Supersolvable posets and fiber-type abelian arrangements

• Christin Bibby ^[1] ; Emanuele Delucchi ^[2]
  1. [1] Louisiana State University

     Louisiana State University

     Estados Unidos

     
  2. [2] SUPSI-IDSIA, University of Applied Arts and Sciences of Southern Switzerland, Lugano, Switzerland
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 5, 2024
• Idioma: inglés
• DOI: 10.1007/s00029-024-00976-w
• Enlaces
  • Texto completo
• Resumen

  • We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups. These bundles are akin to those of Fadell–Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. This is consistent with Terao’s fibration theorem connecting bundles of hyperplane arrangements to Stanley’s lattice supersolvability. We obtain a combinatorially determined class of K(π,1) toric and elliptic arrangements. Under a stronger combinatorial condition, we prove a factorization of the Poincaré polynomial when the Lie group is noncompact. In the case of toric arrangements, this provides an analogue of Falk–Randell’s formula relating the Poincaré polynomial to the lower central series of the fundamental group.

• Referencias bibliográficas
  • Aigner, M.: Combinatorial theory. Springer, Berlin (1997)
  • Bibby, C., Gadish, N.: Combinatorics of orbit configuration spaces. Int. Math. Res. Not 2022, 6903– 6947 (2021)
  • Brieskorn, E.: Sur les groupes de tresses [d’après V. I. Arnol’d]. In Séminaire Bourbaki, 24ème année, Exp. No. 401, pages 21–44. Lecture...
  • Brylawski, T.: Modular constructions for combinatorial geometries. Trans. Am. Math. Soc. 203, 1–44 (1975)
  • Cohen, D.C.: Monodromy of fiber-type arrangements and orbit configuration spaces. Forum Math. 13(4), 505–530 (2001)
  • Cohen, F. R.: Introduction to configuration spaces and their applications. In Braids, volume 19 of Lect. Notes Ser. Inst. Math. Sci. Natl....
  • Crapo, H.H., Rota, G.-C.: On the foundations of combinatorial theory. II: Combinatorial geometries. Stud. Appl. Math. 49, 109–133 (1970)
  • D’Alì, A., Delucchi, E.: Stanley-Reisner rings for symmetric simplicial complexes, G-semimatroids and Abelian arrangements. J. Comb. Algebra...
  • Deligne, P.: Les immeubles des groupes de tresses généralises. Invent. Math. 17, 273–302 (1972)
  • Dowling, T.A.: A class of geometric lattices based on finite groups. J. Comb. Theory Ser. B 14, 61–86 (1973)
  • Delucchi, E., Riedel, S.: Group actions on semimatroids. Adv. Appl. Math. 95, 199–270 (2018)
  • Fadell, E., Neuwirth, L.: Configuration spaces. Math. Scand. 10, 111–118 (1962)
  • Falk, M., Randell, R.: The lower central series of a fiber-type arrangement. Invent. Math. 82(1), 77–88 (1985)
  • Falk, M., Terao, H.: βnbc-bases for cohomology of local systems on hyperplane complements. Trans. Am. Math. Soc. 349(1), 189–202 (1997)
  • Hallam, J.: Applications of quotient posets. Discrete Math. 340(4), 800–810 (2017)
  • Kohno, T.: Série de Poincaré-Koszul associée aux groupes de tresses pures. Invent. Math. 82(1), 57–75 (1985)
  • Liu, Y., Tran, T.N., Yoshinaga, M.: G-Tutte polynomials and abelian Lie group arrangements. Int. Math. Res. Not. IMRN 1, 152–190 (2021)
  • Magnus, W., Karrass, A., Solitar, D.: Combinatorial group theory: presentations of groups in terms of generators and relations. Wiley, New...
  • Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes. Invent. Math. 56(2), 167–189 (1980)
  • Orlik, P., Terao, H.: Arrangements of hyperplanes. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],...
  • Paris, L.: Intersection subgroups of complex hyperplane arrangements. Topol. Appl. 105(3), 319–343 (2000)
  • Pontrjagin, L.: Topological groups. (Translated by Emma Lehmer), volume 2 of Princeton Math. Ser. Princeton University Press, Princeton, NJ,...
  • Stanley, R.P.: Supersolvable lattices. Algebra Univ. 2(1), 197–217 (1972)
  • Stanley, R. P.: Modular elements of geometric lattices. Algebra Universalis, 1:214–217, (1971/72)
  • Terao, H.: Modular elements of lattices and topological fibration. Adv. Math. 62(2), 135–154 (1986)
  • Wachs, M.L., Walker, J.W.: On geometric semilattices. Order 2(4), 367–385 (1986)
  • Xicotencatl Merino, M. A.: Orbit configuration spaces, infinitesimal braid relations in homology and equivariant loop spaces. ProQuest LLC,...
  • Zaslavsky, T.: A combinatorial analysis of topological dissections. Adv. Math. 25(3), 267–285 (1977)
  • Ziegler, G.M.: Binary supersolvable matroids and modular constructions. Proc. Am. Math. Soc. 113(3), 817–829 (1991)