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 $F_n^G(X)/\sn$ with respect to a free action of a group $G$ on $X$, and the graph configuration spaces $F_n^{\Gamma}(X)$ and $F_n^{\Gamma}(X)/H$, where $\Gamma$ is a graph and $H$ is a suitable subgroup of the symmetric group $\sn$. The ordered configuration spaces $F_n(X)$, $F_n^G(X)$, $F_n^{\Gamma}(X)$ are all subsets of the $n$-fold Cartesian product $\prod\limits_1^n\, X$ of $X$ with itself, and satisfy $F_n^G(X)\subset F_n(X) \subset F_n^{\Gamma}(X)\subset \prod\limits_1^n\, X$. If $A$ denotes one of these configuration spaces, the authors analyse the difference between $A$ and $\prod\limits_1^n\, X$ from 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 inclusion $\map{\iota}{A}\prod\limits_1^n\, X]$, the homotopy type of the homotopy fibre $I_{\iota}$ of the map $\iota$ via certain constructions on various spaces that depend on $X$, and the long exact sequence in homotopy of the fibration involving $I_{\iota}$ and arising from the inclusion $\iota$. 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 $\Stk]/G$ of 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.