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)

 
 

 

Group actions on arrangements of
linear subspaces and applications to
configuration spaces


Authors: Sheila Sundaram and Volkmar Welker
Journal: Trans. Amer. Math. Soc. 349 (1997), 1389-1420
MSC (1991): Primary 05E25, 57N65.; Secondary 20C30, 55M35
DOI: https://doi.org/10.1090/S0002-9947-97-01565-1
MathSciNet review: 1340186
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For an arrangement of linear subspaces in ${\mathbb R} ^n$ that is invariant under a finite subgroup of the general linear group $Gl_n({\mathbb R} )$ we develop a formula for the $G$-module structure of the cohomology of the complement ${\mathcal M} _{\mathcal A} $. Our formula specializes to the well known Goresky-MacPherson theorem in case $G = 1$, but for $G \neq 1$ the formula shows that the $G$-module structure of the complement is not a combinatorial invariant. As an application we are able to describe the free part of the cohomology of the quotient space ${\mathcal M} _{\mathcal A} /G$. Our motivating examples are arrangements in ${\mathbb C} ^n$ that are invariant under the action of $S_n$ by permuting coordinates. A particular case is the ``$k$-equal'' arrangement, first studied by Björner, Lovász, and Yao motivated by questions in complexity theory. In these cases ${\mathcal M} _{\mathcal A} $ and ${\mathcal M} _{\mathcal A} /S_n$ are spaces of ordered and unordered point configurations in ${\mathbb C} ^n$ many of whose properties are reduced by our formulas to combinatorial questions in partition lattices. More generally, we treat point configurations in ${\mathbb R} ^d$ and provide explicit results for the ``$k$-equal'' and the ``$k$-divisible'' cases.


