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)



Small rational model of subspace complement

Author: Sergey Yuzvinsky
Journal: Trans. Amer. Math. Soc. 354 (2002), 1921-1945
MSC (2000): Primary 52C35, 05E25
Published electronically: January 8, 2002
MathSciNet review: 1881024
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper concerns the rational cohomology ring of the complement $M$ of a complex subspace arrangement. We start with the De Concini-Procesi differential graded algebra that is a rational model for $M$. Inside it we find a much smaller subalgebra $D$ quasi-isomorphic to the whole algebra. $D$ is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice $L$whence connecting the De Concini-Procesi model with the Goresky-MacPherson formula for the additive structure of $H^*(M)$. The algebra $D$ has a natural integral version that is a good candidate for an integral model of $M$. If the rational local homology of $L$ can be computed explicitly we obtain an explicit presentation of the ring $H^*(M,{\mathbf Q})$. For example, this is done for the cases where $L$ is a geometric lattice and where $M$ is a $k$-equal manifold.

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

  • 1. V. I. Arnold, The cohomology ring of the colored braid group, Mat. Zametki 5 (1969), 227-231 (Math. Notes 5 (1969), 138-140). MR 39:3529
  • 2. A. Björner and J. Walker, A homotopy complementation formula for partially ordered sets, European J. Combin. 4 (1983), 11-19. MR 84f:06003
  • 3. A. Björner and V. Welker, The homology of ``$k$-equal'' manifolds and related partition lattices, Adv. in Math. 110 (1995), 277-313. MR 95m:52029
  • 4. E. Brieskorn, Sur les groupes de tresses, in Séminre Bourbaki 1971/72, Lecture Notes in Math., 317, Springer-Verlag, 1973, pp. 21-44. MR 54:10660
  • 5. C. De Concini and C. Procesi, Wonderful models of subspace arrangements, Selecta Math. (N.S.) 1 (1995), 459-494. MR 97k:14013
  • 6. E. M. Feichtner, Cohomology algebras of subspace arrangements and of classical configuration spaces, Cuvillier-Verlag, Göttingen, 1997 (Doctors Dissertation at TU, Berlin).
  • 7. J. Folkman, The homology groups of a lattice, J. Math. Mech. 15 (1966), 631-636. MR 32:5557
  • 8. G. Gaiffi, Blow-ups and cohomology bases for De Concini-Procesi models of subspace arrangements, Selecta Math. (N.S.) 3 (1997), 315-333. MR 99d:52009
  • 9. M. Goresky and R. MacPherson, Stratified Morse Theory, Part III, Springer-Verlag, 1988. MR 90d:57039
  • 10. J. Morgan, The algebraic topology of smooth algebraic varieties, Publ. Math. IHES 48 (1978), 137-204. MR 878m:55014
  • 11. J. Munkres, Elements of algebraic topology, Addison-Wesley, Menlo Park, CA, 1984. MR 85m:55001
  • 12. P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167-189. MR 81e:32015
  • 13. P. Orlik and H. Terao, Arrangements of hyperplanes, Springer-Verlag, Berlin, 1992. MR 94e:52014
  • 14. D. Quillen, Homotopy properties of the poset of nontrivial $p$-subgroups of a group, Adv. in Math. 28 (1978), 101-128. MR 80k:20049
  • 15. S. Yuzvinsky, Cohomology bases for the DeConcini-Procesi models of hyperplane arrangements and sums over trees, Invent. Math. 127 (1997), 319-335. MR 98m:14020
  • 16. G. Ziegler and R. Zivaljevic, Homotopy types of subspace arrangements via diagrams of spaces, Math. Ann. 295 (1993), 527-548. MR 94c:55018
  • 17. M. De Longueville and C. Schultz, The cohomology rings of complements of subspace arrangements, Math. Ann. 319 (2001), 625-646.
  • 18. P. Deligne, M. Goresky, and R. MacPherson, L'algèbre de cohomologie du complément, dans un espace affine, d'une famille finie de sous-espaces affines, Michigan Math. J. 48 (2000), 121-136. CMP 2001:03
  • 19. S. Yuzvinsky, Rational model of subspace complement on atomic complex, Publ. L'Institut Math. 66 (80) (1999), 157-164. CMP 2000:16

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 52C35, 05E25

Retrieve articles in all journals with MSC (2000): 52C35, 05E25

Additional Information

Sergey Yuzvinsky
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403

Received by editor(s): November 13, 2000
Published electronically: January 8, 2002
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society