenaru and C. Tanasi, Some remarks on geometric simple connectivity, Acta Math. Hung. 81 (1998), no. 1-2, 1-12.
  21. V. Poenaru and C. Tanasi, Equivariant, almost-arborescent representations of open simply-connected 3-manifolds; A finiteness result, Mem. Amer. Math. Soc. 169 (2004), no. 800, 88 pp.
  22. L. C. Siebenmann, Les bisections expliquent le theoreme de Reidemeister-Singer, un retour aux sources, Prepublications mathematiques d'Orsay 80T16, 1980.
  23. S. Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. 69 (1959), 327-344.
  24. S. Smale, Generalized Poincare's conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391-406.
  25. S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387-399.
  26. T. W. Tucker, Non-compact 3-manifolds and the missing boundary problem, Topology 13 (1974), 267-273.
  27. J. H. C. Whitehead, Simplicial spaces, nuclei and m-groups, Proc. Lond. Math. Soc. II. Ser. 45 (1939), 243-327.