Geodesic
Configuration Spaces of Labeled Particles and Finite Eilenberg–MacLane Complexes
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 37-54
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For any Coxeter system $(W,S)$, the group $W$ acts naturally on the complement of the associated complex hyperplane arrangement. By the well-known conjecture, the orbit space of this action is the classifying space of the corresponding Artin group. We describe some properties of configuration spaces of particles labeled by elements of a partial monoid and use them to prove that the orbit space mentioned in the conjecture is the classifying space of the positive Artin monoid. In particular, the conjecture reduces to a problem concerning the group completion of this monoid. 
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N. E. Dobrinskaya. Configuration Spaces of Labeled Particles and Finite Eilenberg–MacLane Complexes. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 37-54. http://geodesic.mathdoc.fr/item/TM_2006_252_a4/

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