Browse > Article
http://dx.doi.org/10.4134/JKMS.j160257

GEOMETRIC RANK AND THE TUCKER PROPERTY  

Otera, Daniele Ettore (Mathematics and Informatics Institute Vilnius University)
Publication Information
Journal of the Korean Mathematical Society / v.54, no.3, 2017 , pp. 807-820 More about this Journal
Abstract
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.
Keywords
handlebody decomposition; singularities; triangulations; Tucker property;
Citations & Related Records
연도 인용수 순위
  • Reference
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.   DOI
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.   DOI
6 D. E. Otera, Topological tameness conditions of spaces and groups: results and developments, Lith. Math. J. 56 (2016), no. 3, 357-376.   DOI
7 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.
8 D. E. Otera and V. Poenaru, Tame combings and easy groups, Forum Math., (to appear).
9 D. E. Otera and V. Poenaru, Finitely presented groups and the Whitehead nightmare, Groups, Geom., Dyn. (to appear).
10 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.
11 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.   DOI
12 D. E. Otera and F. G. Russo, On topological filtrations of groups, Period. Math. Hungar. 72 (2016), no. 2 (2016), 218-223.   DOI
13 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.   DOI
14 V. Poenaru, Killing handles of index one stably and ${\pi}^{\infty}_1$, Duke Math. J. 63 (1991), no. 2, 431-447.   DOI
15 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.   DOI
16 V. Poenaru, Equivariant, locally finite inverse representations with uniformly bounded zipping length, for arbitrary finitely presented groups, Geom. Dedicata 167 (2013), 91-121.   DOI
17 V. Poenaru, Geometric simple connectivity and finitely presented groups, Preprint (2014), arXiv:1404.4283 [math.GT].
18 V. Poenaru and C. Tanasi, Some remarks on geometric simple connectivity, Acta Math. Hung. 81 (1998), no. 1-2, 1-12.   DOI
19 D. E. Otera, An application of Poenaru's zipping theory, Indag. Math. (N.S.) 27 (2016), no. 4, 1003-1012.   DOI
20 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.
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.   DOI
24 S. Smale, Generalized Poincare's conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391-406.   DOI
25 S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387-399.   DOI
26 T. W. Tucker, Non-compact 3-manifolds and the missing boundary problem, Topology 13 (1974), 267-273.   DOI
27 J. H. C. Whitehead, Simplicial spaces, nuclei and m-groups, Proc. Lond. Math. Soc. II. Ser. 45 (1939), 243-327.