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

  • [Ar1] Arnol'd, V.I.: The cohomology ring of the colored braid group. Math. Notes 5, 138-140 (1969) MR 39:3529
  • [Ar2] Arnol'd, V.I.: Topological invariants of algebraic functions. Trans. Moscow Math. Soc. 21, 30-52 (1970) MR 43:1991
  • [Ar3] Arnol'd, V.I.: The spaces of functions with mild singularities. Funktsional. Anal. i Prilozhen. 3, 1-10 (1989). MR 90m:58016 English translation in Functional Anal. Appl. 23 (1989)
  • [Ba] Baclawski, K.: Whitney numbers of geometric lattices. Adv. in Math. 16, 125-138 (1975) MR 52:7933
  • [Bj1] Björner, A.: Shellable and Cohen-Macaulay partially ordered sets. Trans. Amer. Math. Soc. 260, 159-183 (1980) MR 81i:06001
  • [Bj2] Björner, A.: On the homology of geometric lattices, Algebra Universalis 14, 107-128 (1982). MR 83d:05029
  • [Bj3] Björner, A.: Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Adv. in Math. 52, 173-212 (1984). MR 85m:52003
  • [Bj4] Björner, A.: Subspace arrangements. In: Proceedings of the 1st European Congress of Mathematics (Paris 1992), ed. A. Joseph, R. Rentschler, Basel, Boston: Birkhäuser, 321-370 (1994). MR 96h:52012
  • [Bj-Lo-Y] Björner, A., Lovász, L., Yao, A.: Linear decision trees: volume estimates and topological bounds. In: Proc. 24th ACM Symp. on Theory of Computing, pp. 170-177. New York: ACM Press 1992
  • [Bj-Lo] Björner, A., Lovász, L.: Linear decision trees, subspace arrangements and Möbius functions. J. Amer. Math. Soc. 7, 677-706 (1994) MR 95e:52024
  • [Bj-Wa] Björner, A., Wachs, M.: Shellable nonpure complexes and posets, I, Trans. Amer. Math. Soc. 348 1299-1327 (1996). MR 96i:06008
  • [Bj-Wal] Björner, A., Walker, J.W.: A homotopy complementation formula for partially ordered sets. European J. Combin. 4, 11-19 (1983) MR 84f:06003
  • [Bj-We] Björner, A., Welker, V.: Homology of the ``$k$-equal" manifolds and related partition lattices. Adv. in Math. 110(2), 277-313 (1995). MR 95m:52029
  • [Bo-Ka] Bousfield, M., Kan, D. M.: Homotopy Limits, Completions and Localizations. (Lecture Notes in Mathematics, Vol. 304). Berlin, Heidelberg, New York: Springer 1972 MR 51:1825
  • [Bre] Bredon, G.: Compact Transformation Groups. New York, London: Academic Press 1972 MR 54:1265
  • [Bri] Brieskorn, E.: Sur les groupes de tresses. In: Séminaire Bourbaki 1971/72, pp. 21-44. Berlin, Heidelberg, New York: Springer 1973 MR 54:10660
  • [C-H-R] Calderbank, A., Hanlon, P., Robinson, R.: Partitions into even and odd block sizes and some unusual characters of the symmetric groups. Proc. London Math. Soc. 53(3), 288-320 (1986) MR 87m:20042
  • [Co-La-Ma] Cohen, F.L., Lada, T.J., May, J.P., The Homology of Iterated Loop Spaces. (Lecture Notes in Mathematics, Vol. 533). Berlin, Heidelberg, New York: Springer 1976 MR 55:9096
  • [Co-Lu] Cohen, F.L., Lusk, E.L.: Configuration-like spaces and the Borsuk-Ulam Theorem, Proc. Amer. Math. Soc. 56, 313-317 (1976) MR 54:13899
  • [Ep] Epshtein,S.I.: Fundamental groups of coprime polynomials. Functional Anal. Appl. 7,82-83 (1973) MR 49:9865
  • [Fa-Ne] Fadell, E., Neuwirth, L.: Configuration spaces. Math. Scand. 10, 111-118 (1962) MR 25:4537
  • [Fo-Ne] Fox, R., Neuwirth, L.: The braid groups. Math. Scand. 10, 119-126 (1962) MR 27:742
  • [Fu] Fuks, D.: Cohomologies of the braid groups mod 2. Functional Anal. Appl. 4, 143-151 (1970) MR 43:226
  • [Gab-Zis] Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory. (Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 35). Berlin, Heidelberg, New York: Springer 1967 MR 35:1019
  • [Go-MacP] Goresky, M., MacPherson, R.: Stratified Morse Theory. (Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 14). Berlin, Heidelberg, New York: Springer 1988 MR 90d:57039
  • [Gu-Ko-Y] Guest, M., Kozlowski, A., Yamaguchi, K.: The topology of spaces of coprime polynomials. Math. Z. 217, No. 3, 435-446 (1994) MR 95:05
  • [Ha1] Hanlon, P.: The fixed-point partition lattices, Pacific J. Math. 96, 319-341 (1981) MR 83d:06010
  • [Ha2] Hanlon, P.: The generalized Dowling lattices. Trans. AMS 325(1), 1-37 (1991) MR 91h:06011
  • [Hu] Hu, Y.: Homology of subspace arrangements. Proc. Amer. Math. Soc. 122, 285-290 (1994). MR 94k:52020
  • [Je] Jewell, K.: Complements of sphere and subspace arrangements. Topology and its Appl. 56, No. 3, 199-214 (1994) MR 95a:52017
  • [Le-So] Lehrer, G., Solomon, L.: On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes. J. Algebra 104, 410-424 (1986) MR 88a:32017
  • [Macd] Macdonald, I.: Symmetric functions and Hall polynomials. Oxford University Press 1979 MR 84g:05003
  • [Me] Merkov, A.B.: Finite-order invariants of ornaments (Preprint 1994).
  • [Mu] J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley (1984) MR 85m:55001
  • [O-So] Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes. Invent. Math. 56, 167-189 (1980) MR 81e:32015
  • [O-T] Orlik, P., Terao, H.: Arrangements of Hyperplanes. (Grundlehren der mathematischen Wissenschaften, Vol. 300). Berlin, Heidelberg, New York: Springer 1992 MR 94e:52014
  • [Sa] Sagan, B.: Shellability of exponential structures. Order 3, 47-54 (1986) MR 87j:05020
  • [Se] Segal, G.: Classifying spaces and spectral sequences. Publ. Math. I.H.E.S. 34, 105-112 (1968) MR 38:718
  • [So] Solomon, L.: A decomposition of the group algebra of a finite Coxeter group. J. Algebra 9, 220-239 (1968) MR 38:1991
  • [Sp] Spanier, E.: Algebraic Topology. New York: MacGraw-Hill 1966 MR 35:1007
  • [St1] Stanley, R. P.: Exponential structures. Stud. Appl. Math. 59, 73-82 (1978) MR 58:262
  • [St2] Stanley, R. P.: Some aspects of groups acting on finite posets. J. Combin. Theory Ser. A 32, 132-161 (1982) MR 83d:06002
  • [St3] Stanley, R. P.: Enumerative combinatorics I. Monterey, CA: Wadsworth & Brooks / Cole, 1986 MR 87j:05003
  • [Su] Sundaram, S.: The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice. Adv. in Math. 104(2), 225-296 (1994) MR 94:11
  • [Su-Wa] Sundaram, S., Wachs, M.: The homology representations of the k-equal partition lattice. Trans. Amer. Math. Soc., to appear.
  • [tDi] tom Dieck, T.: Transformation Groups, de Gruyter, 1987. MR 89c:57048
  • [Va1] Vassiliev, V.A.: Cohomology of knot spaces. Advances in Soviet Math. 1, 23-69 (1990) MR 92a:57016
  • [Va2] Vassiliev, V.A.: Complements of discriminants of smooth maps : Topology and applications. (Transl. of Math. Monographs, Vol. 98). Providence, RI: Amer. Math. Soc. 1992 MR 94i:57020
  • [Va3] Vassiliev, V.A.: Complexes of connected graphs In: The Gelfand Seminar, 1990-1992, ed. Corwin, L., pp. 223-235, Basel, Boston: Birkhäuser 1993 MR 94h:55032
  • [Vo] Vogt, R.M.: Homotopy limits and colimits. Math. Zeitschrift. 134, 11-52 (1973) MR 48:12516
  • [Wa] Wachs, M.: A basis for the homology of the d-divisible partition lattice. Adv. in Math. 117 (2), 294-318 (1996). CMP 96:07
  • [Zie-\v{Z}] Ziegler, G. M., \v{Z}ivaljevi\'{c}, R.: Homotopy types of subspace arrangements via diagrams of spaces. Math. Annalen 295, 527-548 (1993) MR 94c:55018

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 05E25, 57N65., 20C30, 55M35

Retrieve articles in all journals with MSC (1991): 05E25, 57N65., 20C30, 55M35


Additional Information

Sheila Sundaram
Affiliation: Department of Mathematics and Computer Science, University of Miami, Coral Gables, Florida 33124
Address at time of publication: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
Email: sheila@claude.math.wesleyan.edu

Volkmar Welker
Affiliation: Institute for Experimental Mathematics, Ellernstr. 29, 45326 Essen, Germany
Email: welker@exp-math.uni-essen.de

DOI: https://doi.org/10.1090/S0002-9947-97-01565-1
Keywords: Subspace arrangements, group action, poset homology, configuration spaces, homotopy limits, symmetric functions
Received by editor(s): January 1, 1965
Additional Notes: The author acknowledges support by the DFG while he was visiting scholar at MIT
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society