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Geometric two-dimensional duality groups. (English) Zbl 1347.57005
The present paper deals with connected, finite, aspherical 2-dimensional simplicial complexes which are Cohen-Macaulay, namely complexes \(\Delta\) such that the link of any vertex is non-empty and connected, and the link of any edge is non-empty, too. {
} Since the universal cover of \(\Delta\) is assumed to be contractible, the simplicial complex \(\Delta\) is a finite \(K(G,1)\)-complex, where \(G\) is the group \(\pi_1 (\Delta, v)\), for \(v\) a vertex of \(\Delta\). The main problem is to understand when \(G\) has one end. In such a case the 2-dimensional group \(G\) becomes a duality group, which means that the universal cover \(\widetilde \Delta\) satisfies Poincaré duality. {
} The main result (Theorem 5.6) concerns one-endedness of very general contractible, locally-finite, infinite, 2-dimensional Cohen-Macaulay complexes. As a consequence of this (quite technical) theorem, the author provides necessary and sufficient conditions for \(\widetilde \Delta\) to be one-ended (Corollary 6.13), and simple combinatorial conditions on \(\Delta\) assuring that \(G=\pi_1 (\Delta, v)\) is a 2-dimensional duality group.
MSC:
57M20 Two-dimensional complexes (manifolds) (MSC2010)
20F65 Geometric group theory
55U10 Simplicial sets and complexes in algebraic topology
55U30 Duality in applied homological algebra and category theory (aspects of algebraic topology)
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[1] R. Bieri, Homological dimension of discrete groups . 2nd ed., Queen Mary College Math- ematical Notes, Queen Mary College, Department of Pure Mathematics, London 1981. · Zbl 0357.20027
[2] N. Brady, J. McCammond, and J. Meier, Local-to-asymptotic topology for cocompact CAT.0/ complexes. Topology Appl. 131 (2003), 177-188. · Zbl 1045.20037
[3] R. A.Brualdi, Introductory combinatorics . 5th ed., Pearson Prentice Hall, Upper Saddle River, NJ, 2010. · Zbl 0734.05001
[4] W. Bruns and J. Herzog, Cohen-Macaulay rings . Cambridge Stud. Adv. Math. 39, Cam- bridge University Press, Cambridge 1993. · Zbl 0788.13005
[5] F. J. Fernandez-Lasheras, Fake Surfaces, Thickenings and cohomology of groups. Dis- sertation, SUNY Binghamton, Binghamton 1996.
[6] R. Geoghegan, Topological methods in group theory . Graduate Texts in Math. 243, Springer, New York 2008. · Zbl 1141.57001
[7] P. J. Hilton and U. Stammbach, A course in homological algebra . Graduate Texts in Math. 4, Springer-Verlag, New York-Berlin 1971. · Zbl 0238.18006
[8] J. McCleary, A user’s guide to spectral sequences . 2nd ed., Cambridge Stud. Adv. Math. 58, Cambridge University Press, Cambridge 2001. · Zbl 0959.55001
[9] M. L. Mihalik and S. T. Tschantz, Semistability of amalgamated products and HNN- extensions. Mem. Amer. Math. Soc. 98 (1992), no. 471. · Zbl 0792.20027
[10] C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology . Ergeb. Math. Grenzgeb. 69, Springer-Verlag, Berlin 1972. · Zbl 0477.57003
[11] P. Wright, Formal 3-deformations of 2-polyhedra. Proc. Amer. Math. Soc. 37 (1973), 305-308. · Zbl 0253.57001
[12] E. C. Zeeman, Dihomology III. A generalization of the Poincaré duality for manifolds. Proc. London Math. Soc. (3) 13 (1963), 155-183. · Zbl 0109.41302
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