The 6-property for simplicial complexes and a combinatorial Cartan-Hadamard theorem for manifolds
HTML articles powered by AMS MathViewer
- by J. M. Corson and B. Trace PDF
- Proc. Amer. Math. Soc. 126 (1998), 917-924 Request permission
Abstract:
The 6-property for 2-dimensional simplicial complexes is the condition that every nontrivial circuit in the link of a vertex has length greater than or equal to six. If a compact $n$-manifold $M$ has a 2-dimensional spine with the 6-property, then we show that the interior of $M$ is covered by euclidean $n$-space. In dimension $n=3$, we show further that such a 3-manifold is Haken.References
- Juan M. Alonso and Martin R. Bridson, Semihyperbolic groups, Proc. London Math. Soc. (3) 70 (1995), no. 1, 56–114. MR 1300841, DOI 10.1112/plms/s3-70.1.56
- Stephen G. Brick and Michael L. Mihalik, The QSF property for groups and spaces, Math. Z. 220 (1995), no. 2, 207–217. MR 1355026, DOI 10.1007/BF02572610
- Morton Brown, The monotone union of open $n$-cells is an open $n$-cell, Proc. Amer. Math. Soc. 12 (1961), 812–814. MR 126835, DOI 10.1090/S0002-9939-1961-0126835-6
- J. Corson and B. Trace, Geometry and algebra of nonspherical 2-complexes, J. London Math. Soc. 54 (1996), 180–198.
- Michael W. Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. (2) 117 (1983), no. 2, 293–324. MR 690848, DOI 10.2307/2007079
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
- M. L. Mihalik and S. T. Tschantz, Tame combings of groups, Trans. Amer. Math. Soc. (to appear).
- V. Poénaru, 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. MR 1152227, DOI 10.4310/jdg/1214447807
- C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69, Springer-Verlag, New York-Heidelberg, 1972. MR 0350744, DOI 10.1007/978-3-642-81735-9
- John R. Stallings, Brick’s quasi-simple filtrations for groups and $3$-manifolds, Geometric group theory, Vol. 1 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 181, Cambridge Univ. Press, Cambridge, 1993, pp. 188–203. MR 1238526, DOI 10.1017/CBO9780511661860.017
- John R. Stallings and S. M. Gersten, Casson’s idea about $3$-manifolds whose universal cover is $\textbf {R}^3$, Internat. J. Algebra Comput. 1 (1991), no. 4, 395–406. MR 1154440, DOI 10.1142/S0218196791000274
Additional Information
- J. M. Corson
- Affiliation: Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, Alabama 35487-0350
- Email: jcorson@mathdept.as.ua.edu
- B. Trace
- Affiliation: Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, Alabama 35487-0350
- Email: btrace@mathdept.as.ua.edu
- Received by editor(s): March 26, 1996
- Received by editor(s) in revised form: September 3, 1996
- Communicated by: Ronald A. Fintushel
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 917-924
- MSC (1991): Primary 57M20, 57N10, 20F06
- DOI: https://doi.org/10.1090/S0002-9939-98-04158-6
- MathSciNet review: 1425116