Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On cohomology algebras of complex subspace arrangements

Authors: Eva Maria Feichtner and Günter M. Ziegler
Journal: Trans. Amer. Math. Soc. 352 (2000), 3523-3555
MSC (2000): Primary 52C35, 55N45; Secondary 05B35, 51D25, 57N80
Published electronically: March 2, 2000
MathSciNet review: 1694288
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space. We describe simplicial cochains that generate the cohomology. Among them we distinguish a linear basis, study cup product multiplication, and derive an algebra presentation in terms of generators and relations.

References [Enhancements On Off] (What's this?)

  • [Bj1] A. Björner: Homology and shellability of matroids and geometric lattices; in: Matroid Applications (N. White, ed.); Cambridge University Press, 1992, pp. 226-283. MR 94a:52030
  • [Bj2] A. Björner: Topological methods; in: Handbook of Combinatorics (R. Graham, M. Grötschel, L. Lovász, eds); North-Holland, Amsterdam, 1995, pp. 1819-1872. MR 96m:52012
  • [Bj3] A. Björner: Subspace arrangements; in: Proc. First European Congress of Mathematics, Paris 1992 (A. Joseph et al., eds); Birkhäuser, Basel, 1994, pp. 321-370. MR 96h:52012
  • [Bj4] A. Björner: Nonpure shellability, $f$-vectors, subspace arrangements and complexity; in: Formal Power Series and Algebraic Combinatorics, DIMACS Workshop 1994 (L.J. Billera et al., eds); Amer. Math. Soc., Providence, R.I., 1996, pp. 25-53. MR 96h:05213
  • [BZ] A. Björner, G.M. Ziegler: Combinatorial stratification of complex arrangements; J. Amer. Math. Soc. 5 (1992), 105-149. MR 92k:52022
  • [Br] G.E. Bredon: Topology and Geometry; Graduate Texts in Mathematics 129, Springer-Verlag, 1993. MR 94d:55001
  • [Bn] E. Brieskorn: Sur les groupes de tresses; in: Séminaire Bourbaki, 1971/1972, no. 401; Lecture Notes in Mathematics, vol. 317, Springer-Verlag, 1973, pp. 21-44. MR 54:10660
  • [Bry] T. Brylawski: The broken-circuit complex; Trans. Amer. Math. Soc. 234 (1977), 417-433. MR 80a:05055
  • [CR] H.H. Crapo, G.-C. Rota: On the Foundations of Combinatorial Geometry: Combinatorial Geometries (preliminary edition); MIT Press, Cambridge, MA, 1970. MR 45:74
  • [DP] C. De Concini, C. Procesi: Wonderful models of subspace arrangements; Selecta Math., New Series 1 (1995), 459-494. MR 97k:14013
  • [Fe] E.M. Feichtner: Cohomology algebras of subspace arrangements and of classical configuration spaces; Doctoral thesis, TU Berlin 1997 (Cuvillier Verlag, Göttingen, 1997).
  • [GM] M. Goresky, R. MacPherson: Stratified Morse Theory; Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 14, Springer-Verlag, 1988. MR 90d:57039
  • [MS] J. Milnor, J. Stasheff: Characteristic Classes; Ann. of Math. Studies, Princeton Univ. Press, 1974. MR 55:13428
  • [Mu] J.R. Munkres: Elements of Algebraic Topology; Addison-Wesley, 1984. MR 85m:55001
  • [OS] P. Orlik, L. Solomon: Combinatorics and topology of complements of hyperplanes; Invent. Math. 56 (1980), 167-189. MR 81e:32015
  • [Ox] J. Oxley: Matroid Theory, Oxford University Press, Oxford, 1992. MR 94d:05033
  • [Y] S. Yuzvinsky: Small rational model of subspace complement; preprint, March 1998.
  • [Z] G.M. Ziegler: On the difference between real and complex arrangements; Math. Zeitschrift 212 (1993), 1-11. MR 94f:52017

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 52C35, 55N45, 05B35, 51D25, 57N80

Retrieve articles in all journals with MSC (2000): 52C35, 55N45, 05B35, 51D25, 57N80

Additional Information

Eva Maria Feichtner
Affiliation: Department of Mathematics, MA 7-1, TU Berlin, 10623 Berlin, Germany
Address at time of publication: Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland

Günter M. Ziegler
Affiliation: Department of Mathematics, MA 7-1, TU Berlin, 10623 Berlin, Germany

Received by editor(s): July 8, 1998
Published electronically: March 2, 2000
Additional Notes: The first author was supported by the Graduate School “Algorithmic Discrete Mathematics” in Berlin, DFG grant GRK 219/2-97.
The second author was supported by the DFG Gerhard Hess Prize Zi 475/1-1/2 and by the German-Israeli Foundation grant I-0309-146.06/93.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society