#### Volume 16, issue 5 (2016)

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Homotopy groups of diagonal complements

### Sadok Kallel and Ines Saihi

Algebraic & Geometric Topology 16 (2016) 2949–2980
##### Abstract

For $X$ a connected finite simplicial complex we consider ${\Delta }^{d}\left(X,n\right)$, the space of configurations of $n$ ordered points of $X$ such that no $d+1$ of them are equal, and ${B}^{d}\left(X,n\right)$, 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 ${\Delta }^{d}\left(X,n\right)$ (resp. ${B}^{d}\left(X,n\right)$) 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 ${B}^{d}\left(X,n\right)$ abelianizes as soon as we allow points to collide, ie $d\ge 2$.

 In memory of Abbas Bahri so greatly missed
##### Keywords
diagonal arrangements, homotopy groups, configuration spaces, colimit diagram
Primary: 55Q52
Secondary: 55P10
##### Publication
Revised: 15 January 2016
Accepted: 7 February 2016
Published: 7 November 2016
##### Authors
 Sadok Kallel Department of Mathematics American University of Sharjah Sharjah United Arab Emirates Ines Saihi Ecole nationale supérieure d’ingénieurs de Tunis Université de Tunis 05, Avenue Taha Hussein 1008 Montfleury Tunisia Laboratoire LATAO Faculté des sciences de Tunis Université de Tunis-El Manar