Algebraic & Geometric Topology

Homotopy groups of diagonal complements

Sadok Kallel and Ines Saihi

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For X a connected finite simplicial complex we consider Δd(X,n), the space of configurations of n ordered points of X such that no d + 1 of them are equal, and Bd(X,n), the analogous space of configurations of unordered points. These reduce to the standard configuration spaces of distinct points when d = 1. We describe the homotopy groups of Δd(X,n) (resp. Bd(X,n)) in terms of the homotopy (resp. homology) groups of X through a range which is generally sharp. It is noteworthy that the fundamental group of the configuration space Bd(X,n) abelianizes as soon as we allow points to collide, ie d 2.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 5 (2016), 2949-2980.

Dates
Received: 22 September 2015
Revised: 15 January 2016
Accepted: 7 February 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841235

Digital Object Identifier
doi:10.2140/agt.2016.16.2949

Mathematical Reviews number (MathSciNet)
MR3572355

Zentralblatt MATH identifier
1355.55012

Subjects
Primary: 55Q52: Homotopy groups of special spaces
Secondary: 55P10: Homotopy equivalences

Keywords
diagonal arrangements homotopy groups configuration spaces colimit diagram

Citation

Kallel, Sadok; Saihi, Ines. Homotopy groups of diagonal complements. Algebr. Geom. Topol. 16 (2016), no. 5, 2949--2980. doi:10.2140/agt.2016.16.2949. https://projecteuclid.org/euclid.agt/1510841235


Export citation

References

  • M,A Armstrong, The fundamental group of the orbit space of a discontinuous group, Proc. Cambridge Philos. Soc. 64 (1968) 299–301
  • A Bj örner, M Las Vergnas, B Sturmfels, N White, G,M Ziegler, Oriented matroids, 2nd edition, Encyclopedia of Mathematics and its Applications 46, Cambridge Univ. Press (1999)
  • A Bj örner, V Welker, The homology of “$k$–equal” manifolds and related partition lattices, Adv. Math. 110 (1995) 277–313
  • P,V,M Blagojević, B Matschke, G,M Ziegler, A tight colored Tverberg theorem for maps to manifolds, Topology Appl. 158 (2011) 1445–1452
  • P,V,M Blagojević, G,M Ziegler, Convex equipartitions via equivariant obstruction theory, Israel J. Math. 200 (2014) 49–77
  • C-F B ödigheimer, I Madsen, Homotopy quotients of mapping spaces and their stable splitting, Quart. J. Math. Oxford Ser. 39 (1988) 401–409
  • B Branman, I Kriz, A Pultr, A sequence of inclusions whose colimit is not a homotopy colimit, New York J. Math. 21 (2015) 333–338
  • F Cohen, E,L Lusk, Configuration-like spaces and the Borsuk–Ulam theorem, Proc. Amer. Math. Soc. 56 (1976) 313–317
  • N Dobrinskaya, V Turchin, Homology of non-$k$–overlapping discs, Homology Homotopy Appl. 17 (2015) 261–290
  • C Eyral, Profondeur homotopique et conjecture de Grothendieck, Ann. Sci. École Norm. Sup. 33 (2000) 823–836
  • E,D Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics 1622, Springer (1996)
  • M,A Guest, A Kozlowski, K Yamaguchi, Stable splitting of the space of polynomials with roots of bounded multiplicity, J. Math. Kyoto Univ. 38 (1998) 351–366
  • A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
  • U Helmke, Topology of the moduli space for reachable linear dynamical systems: the complex case, Math. Systems Theory 19 (1986) 155–187
  • T,D Imbo, C Shah Imbo, E,C,G Sudarshan, Identical particles, exotic statistics and braid groups, Phys. Lett. B 234 (1990) 103–107
  • S Kallel, Spaces of particles on manifolds and generalized Poincaré dualities, Q. J. Math. 52 (2001) 45–70
  • S Kallel, R Karoui, Symmetric joins and weighted barycenters, Adv. Nonlinear Stud. 11 (2011) 117–143
  • S Kallel, W Taamallah, The geometry and fundamental group of permutation products and fat diagonals, Canad. J. Math. 65 (2013) 575–599
  • M Khovanov, Real $K(\pi,1)$ arrangements from finite root systems, Math. Res. Lett. 3 (1996) 261–274
  • B Kloeckner, The space of closed subgroups of $\mathbb R^n$ is stratified and simply connected, J. Topol. 2 (2009) 570–588
  • K,H Ko, H,W Park, Characteristics of graph braid groups, Discrete Comput. Geom. 48 (2012) 915–963
  • J,R Munkres, Elements of algebraic topology, Addison–Wesley Publishing Company, Menlo Park, CA (1984)
  • M Nakaoka, Cohomology of symmetric products, J. Inst. Polytech. Osaka City Univ. Ser. A 8 (1957) 121–145
  • S Smale, A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. 8 (1957) 604–610
  • Q Sun, Configuration spaces of singular spaces, PhD thesis, University of Rochester, Ann Arbor, MI (2010) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/871891981 {\unhbox0
  • D Tamaki, Cellular stratified spaces, I: Face categories and classifying spaces, preprint (2011)
  • S Zanos, Méthodes de scindements homologiques en topologie et en géométrie, PhD thesis, Université Lille 1 (2009)