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

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.

Free Cyclic Actions on Surfaces and the Borsuk-Ulam Theorem

Let M and N be topological spaces,let G be a group,and letτ:G×M→M be a proper free action of G.In this paper,we define a Borsuk-Ulam-type property for homotopy classes of maps from M to N with respect to the pair(G,τ)that generalises the classical antipodal Borsuk-Ulam theorem of maps from the n-sphere S^(n) to R^(n).In the cases where M is a finite pathwise-connected CWcomplex,G is a finite,non-trivial Abelian group,τis a proper free cellular action,and N is either R^(2) or a compact surface without boundary different from S^(2) and RP^(2),we give an algebraic criterion involving braid groups to decide whether a free homotopy class β∈[M,N]has the Borsuk-Ulam property.As an application of this criterion,we consider the case where M is a compact surface without boundary equipped with a free actionτof the finite cyclic group Zn.In terms of the orient ability of the orbit space Mof M by the actionτ,the value of n modulo 4 and a certain algebraic condition involving the first homology group of M,we are able to determine if the single homotopy class of maps from M to R^(2) possesses the Borsuk-Ulam property with respect to(Z_(n),τ).Finally,we give some examples of surfaces on which the symmetric group acts,and for these cases,we obtain some partial results regarding the Borsuk-Ulam property for maps whose target is R^(2).