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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.
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)
Full Text: DOI
[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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