## Algebraic & Geometric Topology

### Homotopy groups of diagonal complements

#### 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
Revised: 15 January 2016
Accepted: 7 February 2016
First available in Project Euclid: 16 November 2017

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

#### 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

#### 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)