Geometriae Dedicata

, Volume 147, Issue 1, pp 357–387 | Cite as

Combinatorial cell complexes and Poincaré duality

  • Tathagata Basak
Original Paper


We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces, combinatorial cell complexes (or c.c.c). We define orientability, homology and cohomology of c.c.c’s and develop enough algebraic topology in this setting to prove the Poincaré duality theorem for a c.c.c satisfying suitable regularity conditions. The definitions and proofs are completely finitary and combinatorial in nature.


Combinatorial topology Finite topological space Cell complex Homology Orientability Poincaré duality theorem 

Mathematics Subject Classification (2000)

05E25 06A07 06A11 55U05 55N35 55U10 55U15 57P10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alexandroff P.S.: Discrete raume. Mathematiceskii Sbornik (NS) 2, 501–518 (1937)MATHGoogle Scholar
  2. 2.
    Baez J., Dolan J.: Higher-dimensional algebra III: n-categories and the algebra of opetopes. Adv. Math. 135(2), 145–206 (1998)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barmak J.A., Minian E.G.: Simple homotopy types and finite spaces. Adv. Math. 218, 87–104 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cheng, E., Lauda, A.: Higher-Dimensional Categories: An Illustrated Guide Book, Preprint. Available at (2004)
  5. 5.
    Eilenberg S., Zilber J.A.: Semi-simplicial complexes and singular homology. Ann. Math. (2) 51(3), 499–513 (1950)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ewald G., Shephard G.C.: Stellar subdivisions of boundary complexes of convex polytopes. Math. Ann. 210, 7–16 (1974)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley, New York (1994)MATHGoogle Scholar
  8. 8.
    Kozlov, D.: Combinatorial Algebraic Topology. Springer Verlag, Series: Algorithms and computation in mathematics, vol. 21 (2008)Google Scholar
  9. 9.
    May, J.P.: Simplicial Objects in Algebraic Topology. Chicago lectures in mathematics, University of Chicago Press, Chicago (1993)Google Scholar
  10. 10.
    May, J.P.: Finite Topological Spaces. Notes for REU. Available at (2003)
  11. 11.
    May, J.P.: Finite Spaces and Simplicial Complexes. Notes for REU. Available at (2003)
  12. 12.
    McCord M.C.: Singular homology groups and homotopy groups of finite topological spaces. Duke Math. J. 33, 465–474 (1966)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Retakh, V., Serconek, S., Wilson, R.W.: Koszulity of Splitting Algebras Associated with Cell Complexes, arXiv: 0810:1241 (2008)Google Scholar
  14. 14.
    Street R.: The algebra of oriented simplexes. J. Pure Appl. Algebra 49(3), 283–335 (1987)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Wachs, M.: Poset Topology: Tools and Applications. In: Geometric combinatorics, IAS/Park City Math. Ser. 13, Amer. Math. Soc. Providence, RI, pp. 497–615 (2007)Google Scholar
  16. 16.
    Whitehead J.H.C.: Simple homotopy types. Am. J. Math. 72, 1–57 (1950)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.IPMUUniversity of TokyoKashiwaJapan

Personalised recommendations