A Survey of the Homotopy Properties of Inclusion of Certain Types of Configuration Spaces into the Cartesian Product  

作　　者：Daciberg Lima GONCALVES John GUASCHI 

机构地区：[1]Departamento de Matematica,IME-USP,Rua do Matao 1010,CEP:05508-090,Sao Paulo,SP,Brazil [2]Normandie Universite,UNICAEN,Laboratoire de Mathematiques Nicolas Oresme UMR CNRS 6139,CS 14032,14032 Caen Cedex 5,France

出　　处：《Chinese Annals of Mathematics,Series B》2017年第6期1223-1246,共24页数学年刊（B辑英文版）

基　　金：supported by the CNRS/FAPESP programme no226555(France)and n^(o) 2014/50131-7(Brazil);FAPESP–Fundacao de Amparo a Pesquisa do Estado de Sao Paulo,Projeto Tematico Topologia Algebrica,Geometrica 2012/24454-8(Brazil)for partial support;the Institute for Mathematical Sciences,National University of Singapore

摘　　要：Let X be a topological space.In 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_n.The 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~nX.If A denotes one of these configuration spaces,the authors analyse the difference between A and ∏_1~nXfrom a topological and homotopical point of view.The principal results known in the literature concern the usual configuration spaces.The 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 cases.The authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest.In 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.Let X be a topological space.In 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_n.The 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~nX.If A denotes one of these configuration spaces,the authors analyse the difference between A and ∏_1~nXfrom a topological and homotopical point of view.The principal results known in the literature concern the usual configuration spaces.The 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 cases.The authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest.In 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 

分 类 号：U291.1[交通运输工程—交通运输规划与管理] O1-0[交通运输工程—道路与铁道工程]