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Homotopy groups of diagonal complements

Sadok Kallel and Ines Saihi

Algebraic & Geometric Topology 16 (2016) 2949–2980

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.

In memory of Abbas Bahri so greatly missed

diagonal arrangements, homotopy groups, configuration spaces, colimit diagram
Mathematical Subject Classification 2010
Primary: 55Q52
Secondary: 55P10
Received: 22 September 2015
Revised: 15 January 2016
Accepted: 7 February 2016
Published: 7 November 2016
Sadok Kallel
Department of Mathematics
American University of Sharjah
United Arab Emirates
Ines Saihi
Ecole nationale supérieure d’ingénieurs de Tunis
Université de Tunis
05, Avenue Taha Hussein
1008 Montfleury
Laboratoire LATAO
Faculté des sciences de Tunis
Université de Tunis-El Manar