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A Survey of the Homotopy Properties of Inclusion of Certain Types of configuration spaces into the Cartesian Product

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Chinese Annals of Mathematics,Series B 2017年第6期38卷 1223-1246页

作者： Daciberg Lima GONCALVES John GUASCHI Departamento de Matematica IME-USPRua do Matao 1010CEP:05508-090Sao PauloSPBrazil Normandie Universite UNICAENLaboratoire de Mathematiques Nicolas Oresme UMR CNRS 6139CS 1403214032 Caen Cedex 5France

Let X be a topological *** this survey the authors consider several types of configuration spaces,namely,the classical(usual)configuration spaces F_n(X)and D_n(X),the orbit configuration spaces F_n^G(X)and F_n^G(X)/S_nwith respect to a free action of a group G on X,and the graph configuration spaces F_n~Γ(X)and F_n~Γ(X)/H,whereΓis a graph and H is a suitable subgroup of the symmetric group S_*** ordered configuration spaces F_n(X),F_n^G(X),F_n~Γ(X)are all subsets of the n-fold Cartesian product ∏_1~nX of X with itself,and satisfy F_n^G(X)?F_n(X)?F_n~Γ(X)?∏_1~*** A denotes one of these configuration spaces,the authors analyse the difference between A and ∏_1~nXfrom a topological and homotopical point of *** principal results known in the literature concern the usual configuration *** authors are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusionι:A-→∏_1~nX,the homotopy type of the homotopy fibre I_ιof the mapιvia certain constructions on various spaces that depend on X,and the long exact sequence in homotopy of the fibration involving I_ιand arising from the inclusionι.In this respect,if X is either a surface without boundary,in particular if X is the 2-sphere or the real projective plane,or a space whose universal covering is contractible,or an orbit space S^k/Gof the k-dimensional sphere by a free action of a Lie group G,the authors present recent results obtained by themselves for the first case,and in collaboration with Golasi′nski for the second and third *** authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of *** order to motivate various questions,for the remaining types of configuration spaces,a few of their basic properties are described and proved.A list of open questions and problems is given at the end of the paper.

关键词： configuration space equivariant configuration space Fibration Homotopy fibre K(π,1)space Braid groups

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