Homotopy groups of diagonal complements

Kallel, Sadok; Saihi, Ines

doi:10.2140/agt.2016.16.2949

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Homotopy groups of diagonal complements

Sadok Kallel and Ines Saihi

Algebraic & Geometric Topology 16 (2016) 2949–2980

DOI: 10.2140/agt.2016.16.2949

Abstract

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

In memory of Abbas Bahri so greatly missed

Keywords

diagonal arrangements, homotopy groups, configuration spaces, colimit diagram

Mathematical Subject Classification 2010

Primary: 55Q52

Secondary: 55P10

References

Publication

Received: 22 September 2015

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

Volume 16, issue 5 (2016)