## On cohomology algebras of complex subspace arrangements

HTML articles powered by AMS MathViewer

- by Eva Maria Feichtner and Günter M. Ziegler
- Trans. Amer. Math. Soc.
**352**(2000), 3523-3555 - DOI: https://doi.org/10.1090/S0002-9947-00-02537-X
- Published electronically: March 2, 2000
- PDF | Request permission

## Abstract:

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

- Anders Björner,
*The homology and shellability of matroids and geometric lattices*, Matroid applications, Encyclopedia Math. Appl., vol. 40, Cambridge Univ. Press, Cambridge, 1992, pp. 226–283. MR**1165544**, DOI 10.1017/CBO9780511662041.008 - A. Björner,
*Topological methods*, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819–1872. MR**1373690** - Anders Björner,
*Subspace arrangements*, First European Congress of Mathematics, Vol. I (Paris, 1992) Progr. Math., vol. 119, Birkhäuser, Basel, 1994, pp. 321–370. MR**1341828** - Anders Björner,
*Nonpure shellability, $f$-vectors, subspace arrangements and complexity*, Formal power series and algebraic combinatorics (New Brunswick, NJ, 1994) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 24, Amer. Math. Soc., Providence, RI, 1996, pp. 25–53. MR**1363505**, DOI 10.1090/dimacs/024/02 - Anders Björner and Günter M. Ziegler,
*Combinatorial stratification of complex arrangements*, J. Amer. Math. Soc.**5**(1992), no. 1, 105–149. MR**1119198**, DOI 10.1090/S0894-0347-1992-1119198-9 - Glen E. Bredon,
*Topology and geometry*, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, New York, 1993. MR**1224675**, DOI 10.1007/978-1-4757-6848-0 - Egbert Brieskorn,
*Sur les groupes de tresses [d’après V. I. Arnol′d]*, Séminaire Bourbaki, 24ème année (1971/1972), Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. Exp. No. 401, pp. 21–44 (French). MR**0422674** - Tom Brylawski,
*The broken-circuit complex*, Trans. Amer. Math. Soc.**234**(1977), no. 2, 417–433. MR**468931**, DOI 10.1090/S0002-9947-1977-0468931-6 - Henry H. Crapo and Gian-Carlo Rota,
*On the foundations of combinatorial theory: Combinatorial geometries*, Preliminary edition, The M.I.T. Press, Cambridge, Mass.-London, 1970. MR**0290980** - C. De Concini and C. Procesi,
*Wonderful models of subspace arrangements*, Selecta Math. (N.S.)**1**(1995), no. 3, 459–494. MR**1366622**, DOI 10.1007/BF01589496 - E.M. Feichtner:
*Cohomology algebras of subspace arrangements and of classical configuration spaces*; Doctoral thesis, TU Berlin 1997 (Cuvillier Verlag, Göttingen, 1997). - Mark Goresky and Robert MacPherson,
*Stratified Morse theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. MR**932724**, DOI 10.1007/978-3-642-71714-7 - John W. Milnor and James D. Stasheff,
*Characteristic classes*, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR**0440554** - James R. Munkres,
*Elements of algebraic topology*, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR**755006** - Peter Orlik and Louis Solomon,
*Combinatorics and topology of complements of hyperplanes*, Invent. Math.**56**(1980), no. 2, 167–189. MR**558866**, DOI 10.1007/BF01392549 - James G. Oxley,
*Matroid theory*, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR**1207587** - S. Yuzvinsky:
*Small rational model of subspace complement*; preprint, March 1998. - Günter M. Ziegler,
*On the difference between real and complex arrangements*, Math. Z.**212**(1993), no. 1, 1–11. MR**1200161**, DOI 10.1007/BF02571638

## Bibliographic 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
- Email: feichtne@math.ethz.ch
**Günter M. Ziegler**- Affiliation: Department of Mathematics, MA 7-1, TU Berlin, 10623 Berlin, Germany
- Email: ziegler@math.tu-berlin.de
- 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. - © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 3523-3555 - MSC (2000): Primary 52C35, 55N45; Secondary 05B35, 51D25, 57N80
- DOI: https://doi.org/10.1090/S0002-9947-00-02537-X
- MathSciNet review: 1694288