Geometriae Dedicata

, Volume 176, Issue 1, pp 1–9 | Cite as

Every finite complex has the homology of some \({\mathrm{CAT}(0)}\) cubical duality group

  • Raeyong Kim
Original Paper


We prove that every finite connected simplicial complex has the homology of the classifying space for some \({\mathrm{CAT}(0)}\) cubical duality group. More specifically, for any finite simplicial complex \(X\), we construct a locally \({\mathrm{CAT}(0)}\) cubical complex \(T_{X}\) and an acyclic map \(t_{X} : T_{X} \rightarrow X\) such that \(\pi _{1}(T_{X})\) is a duality group.


The Kan–Thurston theorem Duality groups \(\mathrm{CAT}(0)\) cubical complexes 

Mathematics Subject Classification (2010)

55P20 20F67 (57P10, 20J05) 


  1. 1.
    Baumslag, G., Dyer, E., Heller, A.: The topology of discrete groups. J. Pure Appl. Algebra 16(1), 1–47 (1980)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bieri, R.: Homological Dimension of Discrete Groups. Queen Mary College Mathematical Notes. Queen Mary College, Department of Pure Mathematics, London (1981)Google Scholar
  3. 3.
    Brady, N., Meier, I.: Connectivity at infinity for right angled artin groups. Trans. Am. Math. Soc. 353(1), 117–132 (2001)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bridson, M., Haefliger, A.: Metric Spaces of Non-Positive Curvature, Volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1999)Google Scholar
  5. 5.
    Brown, K.S.: Cohomology of Groups. Springer, Berlin (1982)CrossRefMATHGoogle Scholar
  6. 6.
    Hausmann, J.-C.: Every finite complex has the homology of a duality group. Math. Ann. 275(2), 327–336 (1986)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Kan, D., Thurston, W.: Every connected space has the homology of a \(K(\pi,1)\). Topology 15(3), 253–258 (1976)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Kim, R.: Every finite CW-complex is the classifying space for proper bundles of a virtual poincaré duality group., (2012). arXiv:1209.4846Google Scholar
  9. 9.
    Leary, I.: A metric Kan–Thurston theorem. J. Topol. 6(1), 251–284 (2013)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Leary, I., Nucinkis, B.E.: Every CW-complex is a classifying space for proper bundles. Topology 40(3), 539–550 (2001)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Maunder, C.R.F.: A short proof of a theorem of Kan and Thurston. Bull. Lond. Math. Soc. 13(4), 325–327 (1981)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    McDuff, D.: On the classifying spaces of discrete monoids. Topology 18(4), 313–320 (1979)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

Personalised recommendations