The 6-property for simplicial complexes

and a combinatorial Cartan-Hadamard theorem

for manifolds

Authors:
J. M. Corson and B. Trace

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 -manifold has a 2-dimensional spine with the 6-property, then we show that the interior of is covered by euclidean -space. In dimension , we show further that such a 3-manifold is Haken.

**[AB]**J. M. Alonso and M. R. Bridson,*Semihyperbolic groups*, Proc. London Math. Soc. (3)**70**(1995), 56-114. MR**95j:20033****[BM]**S. G. Brick and M. L. Mihalik,*The qsf property for groups and spaces*, Math. Z.**220**(1995), 207-217. MR**96i:57009****[B]**M. Brown,*The monotone union of open -cells is an open -cell*, Proc. Amer. Math. Soc.**12**(1961), 812-814. MR**23:A4129****[CT]**J. Corson and B. Trace,*Geometry and algebra of nonspherical 2-complexes*, J. London Math. Soc.**54**(1996), 180-198. CMP**96:14****[D]**M. W. Davis,*Groups generated by reflections and aspherical manifolds not covered by Euclidean space*, Ann. of Math.**117**(1983), 293-324. MR**86d:57025****[LS]**R. C. Lyndon and P. E. Schupp,*Combinatorial group theory*, Ergeb. Math., Bd. 89, Springer, New York, 1977. MR**58:28182****[MT]**M. L. Mihalik and S. T. Tschantz,*Tame combings of groups*, Trans. Amer. Math. Soc. (to appear). CMP**96:12****[P]**V. Poénaru,*Almost convex groups, Lipschitz combing, and for universal covering spaces of closed -manifolds*, J. Differential Geom.**35**(1992), 103-130. MR**93d:57032****[RS]**C. P. Rourke and B. J. Sanderson,*Introduction to piecewise-linear topology*, Ergeb. Math., Bd. 69, Springer, New York, 1972. MR**50:3236****[S]**J. R. Stallings,*Brick's quasi simple filtrations for groups and 3-manifolds*, Geometric group theory (G. A. Niblo and M. A. Roller, eds.), vol. 1, Cambridge University Press, Cambridge, 1993, pp. 188-203. MR**94k:57004****[SG]**J. Stallings and S. M. Gersten,*Casson's idea about -manifolds whose universal cover is*, Internat. J. Algebra Comput.**1**(1991), 395-406. MR**93b:57018**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
57M20,
57N10,
20F06

Retrieve articles in all journals with MSC (1991): 57M20, 57N10, 20F06

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

DOI:
https://doi.org/10.1090/S0002-9939-98-04158-6

Keywords:
Manifold,
spine,
universal cover,
6-property,
collapsing

Received by editor(s):
March 26, 1996

Received by editor(s) in revised form:
September 3, 1996

Communicated by:
Ronald A. Fintushel

Article copyright:
© Copyright 1998
American Mathematical Society